INTEGERS 13 (2013)
#A37
GONI: PRIMES REPRESENTED BY BINARY QUADRATIC
FORMS
Pete L. Clark
Department of Mathematics, University of Georgia, Athens, Georgia
pete@math.uga.edu
Jacob Hicks
Department of Mathematics, University of Georgia, Athens, Georgia
jhicks@math.uga.edu
Hans Parshall
Department of Mathematics, University of Georgia, Athens, Georgia
hans@math.uga.edu
Katherine Thompson
Department of Mathematics, University of Georgia, Athens, Georgia
thompson@math.uga.edu
Received: 8/4/12, Revised: 2/28/13, Accepted: 5/2/13, Published: 6/14/13
Abstract
We list 2779 regular primitive positive definite integral binary quadratic forms,
and show that, conditional on the Generalized Riemann Hypothesis, this is the
complete list of regular, positive definite binary integral quadratic forms (up to
SL2 (Z)-equivalence). For each of these 2779 forms we determine the primes that
they represent by elementary combinatorial methods, avoiding Gauss’s genus theory.
The key intermediate result is a Small Multiple Theorem for representations of
primes by integral binary forms.
1. Introduction
An imaginary quadratic discriminant is a negative integer ∆ which is 0 or 1
modulo 4. For a given imaginary quadratic disciminant ∆, let C(∆) be the set of
SL2 (Z)-equivalence classes of primitive positive definite integral binary quadratic
forms of discriminant ∆. Then C(∆) is a finite set [3, Thm. 2.13] which, when
endowed with Gauss’s composition law, becomes a finite abelian group, the class
group of discriminant ∆ [3, Thm. 3.9].
Thus a form q of discriminant ∆ determines an element [q] ∈ C(∆). A quadratic
form q is ambiguous if [q]2 = 1. For a q = �A, B, C�, the form q = �A, −B, C�
represents the inverse of [q] in C(∆) [3, Thm. 3.9]. Note that q and q are GL2 (Z)equivalent: q(x, y) = q(x, −y), so q and q represent the same integers.
INTEGERS: 13 (2013)
2
A discriminant ∆ is idoneal if every q ∈ C(∆) is ambiguous; this holds if and
only if C(∆) ∼
= (Z/2Z)r for some r ∈ N. A quadratic form is idoneal if its discriminant is idoneal. A discriminant ∆ is bi-idoneal if C(∆) ∼
= (Z/4Z) ⊕ (Z/2Z)r
for some r ∈ N. A quadratic form q is bi-idoneal if ∆ is bi-idoneal and q is not
ambiguous.
A full congruence class of primes is the set of all primes p � 2∆ with p ≡ n
(mod N ) for fixed coprime positive integers n and N . We say q is regular if the
set of primes p � 2∆ represented by q is a union of full congruence classes.
Recall Fermat’s Two Squares Theorem: an odd prime p is of the form x2 + y 2 if and
only if p ≡ 1 (mod 4). In our terminology then the form q(x, y) = x2 + y 2 is regular. Indeed, much classical work on quadratic forms can be phrased as showing that
certain specific binary quadratic forms represent full congruence classes of primes,
or are regular. Among primitive, positive definite, integral binary quadratic forms,
how many are regular? How many represent full congruence classes of primes? Remarkably, this problem has recently been solved (conditionally on GRH) but the
answer does not appear explicitly in the literature. Here it is:
Theorem 1. Let q be a primitive, positive definite integral binary quadratic form.
a) The following are equivalent:
(i) q is regular.
(ii) q represents a full congruence class of primes.
(iii) q is either idoneal or bi-idoneal.
b) There are at least 425 and at most 432 imaginary quadratic discriminants which
are either idoneal or bi-idoneal. These 425 known discriminants give rise to precisely
2779 SL2 (Z)-equivalence classes of regular forms: see Table 1.
c) The list of idoneal and bi-idoneal discriminants of part b) is complete among
all imaginary quadratic discriminants ∆ with |∆| ≤ 80604484. Assuming the Riemann Hypothesis for Dedekind zeta functions of imaginary quadratic fields, there
are precisely 425 imaginary discriminants which are idoneal or bi-idoneal.
For these 2779 regular forms, it is natural to ask for explicit congruence conditions, as in Fermat’s Two Squares Theorem. The following result accomplishes
this.
Theorem 2. Let q = �A, B, C� be one of the 2779 primitive, positive definite
integral binary quadratic forms in Table 1, and let ∆ = B 2 −4AC be the discriminant
of q. For a prime p � 2∆, the following are equivalent:
a) The form q integrally represents p: there are x, y ∈ Z with q(x, y) = p.
b) All of the following conditions hold:
(i) ( ∆
p ) = 1.
INTEGERS: 13 (2013)
3
p
A
(ii) For each odd prime m | ∆, if m � A, then ( m
) = (m
), and if m � C, then
p
C
( m ) = ( m ).
(iii) If 16 | ∆ and 2 � A, then p ≡ A (mod 4). If 16 | ∆ and 2 � C, then
p ≡ C (mod 4).
(iv) If 32 | ∆ and 2 � A, then p ≡ A (mod 8). If 32 | ∆ and 2 � C, then
p ≡ C (mod 8).
We will prove Theorem 1 by deducing it from Gauss’s genus theory together with
results of Meyer, Weinberger, Louboutin, Kaplan-Williams and Voight. We do this
mostly for completeness and perspective. Our main goal is quite different: we will
give a new proof of Theorem 2 using none of Gauss’s genus theory but instead using
elementary ideas from the Geometry of Numbers. Our methods build on the
classical proof of the Two Squares Theorem via Minkowski’s Convex Body Theorem and its recent generalization to the 65 principal idoneal forms x2 + Dy 2 of
T. Hagedorn [5], although we find it simpler to use (sharp) bounds on minima of
binary quadratic forms going back to Lagrange and Legendre.
We may compare the two methods as follows: let q be a binary form of discriminant ∆, and let p � 2∆ be a prime. To analyze the question of whether q
represents p, genus theory begins with the observation that ( ∆
p ) = 1 if and only if
�
some q ∈ C(∆) represents p and attempts to rule out the representation of p by all
forms q � �= q. Our method begins with a small multiple theorem: if ( ∆
p ) = 1, then q
represents some multiple kp of p with k bounded in terms of ∆ and via a combination of elimination and reduction attempts to show that we may take k = 1. Our
method is more computational – at present it is more a technique than a theory –
and the reasons for its success in all 2779 cases are rather mysterious! However, our
technique can be used in settings where genus theory does not apply: in [2] and [8]
some of us use these ideas to establish universality of most (but not all) of the 112
positive definite quaternary universal forms of square discriminant. In [1], the first
author extends the method to a technique for proving representation theorems for
quadratic forms in 2d variables over a normed Dedekind domain.
2. Proof of Theorem 1
2.1. Part a)
(i) =⇒ (ii): By [3, Thm. 9.12], q represents infinitely many prime numbers.
Having established this, the implication is immediate.
(ii) =⇒ (iii): Suppose that there are coprime integers n and N such that for all
primes p, if p � 2∆ and p ≡ n (mod N ), then q represents p. By [9, Thm. 2], if q is
ambiguous then ∆ is idoneal hence so is q; whereas if q is not ambiguous then ∆ is
bi-idoneal and hence – since q is not ambiguous – so is q.
INTEGERS: 13 (2013)
4
(iii) =⇒ (i): Let G(∆) = C(∆)/C(∆)2 , and let r : C(∆) → G(∆) be the quotient
map. The fibers of r are called genera; they are cosets of C(∆)2 , the principal
genus. Let c = #C(∆)/#G(∆). Thus ∆ is idoneal if and only if c = 1 and
bi-idoneal if and only if c = 2. For q ∈ C(∆), we define g(q) to be the set of
n ∈ (Z/∆Z)× which are represented by q. We will need the following tenets of
genus theory:
• For all q, q � ∈ C(∆), g(q) = g(q � ) ⇐⇒ r(q) = r(q � ) [3, pp. 53-54].
• If q ∈ C(∆)2 , then g(q) is a subgroup, H, of (Z/∆Z)× [3, Lem. 2.24, Thm. 3.15].
• For all q ∈ C(∆), g(q) is a coset of H in (Z/∆Z)× [3, Lem. 2.24, Thm. 3.15].
• Let n be a positive integer which
� � is relatively prime to 2∆. Then there is q ∈ C(∆)
representing n if and only if ∆
n = 1 [3, Thm. 2.16].
� �
In particular, let p � 2∆ be an odd prime. Then if ∆
= −1, no q ∈ C(∆) reprep
� �
sents p, whereas if ∆
= 1, then some q ∈ C(∆) represents p, and if q, q � ∈ C(∆)
p
both represent p, then r(q) = r(q � ).
Suppose ∆ is idoneal, let q ∈ C(∆), and�let�p � 2∆ be a prime. If q represents p
then p ∈ g(q); conversely, if p ∈ g(q) then ∆
= 1, so some q � ∈ C(∆) represents
p
�
p and any such q must lie in g(q). But since ∆ is idoneal, c = 1, and q is the only
form in r(q). Thus q represents p if and only if p ∈ g(q), so q is regular.
Suppose ∆ is bi-idoneal, let q ∈ C(∆) be a nonambiguous form, and let p � 2∆
be a prime. As above, if q represents p then p ∈ g(q); conversely, if p ∈ g(q) then
some q � ∈ r(q) repesents p. But since c = 2, r(q) = {[q], [q]} = {[q], [q]−1 }, and q
and q represent the same primes. Thus q represents p if and only if p ∈ g(q), so q
is regular.
Remark. The implication (ii) =⇒ (iii) for fundamental discriminants was first
proven by Kusaba [10], using methods of class field theory. In [9] the general case
is proved, and their proof uses Gauss’s genus theory together with a theorem of
Meyer [12]: if n and N are coprime positive integers, p ≡ n (mod N ), p � 2∆ is a
prime number and q ∈ C(∆) represents p, then q represents infinitely many prime
numbers q ≡ n (mod N ). See [6] for a proof of Meyer’s theorem and a second proof
of (ii) =⇒ (iii), both using class field theory.
2.2. Part b)
That the total number of idoneal and bi-idoneal discriminants lies between 425 and
432 is [14, Thm. 8.2].
2.3. Part c)
This is [14, Prop. 5.1] and [14, Thm. 8.6]. The latter result builds on work of
Weinberger [15] and Louboutin [11].
INTEGERS: 13 (2013)
5
3. A Small Multiple Theorem
Let q = �A, B, C� be a real binary quadratic form with discriminant ∆ �= 0. Recall:
• If ∆ > 0, then q is indefinite: it assumes both positive and negative values.
• If ∆ < 0 and A, C > 0, then q is positive definite: it assumes only positive
values except at (x, y) = (0, 0).
• If ∆ < 0 and A, C < 0, then q is negative definite: it assumes only negative
values except at (x, y) = (0, 0). Since q is negative definite if and only if −q is
positive definite, negative definite forms do not require separate consideration.
Theorem 3. Let q = �A, B, C� be a binary form over R with discriminant �
∆.
a) If ∆ < 0, there are integers x and y, not both zero, such that |q(x, y)| ≤ |∆|
.
� 3
b) If ∆ > 0, there are integers x and y, not both zero, such that |q(x, y)| ≤ ∆
5.
Proof. The core of the proof is the following “reduction lemma”: if x0 , y0 are coprime integers with q(x0 , y0 ) = M �= 0, then there are b, c ∈ R such that q is
SL2 (Z)-equivalent to M x2 + bxy + cy 2 with −|M | < b ≤ |M |. For the details, see
e.g. [7, Thm. 453, Thm. 454].
A lattice Λ ⊂ RN is the set of all Z-linear combinations of an R-basis b =
{v1 , . . . , vN } for RN . If Mb ∈ MN (R) is the matrix with columns v1 , . . . , vN , then
Λ = Mb ZN .
Proposition 1. Let q = �A, B, C� be an integral form of discriminant ∆. Let p be
2
an odd prime with ( ∆
p ) = 1. Then there is an index p sublattice Λp ⊂ Z such that
for all (x, y) ∈ Λp , q(x, y) ≡ 0 (mod p).
�
�
1 0
Proof. If p | A, take Mp =
and Λp = Mp Z2 . If p � A, by the quadratic
0 p
�
�
p r
2
formula in Z/pZ, there is r ∈ Z with Ar + Br + C ≡ 0 (mod p); set Mp =
0 1
2
and Λp = Mp Z . In either case, q(x, y) ≡ 0 (mod p) for all (x, y) ∈ Λp .
Theorem 4. Let q = �A, B, C� be an integral form of discriminant ∆. Let p be an
odd prime with ( ∆
p ) = 1.
�
a) If q is positive definite, there are x, y, k ∈ Z with q(x, y) = kp and 1 ≤ k ≤ |∆|
3 .
�
∆
b) If q is indefinite, there are x, y, k ∈ Z with q(x, y) = kp and 1 ≤ |k| ≤
5.
Proof. By Proposition 1, there is an index p sublattice Λp = Mp Z2 ⊂ Z2 with
q(x, y) ≡ 0 (mod p) for all (x, y) ∈ Λp . Thus the quadratic form q � (x, y) =
q(Mp (x, y)) has discriminant (det Mp )2 ∆ = p2 ∆ and is such that q � (x, y) ≡ 0
(mod p) for all (x, y) ∈ Z2 . Apply Theorem 3 to q � : if q is positive definite, there
6
INTEGERS: 13 (2013)
��
|∆|
3
�
are integers x and y, not both zero, such that |q(Mp (x, y)| = |q (x, y)| ≤
p.
�
Thus q(x, y) = kp with 1 ≤ |k| ≤ |∆|
3 ; since q is positive definite, k > 0. If ∆ > 0,
�
there
are
integers
x
and
y,
not
both
zero,
� such that |q(Mp (x, y)| = |q (x, y)| ≤
�� �
∆
∆
5 p, so q(x, y) = kp with 1 ≤ |k| ≤
5.
�
Remark. Taking q = �1, 1, 1� (resp. �1, 1, −1�) shows that the bound in Theorem
4a) (resp. Theorem 4b) is sharp.
�
Remark. Let q = �A, B, C� be positive definite with |∆| < 12. Then |∆|
3 < 2,
� �
and Theorem 4 takes the form: every odd prime p with ∆
= 1 is Z-represented by
p
q. It is easy to see that these are the only odd primes p � 2∆ which are represented
by q (c.f. Proposition 2), so this proves Theorem 2 for these forms, namely for
�1, 1, 1�, �1, 0, 1�, �1, 1, 2�, �1, 0, 2�, and �1, 1, 3�.
4. 2779 Regular Forms
In this section we will use Theorem 4 to prove Theorem 2.
Henceforth “forms” are primitive, positive definite integral binary quadratic forms.
4.1. Necessity
Proposition 2. Let q = �A, B, C� be a form with discriminant ∆. Let p be an
odd prime not dividing ∆. Suppose there exist x, y ∈ Z with q(x, y) = p. Then p
satisfies conditions (i) - (iv) from Theorem 2.
Proof. Via the discriminant-preserving transformation �A, B, C� �→ �C, B, A� we
may assume in m � A in part (ii) and 2 � A in parts (iii) and (iv); otherwise, q would
not be primitive.
(i) If both x and y were divisible by p, this would imply p2 | q(x, y) = p, a contradiction. If p � y, then we have A(xy −1 )2 + B(xy −1 ) + C ≡ 0 (mod p). Let r ∈ Z
with r ≡ xy −1 (mod p). Then
(2Ar + B)2 = 4A(Ar2 + Br + C) + B 2 − 4AC ≡ ∆ (mod p)
As p � ∆, we conclude ( ∆
p ) = 1. The case p � x follows similarly.
(ii) Let m be an odd prime such that m | ∆ and m � A. Via a change of variables
we can diagonalize q over Z/mZ as �A, 0, C − B 2 (4A)−1 �, so there are w, z ∈ Z with
p = q(x, y) ≡ Aw2 + (C − B 2 (4A)−1 )z 2 (mod m) .
INTEGERS: 13 (2013)
7
Multiplying by 4A gives 4Ap ≡ 4A2 w2 (mod m). Hence p ≡ Aw2 (mod m). It
p
A
follows that ( m
) = (m
).
(iii) Suppose 2 � A and ∆ ≡ 0 (mod 16). We have B 2 ≡ 4AC (mod 16), so B = 2B0
for some B0 ∈ Z. Then 4(B02 − AC) ≡ 0 (mod 16), so B02 − AC ≡ 0 (mod 4).
Case 1: B0 is odd. Then A ≡ C ≡ ±1 (mod 4). Now, Ax2 + 2B0 xy + Cy 2 = p, so
x2 + y 2 ≡ p ≡ 1 (mod 2), and x �≡ y (mod 2). If y ≡ 0 (mod 2), p ≡ A (mod 4)
as claimed. Similarly if x ≡ 0 (mod 2), p ≡ C (mod 4). But since A ≡ C (mod 4),
p ≡ A (mod 4) as claimed.
Case 2: B0 is even. Then AC ≡ 0 (mod 4). As 2 � A, C ≡ 0 (mod 4). Hence,
Ax2 ≡ p (mod 4), and so p ≡ A (mod 4) as claimed.
(iv) Suppose 2 � A and ∆ ≡ 0 (mod 32). Put B = 2B0 , so B02 − AC ≡ 0 (mod 8).
Case 1: B0 is odd, Then A ≡ C (mod 2) and in fact A ≡ C (mod 8). Thus
x2 + y 2 ≡ p ≡ 1 (mod 2), so x �≡ y (mod 2). If y ≡ 0 (mod 2), set y = 2y0 . Then
Ax2 +4y0 (B0 x+Cy0 ) = p. If y0 is even, then Ax2 ≡ A ≡ p (mod 8). If instead y0 is
odd, then since B0 , x, and C are odd, B0 x + Cy0 is even and Ax2 ≡ A ≡ p (mod 8).
Similarly if x ≡ 0 (mod 2), then p ≡ C ≡ A (mod 8).
Case 2: B0 is even. Put B0 = 2B1 and C = 4C0 , so B12 ≡ AC0 (mod 2) and
p = Ax2 + Bxy + Cy 2 = Ax2 + 4y(B1 x + C0 y).
Thus x is odd and x2 ≡ 1 (mod 8). If y is even, then p ≡ Ax2 ≡ A (mod 8). If y
is odd then either B1 ≡ C0 ≡ 0 (mod 2) so p ≡ Ax2 ≡ A (mod 8) or B1 ≡ C0 ≡ 1
(mod 2), so B1 x + C0 y is even and once again p ≡ Ax2 ≡ A (mod 8).
4.2. Sufficiency
Our proof that (b) implies (a) in Theorem 2 is handled individually for each of the
2779 forms. For each form, we apply a three step process. First, we use Theorem 4
to demonstrate that our form represents a small multiple of a prime. In the second
step, we eliminate certain multiples from consideration. In the final step, we reduce
the remaining multiples to find a representation of p.
Example. Consider q = �3, 3, 5� with ∆ = −51. Let p be an odd prime not dividing
∆ that satisfies conditions (i) - (iv) of Theorem 2.
Step 1. From condition (i) of Theorem 2, ( ∆
) = 1. Apply Theorem 4: there are
� p
x, y, k ∈ Z with q(x, y) = kp and 1 ≤ k ≤ 51
3 = 4.123 . . ..
Step 2 (Elimination). We will show that the cases k = 2 and k = 3 cannot
occur.
• Suppose q(x, y) = 2p. Then x and y are both even, so q(x, y) = 2p ≡ 0 (mod 4),
contradicting the fact that p is odd.
• Suppose q(x, y) = 3p. Then q(x, y) ≡ 5y 2 ≡ 0 (mod 3), so 3 | y. Hence,
INTEGERS: 13 (2013)
8
� �
q(x, y) ≡ 3x2 ≡ 3p (mod 9), so p3 = 1. As 3 | ∆, from condition (ii) of Theorem
2, ( p3 ) = ( 53 ) = −1: contradiction.
Step 3 (Reduction). Note that we cannot hope to eliminate the possibility of
k = 4: indeed, we want to show that there are x, y ∈ Z such that q(x, y) = p, and
then necessarily q(2x, 2y) = 4p. (A similar argument will be needed for any value of
k which is a perfect square). We must instead argue that a representation of 4p by
q implies a representation of p by q. In this case, this is easy: suppose q(x, y) = 4p.
Then as above x and y are both even, so q( x2 , y2 ) = p.
In Lemmas 1 and 2, we collect a number of congruence restrictions that apply
assuming a form q represents kp. In particular, for our 2779 forms, we use Lemma
1 in the elimination step and Lemma 2 in the reduction step.
Lemma 1 (Elimination). Let q = �A, B, C� be a form of discriminant ∆. Let
p � 2∆ be a prime. Suppose there are x, y, k ∈ Z, k ≥ 1, with q(x, y) = kp.
a) Let a ∈ Z, a > 1. Suppose 2a+2 | ∆ and 2a | B. If p ≡ A (mod 2a ), then k is a
square modulo 2a .
b) If k is even, A, C are odd, B ≡ 0 (mod 4) and A + C �≡ 2 (mod 4), then 4 | k.
p
A
c) Let m be an odd prime dividing ∆. If ( m
) = (m
), then k is a square modulo m.
∆
d) Let m be an odd prime dividing k. If ( m
) = −1 or m2 | ∆, then m2 | k.
p
A
e) Let m be an odd prime dividing gcd(∆, k) such that m2 � k. If ( m
) = (m
) then
k/m
−∆/m
( m ) = ( m ).
�
�
Proof. a) Since ∆ ≡ B 2 ≡ 0 mod 2a+2 , and A is odd, 2a | C. Then kp ≡ Ax2 ≡
px2 (mod 2a ), and since p is odd, this implies k ≡ x2 (mod 2a ).
b) We have q(x, y) ≡ Ax2 + Cy 2 ≡ A(x2 − y 2 ) ≡ kp (mod 4). Since k is even, x ≡ y
(mod 2) and thus kp ≡ A(x2 − y 2 ) ≡ 0 (mod 4). Since p is odd, 4 | k.
c) Via a change of variables we can diagonalize q over Z/mZ as �A, 0, C −B 2 (4A)−1 �,
so there are w, z ∈ Z with kp = q(x, y) ≡ Aw2 + (C − B 2 (4A)−1 )z 2 (mod m) . Thus,
p
A
4Akp ≡ 4A2 w2 (mod m), implying kp ≡ Aw2 (mod m). As ( m
) = (m
) �= 0, k is a
square modulo m.
∆
d) Suppose first that ( m
) = −1. We have q(x, y) ≡ 0 (mod m). If m � y, then
−1
q(xy , 1) ≡ 0 (mod m), so ∆ is a square modulo m: contradiction. So m | y.
Then Ax2 ≡ 0 (mod m), and m � A, since otherwise ∆ ≡ B 2 (mod m). Hence
2
m | x. Then m2 | q(x, y) = kp, and since ( ∆
p ) = 1, we have p �= m and m | k.
Next suppose m2 | ∆. If m | gcd(A, C), since m | ∆ we would also have m |
B, contradicting the primitivity of q. We may assume without loss of generality
that m � A. As B 2 − 4AC ≡ 0 (mod m), C ≡ B 2 (4A)−1 (mod m). Hence,
INTEGERS: 13 (2013)
9
Ax2 + Bxy + B 2 (4A)−1 y 2 ≡ 0 (mod m), so by multiplying through by 4A,
4A2 x2 + 4ABxy + B 2 y 2 ≡ (2Ax + By)2 ≡ 0 (mod m) .
�
�
2 2
Since m is prime, 2Ax+By
≡
0 (mod m), so 4A2 x2 +4ABxy+B
y ≡ 0 mod m2 .
�
�
�
�
As B 2 − 4AC ≡ 0 mod m2 , we have B 2 (4A)−1 ≡ C mod m2 . Then
�
�
4Akp ≡ 4A2 x2 + 4ABxy + B 2 y 2 ≡ 0 mod m2 .
Since p � ∆, m �= p. Then m does not divide 4Ap, so m2 | k.
e) Since m | ∆ and p � ∆, m �= p. We may write ∆ = �m∆0 and
� k = mk0 with
∆0 , k0 ∈ Z and m � k0 . Then Ax2 + Bxy + Cy 2 ≡ mk0 p mod m2 . As in part d),
�
�
Ax2 + Bxy + (B 2 (4A)−1 )y 2 ≡ 0 mod m2 .
�
�
Subtracting gives (C − B 2 (4A−1 ))y 2 ≡ mk0 p mod m2 . Since gcd(m, k0 p) = 1, it
follows that m � y. Multiplying through by 4A, we get
�
�
−m∆0 y 2 ≡ (4AC − B 2 )y 2 ≡ 4Amk0 p mod m2 .
�
�
Then (4Ak0 p + ∆0 y 2 )m ≡ 0 mod m2 , so 4Apk0 ≡ −∆0 y 2 (mod m). It follows
2
−∆0 y
4Apk0
p
k0
k0
A
0
that ( −∆
m ) = ( m ) = ( m ) ≡ ( m )( m )( m ) = ( m ).
Lemma 2 (Reduction). Let q = �A, B, C� have discriminant ∆. Let p be an odd
prime not dividing ∆. Suppose there exist x, y, k ∈ Z with q(x, y) = kp and k ≥ 1.
a) Let a ∈ Z with a ≥ 1. If p ≡ A (mod 2a ), then q(x, y) ≡ Ak (mod 2a k).
b) Let a ∈ Z with a ≥ 0, and let m | ∆ be an odd prime. If m2a | k, m2a+1 � k, and
2a
2a
p
A
(m
) = (m
), then we have ( q(x,y)/m
) = ( Ak/m
).
m
m
Proof. For a), write p = 2a � + A. Then q(x, y) ≡ k(2a � + A) ≡ Ak (mod 2a k). For
2a
0p
0
b), write k = m2a k0 . Then ( q(x,y)/m
) = ( km
) = ( Ak
m
m ).
4.3. Proof of Theorem 2
(a) =⇒ (b): This is Proposition 2.
(b) =⇒ (a): Let q = �A, B, C� be one of the 2779 regular forms, and let p � 2∆ be
a prime satisfying conditions (i) - (iv) from Theorem 2.
Step 1. Using condition �
(i), Theorem 4 implies there exist x, y, k ∈ Z such that
q(x, y) = kp with 1 ≤ k ≤ |∆|
3 .
�
Step 2 (Elimination). For each k ∈ {2, . . . , � |∆|
3 �}, assume q(x, y) = kp. If k
does not satisfy the conditions imposed on it by Lemma 1, we have a contradiction.
INTEGERS: 13 (2013)
10
We similarly have a contradiction if k does not satisfy the conditions imposed on it
by applying Lemma 1 to the equivalent forms q(y, x) = �C, B, A� and q(x + y, x +
2y) = �A + B + C, 2A + 3B + 4C, A + 2B + 4C� representing kp. We eliminate these
k from consideration.
�
Step 3 (Reduction). For each k ∈ {2, . . . , � |∆|
3 �} that was not eliminated in
Step 2, assume q(x, y) = kp. Using a computer, we have verified that this assumption leads to a representation of p by q in every case. Our algorithm is as follows.
First, we construct the finite set of matrices
��
�
�
�
a b
�
M=
∈ M2 (Z) � a ≥ 0, q(a, c) = kA and q(b, d) = kC
c d
�
�
a b
by enumerating the representations of kA and kC by q. Given M =
∈ M,
c d
q(M (x, y)) = kAx2 + (2abA + (ad + bc)B + 2cdC)xy + kCy 2 .
In particular, q(M (x, y)) = kq(x, y) whenever 2abA + (ad + bc)B + 2cdC = kB. By
iterating over M and checking this condition, we verify that there exists some M ∈
M such that q(M (x, y)) = kq(x, y). Fixing such an M , we further check whether
for each (x, y) ∈ Z2 with q(x, y) ≡ 0 (mod k) that also satisfies the congruence
restrictions imposed by Lemma 2, the pair (x0 , y0 ) = M (x, y) satisfies x0 ≡ y0 ≡
0 (mod k). It suffices to check this condition modulo k∆ by an exhaustive search.
In every case we’ve considered, this search successfully produces such an M ∈
M. Once such an M has been found, we can set x0 = kw and y0 = kz. Then
q(M (x, y)) = q(kw, kz) = k2 p, so q(w, z) = p. Therefore, we’ve shown that q
represents p.
Example. Consider q = �2, 1, 7� with ∆ = −55. Let p be an odd prime not dividing
∆ that satisfies conditions (i) - (iv) of Theorem 2.
Step 1. From condition (i) of Theorem 2, ( ∆
p )�= 1. Thus, applying Theorem 4
yields x, y, k ∈ Z with q(x, y) = kp and 1 ≤ k ≤ 55
3 = 4.28 . . .. .
Step 2 (Elimination). By Lemma 1(c), k is a square modulo 5. As ( 25 ) = ( 35 ) =
−1, k ∈ {1, 4}.
Step 3 (Reduction). Suppose q(x, y) = 4p. One might try to argue, as in the
example in Section 4.2, that both x and y are even. However, this need not be the
case: e.g. q represents 7 and q(3, 1) = 4 · 7. Applying the algorithm described above
11
INTEGERS: 13 (2013)
we obtain
��
M=
�
� �
� �
� �
1 −3
1 −1
1
0
1
,
,
,
−1 −1
−1 2
−1 −2
−1
�
� �
� �
� �
� �
2 −3
2 −1
2 0
2 0
2
,
,
,
,
0 −1
0 2
0 −2
0 2
0
� �
0
1
,
2
−1
� �
1
2
,
−2
0
� �
�
1
1 3
,
,
−2
−1 1
��
3
.
1
�
1 −3
Set M =
. Set (x0 , y0 ) = M (x, y) = (x − 3y, −x − y) and note q(x0 , y0 ) =
−1 −1
4q(x, y) = 16p. If we knew x0 ≡ y0 ≡ 0 (mod 4), then we could divide through
by 4 to obtain an integer representation of p. Certainly we need only consider
(x, y) ∈ Z2 with q(x, y) ≡ 0 (mod 4). Further, since we’re assuming ( p5 ) = ( 25 ) = −1
p
2
and ( 11
) = ( 11
) = −1, condition (ii) of Theorem 2 implies we need only consider
q(x,y)
4p
4p
2
(x, y) ∈ Z with ( q(x,y)
5 ) = ( 5 ) = −1 and ( 11 ) = ( 11 ) = −1. By an exhaustive
2
search modulo 220, we verify the only such (x, y) ∈ Z yield x0 ≡ y0 ≡ 0 (mod 4).
Setting x0 = 4w and y0 = 4z, we have q(x0 , y0 ) = 32w2 + 16wz + 224z 2 = 16p.
Dividing through by 16, we see q(w, z) = 2w2 + wz + 7z 2 = p. Therefore, we’ve
shown that q represents p.
Acknowledgements. This work was done in the context of a VIGRE Research
Group at the University of Georgia during the 2011-2012 academic year. The
group was led by the first author, with participants the other three authors together with Christopher Drupieski (postdoc), Brian Bonsignore, Harrison Chapman, Lauren Huckaba, David Krumm, Allan Lacy Mora, Nham Ngo, Alex Rice,
James Stankewicz, Lee Troupe, Nathan Walters (doctoral students) and Jun Zhang
(master’s student).
References
[1] P.L. Clark, Geometry of numbers explained, in preparation.
[2] P.L. Clark, J. Hicks, K. Thompson and N. Walters, GoNII: Universal quaternary quadratic
forms. Integers 12 (2012), A50, 16 pp.
[3] D.A. Cox, Primes of the Form x2 + ny 2 , John Wiley & Sons Inc., 1989.
[4] C.F. Gauss, Disquisitiones Arithmeticae (English Edition), trans. A.A. Clarke, SpringerVerlag, 1986.
[5] T.R. Hagedorn, Primes of the Form x2 + ny 2 and the Geometry of (Convenient) Numbers,
preprint.
[6] F. Halter-Koch, Representation of prime powers in arithmetical progressions by binary
quadratic forms. Les XXIIèmes Journées Arithmetiques (Lille, 2001). J. Théor. Nombres Bordeaux 15 (2003), no. 1, 141-149.
12
INTEGERS: 13 (2013)
[7] G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers. Sixth edition.
Revised by D. R. Heath-Brown and J. H. Silverman. Oxford, 2008.
[8] J. Hicks and K. Thompson, GoNIII: More universal quaternary quadratic forms, in preparation.
[9] P. Kaplan and K.S. Williams, Representation of primes in arithmetic progression by binary
quadratic forms. J. Number Theory (1993), 61-67.
[10] T. Kusaba, Remarque sur la distribution des nombres premiers. C. R. Acad. Sci. Paris Sér.
A-B 265 (1967), A405-A407.
[11] S. Louboutin, Minorations (sous l’hypothèse de Riemann généralisée) des nombres de classes
des corps quadratiques imaginaires. Application. C. R. Acad. Sci. Paris Sér. I Math. 310 (1990),
no. 12, 795-800.
[12] A. Meyer, Über einen satz von Dirichlet. J. Reine Angew. Math. 103 (1888), 98–117.
[13] W. A. Stein et al., Sage Mathematics Software (Version 4.7.1), The Sage Development Team,
2011, http://www.sagemath.org
[14] J. Voight, Quadratic forms that represent almost the same primes. Math. Comp. 76 (2007),
1589-1617.
[15] P.J. Weinberger, Exponents of the class groups of complex quadratic fields. Acta Arith. 22
(1973), 117-124.
Appendix
In Table 1, we list the reduced representative for each of the 2779 SL2 (Z) equivalence
classes of regular forms. The discriminants were calculated by Voight in [14]. We
redid this calculation, and in so doing found a minor error of tabulation which
Voight confirmed. The forms were generated using Sage.
Table 1: Representatives for 2779 SL2 (Z)-equivalence Classes of Regular Forms
|∆|
3
11
16
24
32
36
40
51
55
63
68
75
84
91
96
100
115
120
128
132
�A, B, C�
�1, 1, 1�
�1, 1, 3�
�1, 0, 4�
�1, 0, 6�
�1, 0, 8�
�1, 0, 9�
�2, 0, 5�
�1, 1, 13�
�2, ±1, 7�
�2, ±1, 8�
�3, ±2, 6�
�3, 3, 7�
�3, 0, 7�
�1, 1, 23�
�4, 4, 7�
�1, 0, 25�
�1, 1, 29�
�3, 0, 10�
�3, ±2, 11�
�6, 6, 7�
|∆|
4
12
19
24
32
36
43
51
56
64
72
80
84
91
96
100
115
120
132
136
�A, B, C�
�1, 0, 1�
�1, 0, 3�
�1, 1, 5�
�2, 0, 3�
�3, 2, 3�
�2, 2, 5�
�1, 1, 11�
�3, 3, 5�
�3, ±2, 5�
�1, 0, 16�
�1, 0, 18�
�3, ±2, 7�
�5, 4, 5�
�5, 3, 5�
�5, 2, 5�
�2, 2, 13�
�5, 5, 7�
�5, 0, 6�
�1, 0, 33�
�5, ±2, 7�
|∆|
7
15
20
27
35
39
48
52
60
64
72
84
88
96
99
112
120
123
132
144
�A, B, C�
�1, 1, 2�
�1, 1, 4�
�1, 0, 5�
�1, 1, 7�
�1, 1, 9�
�2, ±1, 5�
�1, 0, 12�
�1, 0, 13�
�1, 0, 15�
�4, 4, 5�
�2, 0, 9�
�1, 0, 21�
�1, 0, 22�
�1, 0, 24�
�1, 1, 25�
�1, 0, 28�
�1, 0, 30�
�1, 1, 31�
�2, 2, 17�
�5, ±4, 8�
|∆|
8
15
20
28
35
40
48
52
60
67
75
84
88
96
99
112
120
123
132
147
�A, B, C�
�1, 0, 2�
�2, 1, 2�
�2, 2, 3�
�1, 0, 7�
�3, 1, 3�
�1, 0, 10�
�3, 0, 4�
�2, 2, 7�
�3, 0, 5�
�1, 1, 17�
�1, 1, 19�
�2, 2, 11�
�2, 0, 11�
�3, 0, 8�
�5, 1, 5�
�4, 0, 7�
�2, 0, 15�
�3, 3, 11�
�3, 0, 11�
�1, 1, 37�
13
INTEGERS: 13 (2013)
Table 1: Representatives for 2779 SL2 (Z)-equivalence Classes of Regular Forms
|∆|
147
156
160
168
180
187
192
195
208
224
228
235
240
260
267
276
280
288
308
312
315
328
340
352
360
372
384
403
408
420
420
435
448
456
475
480
480
483
520
528
532
552
555
568
580
595
603
616
627
640
660
660
672
672
708
715
720
736
760
772
795
819
832
840
�A, B, C�
�3, 3, 13�
�5, ±2, 8�
�7, 6, 7�
�3, 0, 14�
�2, 2, 23�
�1, 1, 47�
�4, 4, 13�
�5, 5, 11�
�7, ±4, 8�
�5, ±4, 12�
�6, 6, 11�
�5, 5, 13�
�5, 0, 12�
�3, ±2, 22�
�1, 1, 67�
�7, ±2, 10�
�7, 0, 10�
�8, 8, 11�
�6, ±2, 13�
�6, 0, 13�
�9, 9, 11�
�7, ±6, 13�
�2, 2, 43�
�4, 4, 23�
�7, ±2, 13�
�2, 2, 47�
�7, ±6, 15�
�1, 1, 101�
�3, 0, 34�
�3, 0, 35�
�10, 10, 13�
�1, 1, 109�
�1, 0, 112�
�5, ±2, 23�
�7, ±1, 17�
�5, 0, 24�
�12, 12, 13�
�11, 1, 11�
�1, 0, 130�
�7, ±2, 19�
�7, 0, 19�
�7, ±6, 21�
�5, 5, 29�
�11, ±2, 13�
�11, ±6, 14�
�7, 7, 23�
�9, ±3, 17�
�10, ±8, 17�
�3, 3, 53�
�11, ±8, 16�
�2, 2, 83�
�10, 10, 19�
�1, 0, 168�
�8, 0, 21�
�1, 0, 177�
�1, 1, 179�
�7, ±6, 27�
�11, ±10, 19�
�10, 0, 19�
�11, ±8, 19�
�3, 3, 67�
�9, ±3, 23�
�11, ±2, 19�
�5, 0, 42�
|∆|
148
160
163
168
180
187
192
195
219
228
232
240
252
260
267
280
288
291
312
315
320
336
340
352
360
372
387
403
408
420
420
435
448
456
480
480
483
504
520
528
532
552
555
576
592
595
612
624
627
651
660
660
672
672
708
715
720
760
763
792
795
820
840
840
�A, B, C�
�1, 0, 37�
�1, 0, 40�
�1, 1, 41�
�6, 0, 7�
�5, 0, 9�
�7, 3, 7�
�7, 2, 7�
�7, 1, 7�
�5, ±1, 11�
�1, 0, 57�
�1, 0, 58�
�1, 0, 60�
�8, ±6, 9�
�6, ±2, 11�
�3, 3, 23�
�1, 0, 70�
�1, 0, 72�
�5, ±3, 15�
�1, 0, 78�
�1, 1, 79�
�3, ±2, 27�
�5, ±2, 17�
�5, 0, 17�
�8, 0, 11�
�9, ±6, 11�
�3, 0, 31�
�9, ±3, 11�
�11, 9, 11�
�6, 0, 17�
�5, 0, 21�
�11, 8, 11�
�3, 3, 37�
�4, 4, 29�
�10, ±8, 13�
�1, 0, 120�
�8, 0, 15�
�1, 1, 121�
�5, ±4, 26�
�2, 0, 65�
�8, ±4, 17�
�13, 12, 13�
�11, ±8, 14�
�13, 11, 13�
�5, ±2, 29�
�8, ±4, 19�
�13, 9, 13�
�7, ±2, 22�
�5, ±4, 32�
�11, 11, 17�
�5, ±3, 33�
�3, 0, 55�
�11, 0, 15�
�3, 0, 56�
�8, 8, 23�
�2, 2, 89�
�5, 5, 37�
�8, ±4, 23�
�1, 0, 190�
�13, ±11, 17�
�9, ±6, 23�
�5, 5, 41�
�11, ±4, 19�
�1, 0, 210�
�6, 0, 35�
|∆|
148
160
168
171
180
192
195
196
220
228
232
240
256
264
275
280
288
292
312
315
320
336
340
352
363
372
388
408
420
420
427
435
448
468
480
480
483
504
520
532
544
555
564
576
595
600
612
624
627
651
660
660
672
672
708
715
723
760
768
792
795
820
840
840
�A, B, C�
�2, 2, 19�
�4, 4, 11�
�1, 0, 42�
�5, ±3, 9�
�7, 4, 7�
�1, 0, 48�
�1, 1, 49�
�5, ±2, 10�
�7, ±2, 8�
�2, 2, 29�
�2, 0, 29�
�3, 0, 20�
�5, ±2, 13�
�5, ±4, 14�
�3, ±1, 23�
�2, 0, 35�
�4, 4, 19�
�7, ±4, 11�
�2, 0, 39�
�5, 5, 17�
�7, ±4, 12�
�8, ±4, 11�
�10, 10, 11�
�8, 8, 13�
�7, ±1, 13�
�6, 6, 17�
�7, ±2, 14�
�1, 0, 102�
�1, 0, 105�
�6, 6, 19�
�1, 1, 107�
�5, 5, 23�
�7, 0, 16�
�7, ±6, 18�
�3, 0, 40�
�8, 8, 17�
�3, 3, 41�
�10, ±4, 13�
�5, 0, 26�
�1, 0, 133�
�5, ±4, 28�
�1, 1, 139�
�5, ±4, 29�
�9, ±6, 17�
�1, 1, 149�
�7, ±4, 22�
�11, ±2, 14�
�11, ±6, 15�
�13, 7, 13�
�11, ±3, 15�
�5, 0, 33�
�13, 4, 13�
�4, 4, 43�
�12, 12, 17�
�3, 0, 59�
�11, 11, 19�
�11, ±5, 17�
�2, 0, 95�
�7, ±4, 28�
�13, ±12, 18�
�15, 15, 17�
�13, ±8, 17�
�2, 0, 105�
�7, 0, 30�
|∆|
155
160
168
180
184
192
195
203
224
228
235
240
259
264
276
280
288
308
312
315
323
340
352
355
372
384
400
408
420
420
427
435
448
468
480
480
483
507
520
532
544
555
564
580
595
600
616
627
640
660
660
667
672
672
708
715
736
760
768
795
819
832
840
840
�A, B, C�
�3, ±1, 13�
�5, 0, 8�
�2, 0, 21�
�1, 0, 45�
�5, ±4, 10�
�3, 0, 16�
�3, 3, 17�
�3, ±1, 17�
�3, ±2, 19�
�3, 0, 19�
�1, 1, 59�
�4, 0, 15�
�5, ±1, 13�
�7, ±4, 10�
�5, ±2, 14�
�5, 0, 14�
�8, 0, 9�
�3, ±2, 26�
�3, 0, 26�
�7, 7, 13�
�3, ±1, 27�
�1, 0, 85�
�1, 0, 88�
�7, ±3, 13�
�1, 0, 93�
�5, ±4, 20�
�8, ±4, 13�
�2, 0, 51�
�2, 2, 53�
�7, 0, 15�
�7, 7, 17�
�11, 7, 11�
�11, 6, 11�
�9, ±6, 14�
�4, 4, 31�
�11, 2, 11�
�7, 7, 19�
�7, ±5, 19�
�10, 0, 13�
�2, 2, 67�
�7, ±4, 20�
�3, 3, 47�
�10, ±6, 15�
�7, ±6, 22�
�5, 5, 31�
�11, ±4, 14�
�5, ±2, 31�
�1, 1, 157�
�7, ±2, 23�
�1, 0, 165�
�6, 6, 29�
�11, ±9, 17�
�7, 0, 24�
�13, 2, 13�
�6, 6, 31�
�13, 13, 17�
�5, ±2, 37�
�5, 0, 38�
�13, ±8, 16�
�1, 1, 199�
�5, ±1, 41�
�7, ±6, 31�
�3, 0, 70�
�10, 0, 21�
14
INTEGERS: 13 (2013)
Table 1: Representatives for 2779 SL2 (Z)-equivalence Classes of Regular Forms
|∆|
840
868
900
915
928
960
960
987
1008
1012
1035
1056
1092
1092
1120
1120
1128
1140
1152
1155
1155
1227
1248
1248
1275
1312
1320
1320
1344
1360
1380
1380
1395
1428
1428
1435
1440
1443
1488
1540
1540
1560
1600
1632
1632
1659
1680
1683
1716
1768
1780
1824
1827
1848
1848
1860
1920
1992
1995
1995
2016
2035
2040
2080
�A, B, C�
�14, 0, 15�
�13, ±4, 17�
�13, ±6, 18�
�11, ±3, 21�
�8, 8, 31�
�1, 0, 240�
�12, 12, 23�
�11, ±5, 23�
�11, ±2, 23�
�17, 12, 17�
�7, ±1, 37�
�13, ±6, 21�
�1, 0, 273�
�7, 0, 39�
�1, 0, 280�
�8, 0, 35�
�11, ±4, 26�
�7, ±6, 42�
�11, ±6, 27�
�5, 5, 59�
�17, 1, 17�
�11, ±7, 29�
�1, 0, 312�
�8, 8, 41�
�11, ±1, 29�
�7, ±2, 47�
�3, 0, 110�
�11, 0, 30�
�5, ±4, 68�
�8, ±4, 43�
�3, 0, 115�
�15, 0, 23�
�17, ±13, 23�
�1, 0, 357�
�7, 0, 51�
�1, 1, 359�
�7, ±4, 52�
�11, ±3, 33�
�17, ±12, 24�
�5, 0, 77�
�14, 14, 31�
�14, ±8, 29�
�17, ±10, 25�
�8, 0, 51�
�23, 22, 23�
�15, ±9, 29�
�11, ±6, 39�
�9, ±3, 47�
�17, ±16, 29�
�22, ±16, 23�
�19, ±14, 26�
�13, ±10, 37�
�19, ±15, 27�
�6, 0, 77�
�21, 0, 22�
�21, ±18, 26�
�17, ±16, 32�
�23, ±20, 26�
�7, 7, 73�
�23, 11, 23�
�20, ±12, 27�
�19, ±13, 29�
�21, ±12, 26�
�4, 4, 131�
|∆|
852
880
912
928
952
960
960
987
1012
1027
1035
1056
1092
1092
1120
1120
1128
1140
1152
1155
1155
1240
1248
1248
1275
1312
1320
1320
1344
1360
1380
1380
1408
1428
1428
1435
1440
1443
1507
1540
1540
1560
1632
1632
1635
1672
1680
1716
1752
1771
1792
1824
1848
1848
1860
1920
1947
1995
1995
2016
2020
2040
2067
2080
�A, B, C�
�7, ±4, 31�
�7, ±4, 32�
�8, ±4, 29�
�1, 0, 232�
�11, ±4, 22�
�3, 0, 80�
�15, 0, 16�
�13, ±1, 19�
�1, 0, 253�
�7, ±3, 37�
�9, ±3, 29�
�15, ±12, 20�
�2, 2, 137�
�13, 0, 21�
�4, 4, 71�
�8, 8, 37�
�13, ±4, 22�
�11, ±2, 26�
�16, ±8, 19�
�7, 7, 43�
�19, 17, 19�
�11, ±6, 29�
�3, 0, 104�
�12, 12, 29�
�13, ±5, 25�
�13, ±12, 28�
�5, 0, 66�
�15, 0, 22�
�11, ±8, 32�
�11, ±2, 31�
�5, 0, 69�
�19, 8, 19�
�13, ±10, 29�
�2, 2, 179�
�14, 14, 29�
�5, 5, 73�
�9, ±6, 41�
�17, ±11, 23�
�13, ±1, 29�
�7, 0, 55�
�22, 22, 23�
�17, ±2, 23�
�1, 0, 408�
�8, 8, 53�
�11, ±9, 39�
�7, ±6, 61�
�13, ±6, 33�
�5, ±2, 86�
�13, ±4, 34�
�5, ±3, 89�
�11, ±10, 43�
�15, ±6, 31�
�1, 0, 462�
�7, 0, 66�
�7, ±4, 67�
�11, ±4, 44�
�13, ±9, 39�
�1, 1, 499�
�15, 15, 37�
�5, ±2, 101�
�11, ±2, 46�
�7, ±2, 73�
�11, ±1, 47�
�5, 0, 104�
|∆|
852
880
912
928
952
960
960
1003
1012
1032
1056
1060
1092
1092
1120
1120
1131
1140
1155
1155
1204
1240
1248
1248
1288
1320
1320
1332
1344
1380
1380
1387
1408
1428
1428
1435
1440
1467
1540
1540
1555
1560
1632
1632
1635
1672
1680
1716
1752
1771
1792
1824
1848
1848
1860
1920
1947
1995
1995
2016
2020
2040
2067
2080
�A, B, C�
�14, ±10, 17�
�13, ±2, 17�
�11, ±10, 23�
�4, 4, 59�
�13, ±6, 19�
�4, 4, 61�
�16, 16, 19�
�11, ±3, 23�
�2, 2, 127�
�7, ±2, 37�
�5, ±2, 53�
�7, ±2, 38�
�3, 0, 91�
�14, 14, 23�
�5, 0, 56�
�17, 6, 17�
�5, ±3, 57�
�13, ±2, 22�
�1, 1, 289�
�11, 11, 29�
�5, ±4, 61�
�17, ±16, 22�
�4, 4, 79�
�13, 0, 24�
�13, ±8, 26�
�1, 0, 330�
�6, 0, 55�
�9, ±6, 38�
�15, ±6, 23�
�1, 0, 345�
�6, 6, 59�
�13, ±11, 29�
�16, ±8, 23�
�3, 0, 119�
�17, 0, 21�
�7, 7, 53�
�11, ±10, 35�
�9, ±3, 41�
�1, 0, 385�
�10, 10, 41�
�17, ±3, 23�
�19, ±6, 21�
�3, 0, 136�
�12, 12, 37�
�13, ±9, 33�
�14, ±8, 31�
�19, ±12, 24�
�10, ±2, 43�
�17, ±4, 26�
�13, ±7, 35�
�16, ±8, 29�
�20, ±4, 23�
�2, 0, 231�
�11, 0, 42�
�13, ±8, 37�
�13, ±2, 37�
�17, ±5, 29�
�3, 3, 167�
�19, 19, 31�
�13, ±8, 40�
�22, ±2, 23�
�13, ±12, 42�
�19, ±17, 31�
�8, 0, 65�
|∆|
868
900
915
928
955
960
960
1008
1012
1032
1056
1060
1092
1092
1120
1120
1131
1140
1155
1155
1204
1243
1248
1248
1288
1320
1320
1332
1344
1380
1380
1395
1411
1428
1428
1435
1440
1488
1540
1540
1560
1600
1632
1632
1659
1680
1683
1716
1768
1780
1824
1827
1848
1848
1860
1920
1992
1995
1995
2016
2035
2040
2080
2080
�A, B, C�
�11, ±10, 22�
�9, ±6, 26�
�7, ±3, 33�
�8, 0, 29�
�7, ±5, 35�
�5, 0, 48�
�17, 14, 17�
�9, ±6, 29�
�11, 0, 23�
�14, ±12, 21�
�7, ±6, 39�
�14, ±2, 19�
�6, 6, 47�
�17, 8, 17�
�7, 0, 40�
�19, 18, 19�
�15, ±3, 19�
�14, ±6, 21�
�3, 3, 97�
�15, 15, 23�
�10, ±6, 31�
�17, ±7, 19�
�8, 0, 39�
�19, 14, 19�
�17, ±2, 19�
�2, 0, 165�
�10, 0, 33�
�18, ±6, 19�
�17, ±4, 20�
�2, 2, 173�
�10, 10, 37�
�13, ±3, 27�
�5, ±3, 71�
�6, 6, 61�
�19, 4, 19�
�19, 3, 19�
�13, ±4, 28�
�8, ±4, 47�
�2, 2, 193�
�11, 0, 35�
�7, ±6, 57�
�13, ±8, 32�
�4, 4, 103�
�17, 0, 24�
�5, ±1, 83�
�8, ±4, 53�
�7, ±5, 61�
�15, ±12, 31�
�11, ±6, 41�
�13, ±12, 37�
�5, ±4, 92�
�17, ±3, 27�
�3, 0, 154�
�14, 0, 33�
�14, ±10, 35�
�16, ±8, 31�
�13, ±6, 39�
�5, 5, 101�
�21, 21, 29�
�19, ±6, 27�
�7, ±3, 73�
�14, ±12, 39�
�1, 0, 520�
�8, 8, 67�
15
INTEGERS: 13 (2013)
Table 1: Representatives for 2779 SL2 (Z)-equivalence Classes of Regular Forms
|∆|
2080
2088
2100
2112
2128
2163
2208
2244
2272
2280
2340
2368
2400
2436
2451
2464
2496
2520
2580
2632
2640
2688
2715
2760
2772
2788
2880
2907
3003
3003
3040
3040
3060
3108
3168
3172
3192
3220
3315
3315
3355
3360
3360
3360
3360
3432
3480
3520
3588
3627
3640
3648
3795
3808
3828
3840
3843
4020
4032
4123
4128
4180
4260
4368
�A, B, C�
�13, 0, 40�
�18, ±12, 31�
�22, ±10, 25�
�21, ±18, 29�
�13, ±2, 41�
�17, ±9, 33�
�21, ±12, 28�
�10, ±6, 57�
�13, ±4, 44�
�14, ±4, 41�
�19, ±4, 31�
�23, ±22, 31�
�11, ±8, 56�
�10, ±2, 61�
�15, ±3, 41�
�20, ±4, 31�
�11, ±10, 59�
�17, ±8, 38�
�17, ±2, 38�
�23, ±6, 29�
�24, ±12, 29�
�16, ±8, 43�
�21, ±15, 35�
�13, ±10, 55�
�17, ±4, 41�
�23, ±8, 31�
�23, ±8, 32�
�27, ±15, 29�
�3, 3, 251�
�21, 21, 41�
�4, 4, 191�
�19, 0, 40�
�11, ±8, 71�
�13, ±8, 61�
�13, ±2, 61�
�23, ±18, 38�
�31, ±30, 33�
�26, ±2, 31�
�3, 3, 277�
�17, 17, 53�
�23, ±7, 37�
�5, 0, 168�
�12, 12, 73�
�24, 0, 35�
�31, 22, 31�
�31, ±28, 34�
�26, ±24, 39�
�13, ±4, 68�
�17, ±4, 53�
�11, ±5, 83�
�31, ±24, 34�
�32, ±24, 33�
�17, ±9, 57�
�13, ±12, 76�
�14, ±6, 69�
�17, ±6, 57�
�17, ±13, 59�
�31, ±14, 34�
�29, ±12, 36�
�29, ±13, 37�
�28, ±4, 37�
�31, ±6, 34�
�31, ±24, 39�
�17, ±16, 68�
|∆|
2080
2100
2112
2115
2139
2208
2212
2244
2275
2280
2340
2392
2400
2436
2464
2475
2496
2520
2580
2640
2667
2688
2755
2760
2772
2832
2880
2968
3003
3003
3040
3040
3060
3108
3168
3192
3220
3243
3315
3315
3360
3360
3360
3360
3432
3480
3507
3520
3588
3640
3648
3712
3795
3808
3828
3840
4020
4032
4048
4128
4180
4260
4323
4368
�A, B, C�
�20, 20, 31�
�11, ±10, 50�
�7, ±4, 76�
�9, ±3, 59�
�5, ±1, 107�
�7, ±2, 79�
�17, ±10, 34�
�15, ±6, 38�
�19, ±9, 31�
�17, ±10, 35�
�22, ±6, 27�
�7, ±4, 86�
�21, ±6, 29�
�15, ±12, 43�
�5, ±4, 124�
�23, ±3, 27�
�15, ±12, 44�
�18, ±12, 37�
�19, ±2, 34�
�8, ±4, 83�
�17, ±11, 41�
�17, ±10, 41�
�13, ±1, 53�
�22, ±12, 33�
�26, ±6, 27�
�8, ±4, 89�
�27, ±24, 32�
�13, ±10, 59�
�7, 7, 109�
�29, 19, 29�
�5, 0, 152�
�20, 20, 43�
�18, ±6, 43�
�22, ±18, 39�
�19, ±10, 43�
�11, ±8, 74�
�11, ±6, 74�
�17, ±15, 51�
�5, 5, 167�
�29, 7, 29�
�1, 0, 840�
�7, 0, 120�
�15, 0, 56�
�24, 24, 41�
�17, ±6, 51�
�13, ±2, 67�
�13, ±9, 69�
�17, ±4, 52�
�22, ±14, 43�
�11, ±10, 85�
�11, ±2, 83�
�16, ±8, 59�
�19, ±9, 51�
�19, ±12, 52�
�21, ±6, 46�
�19, ±6, 51�
�13, ±6, 78�
�9, ±6, 113�
�8, ±4, 127�
�7, ±4, 148�
�17, ±6, 62�
�13, ±2, 82�
�19, ±3, 57�
�23, ±18, 51�
|∆|
2080
2100
2112
2115
2139
2208
2212
2244
2275
2280
2340
2392
2400
2436
2464
2475
2496
2520
2580
2640
2667
2688
2755
2760
2772
2832
2880
2968
3003
3003
3040
3040
3060
3108
3168
3192
3220
3243
3315
3315
3360
3360
3360
3360
3432
3480
3507
3520
3588
3640
3648
3712
3795
3808
3828
3840
4020
4032
4048
4128
4180
4260
4323
4368
�A, B, C�
�23, 6, 23�
�17, ±12, 33�
�17, ±8, 32�
�13, ±11, 43�
�15, ±9, 37�
�11, ±6, 51�
�19, ±12, 31�
�19, ±6, 30�
�23, ±5, 25�
�21, ±18, 31�
�23, ±12, 27�
�14, ±4, 43�
�25, ±20, 28�
�23, ±18, 30�
�17, ±16, 40�
�25, ±15, 27�
�20, ±12, 33�
�19, ±8, 34�
�22, ±18, 33�
�13, ±8, 52�
�23, ±1, 29�
�23, ±16, 32�
�17, ±13, 43�
�26, ±16, 29�
�27, ±24, 31�
�24, ±12, 31�
�27, ±12, 28�
�26, ±16, 31�
�11, 11, 71�
�31, 29, 31�
�8, 0, 95�
�29, 18, 29�
�22, ±14, 37�
�26, ±18, 33�
�23, ±12, 36�
�17, ±2, 47�
�13, ±2, 62�
�19, ±5, 43�
�13, 13, 67�
�31, 23, 31�
�3, 0, 280�
�8, 0, 105�
�20, 20, 47�
�28, 28, 37�
�19, ±8, 46�
�19, ±4, 46�
�23, ±9, 39�
�28, ±20, 35�
�33, ±30, 34�
�17, ±10, 55�
�23, ±20, 44�
�31, ±16, 32�
�29, ±27, 39�
�29, ±22, 37�
�23, ±6, 42�
�23, ±22, 47�
�17, ±14, 62�
�11, ±4, 92�
�17, ±10, 61�
�21, ±18, 53�
�23, ±12, 47�
�23, ±8, 47�
�23, ±1, 47�
�24, ±12, 47�
|∆|
2088
2100
2112
2128
2163
2208
2244
2272
2280
2340
2368
2400
2436
2451
2464
2496
2520
2580
2632
2640
2688
2715
2760
2772
2788
2880
2907
3003
3003
3040
3040
3060
3108
3168
3172
3192
3220
3315
3315
3355
3360
3360
3360
3360
3432
3480
3520
3588
3627
3640
3648
3795
3808
3828
3840
3843
4020
4032
4123
4128
4180
4260
4368
4420
�A, B, C�
�9, ±6, 59�
�19, ±16, 31�
�19, ±4, 28�
�8, ±4, 67�
�11, ±9, 51�
�17, ±6, 33�
�5, ±4, 113�
�11, ±4, 52�
�7, ±4, 82�
�11, ±6, 54�
�19, ±8, 32�
�7, ±6, 87�
�5, ±2, 122�
�5, ±3, 123�
�19, ±14, 35�
�5, ±2, 125�
�9, ±6, 71�
�11, ±4, 59�
�19, ±16, 38�
�19, ±18, 39�
�13, ±4, 52�
�7, ±1, 97�
�11, ±10, 65�
�13, ±6, 54�
�19, ±10, 38�
�7, ±2, 103�
�27, ±21, 31�
�1, 1, 751�
�13, 13, 61�
�1, 0, 760�
�8, 8, 97�
�9, ±6, 86�
�11, ±4, 71�
�9, ±6, 89�
�19, ±18, 46�
�22, ±8, 37�
�22, ±6, 37�
�1, 1, 829�
�15, 15, 59�
�13, ±5, 65�
�4, 4, 211�
�8, 8, 107�
�21, 0, 40�
�29, 2, 29�
�23, ±8, 38�
�23, ±4, 38�
�7, ±6, 127�
�11, ±8, 83�
�9, ±3, 101�
�22, ±12, 43�
�29, ±8, 32�
�13, ±1, 73�
�11, ±8, 88�
�7, ±6, 138�
�16, ±8, 61�
�9, ±3, 107�
�26, ±6, 39�
�23, ±4, 44�
�17, ±5, 61�
�23, ±14, 47�
�29, ±24, 41�
�26, ±2, 41�
�8, ±4, 137�
�7, ±2, 158�
16
INTEGERS: 13 (2013)
Table 1: Representatives for 2779 SL2 (Z)-equivalence Classes of Regular Forms
|∆|
4420
4440
4452
4480
4488
4512
4515
4680
4740
4788
4960
4992
5083
5115
5152
5160
5187
5208
5280
5280
5280
5280
5355
5412
5440
5460
5460
5460
5460
5520
5712
5952
6160
6195
6240
6240
6307
6420
6435
6528
6580
6612
6688
6708
6720
6720
6820
6840
7008
7035
7072
7140
7140
7315
7392
7392
7392
7392
7395
7480
7540
7755
7968
7995
�A, B, C�
�14, ±2, 79�
�19, ±14, 61�
�17, ±6, 66�
�17, ±12, 68�
�26, ±20, 47�
�13, ±8, 88�
�19, ±11, 61�
�18, ±12, 67�
�22, ±10, 55�
�13, ±10, 94�
�17, ±2, 73�
�19, ±10, 67�
�31, ±1, 41�
�35, ±25, 41�
�31, ±26, 47�
�34, ±12, 39�
�33, ±15, 41�
�38, ±32, 41�
�5, 0, 264�
�12, 12, 113�
�24, 24, 61�
�41, 38, 41�
�31, ±15, 45�
�39, ±36, 43�
�32, ±8, 43�
�5, 0, 273�
�13, 0, 105�
�26, 26, 59�
�42, 42, 43�
�19, ±16, 76�
�19, ±8, 76�
�29, ±14, 53�
�23, ±2, 67�
�31, ±25, 55�
�17, ±4, 92�
�28, ±12, 57�
�23, ±15, 71�
�33, ±24, 53�
�37, ±15, 45�
�37, ±24, 48�
�34, ±30, 55�
�37, ±14, 46�
�37, ±34, 53�
�37, ±10, 46�
�31, ±10, 55�
�39, ±12, 44�
�38, ±18, 47�
�43, ±30, 45�
�43, ±42, 51�
�33, ±15, 55�
�41, ±12, 44�
�26, ±6, 69�
�39, ±6, 46�
�37, ±23, 53�
�7, 0, 264�
�12, 12, 157�
�28, 28, 73�
�47, 38, 47�
�35, ±5, 53�
�41, ±8, 46�
�41, ±2, 46�
�35, ±15, 57�
�39, ±12, 52�
�37, ±21, 57�
|∆|
4420
4440
4452
4480
4488
4512
4515
4680
4740
4788
4960
4992
5115
5152
5160
5187
5208
5280
5280
5280
5280
5355
5412
5440
5460
5460
5460
5460
5467
5520
5712
5952
6160
6195
6240
6240
6420
6435
6528
6580
6612
6688
6708
6720
6720
6820
6840
7008
7035
7072
7140
7140
7315
7392
7392
7392
7392
7395
7480
7540
7755
7968
7995
8008
�A, B, C�
�19, ±8, 59�
�22, ±20, 55�
�22, ±6, 51�
�19, ±2, 59�
�29, ±6, 39�
�31, ±18, 39�
�23, ±19, 53�
�23, ±14, 53�
�29, ±4, 41�
�18, ±6, 67�
�23, ±10, 55�
�29, ±24, 48�
�7, ±3, 183�
�13, ±10, 101�
�13, ±12, 102�
�11, ±7, 119�
�19, ±6, 69�
�1, 0, 1320�
�8, 0, 165�
�15, 0, 88�
�33, 0, 40�
�9, ±3, 149�
�13, ±10, 106�
�11, ±4, 124�
�1, 0, 1365�
�6, 6, 229�
�14, 14, 101�
�30, 30, 53�
�19, ±9, 73�
�24, ±12, 59�
�24, ±12, 61�
�32, ±24, 51�
�31, ±28, 56�
�33, ±3, 47�
�19, ±12, 84�
�29, ±16, 56�
�11, ±2, 146�
�9, ±3, 179�
�16, ±8, 103�
�11, ±8, 151�
�17, ±16, 101�
�7, ±2, 239�
�23, ±10, 74�
�11, ±10, 155�
�32, ±24, 57�
�19, ±18, 94�
�9, ±6, 191�
�13, ±8, 136�
�11, ±7, 161�
�11, ±10, 163�
�13, ±6, 138�
�29, ±20, 65�
�13, ±11, 143�
�1, 0, 1848�
�8, 0, 231�
�21, 0, 88�
�33, 0, 56�
�7, ±5, 265�
�19, ±14, 101�
�17, ±12, 113�
�7, ±1, 277�
�13, ±12, 156�
�19, ±17, 109�
�17, ±4, 118�
|∆|
4420
4440
4452
4480
4488
4512
4515
4680
4740
4788
4960
4992
5115
5152
5160
5187
5208
5280
5280
5280
5280
5355
5412
5440
5460
5460
5460
5460
5467
5520
5712
5952
6160
6195
6240
6240
6420
6435
6528
6580
6612
6688
6708
6720
6720
6820
6840
7008
7035
7072
7140
7140
7315
7392
7392
7392
7392
7395
7480
7540
7755
7968
7995
8008
�A, B, C�
�35, ±30, 38�
�33, ±24, 38�
�33, ±6, 34�
�32, ±16, 37�
�31, ±10, 37�
�33, ±30, 41�
�29, ±3, 39�
�31, ±30, 45�
�33, ±12, 37�
�26, ±10, 47�
�29, ±12, 44�
�32, ±16, 41�
�17, ±11, 77�
�17, ±4, 76�
�17, ±12, 78�
�17, ±7, 77�
�23, ±6, 57�
�3, 0, 440�
�8, 8, 167�
�20, 20, 71�
�37, 14, 37�
�13, ±1, 103�
�23, ±4, 59�
�31, ±4, 44�
�2, 2, 683�
�7, 0, 195�
�15, 0, 91�
�35, 0, 39�
�31, ±19, 47�
�37, ±20, 40�
�29, ±28, 56�
�32, ±8, 47�
�40, ±20, 41�
�37, ±13, 43�
�21, ±12, 76�
�35, ±30, 51�
�22, ±2, 73�
�17, ±5, 95�
�23, ±2, 71�
�17, ±4, 97�
�23, ±14, 74�
�28, ±12, 61�
�29, ±22, 62�
�13, ±12, 132�
�32, ±8, 53�
�29, ±16, 61�
�18, ±12, 97�
�17, ±8, 104�
�23, ±7, 77�
�23, ±14, 79�
�19, ±2, 94�
�37, ±36, 57�
�29, ±15, 65�
�3, 0, 616�
�8, 8, 233�
�24, 0, 77�
�43, 2, 43�
�21, ±9, 89�
�23, ±8, 82�
�23, ±2, 82�
�19, ±15, 105�
�23, ±6, 87�
�23, ±3, 87�
�29, ±24, 74�
|∆|
4440
4452
4480
4488
4512
4515
4680
4740
4788
4960
4992
5083
5115
5152
5160
5187
5208
5280
5280
5280
5280
5355
5412
5440
5460
5460
5460
5460
5520
5712
5952
6160
6195
6240
6240
6307
6420
6435
6528
6580
6612
6688
6708
6720
6720
6820
6840
7008
7035
7072
7140
7140
7315
7392
7392
7392
7392
7395
7480
7540
7755
7968
7995
8008
�A, B, C�
�11, ±2, 101�
�11, ±6, 102�
�16, ±8, 71�
�13, ±6, 87�
�11, ±8, 104�
�13, ±3, 87�
�9, ±6, 131�
�11, ±10, 110�
�9, ±6, 134�
�11, ±10, 115�
�16, ±8, 79�
�19, ±3, 67�
�21, ±3, 61�
�19, ±4, 68�
�26, ±12, 51�
�29, ±27, 51�
�37, ±34, 43�
�4, 4, 331�
�11, 0, 120�
�24, 0, 55�
�40, 40, 43�
�23, ±21, 63�
�26, ±10, 53�
�32, ±24, 47�
�3, 0, 455�
�10, 10, 139�
�21, 0, 65�
�37, 4, 37�
�8, ±4, 173�
�8, ±4, 179�
�17, ±10, 89�
�8, ±4, 193�
�11, ±3, 141�
�7, ±2, 223�
�23, ±4, 68�
�19, ±1, 83�
�31, ±20, 55�
�19, ±5, 85�
�32, ±16, 53�
�22, ±14, 77�
�34, ±18, 51�
�31, ±16, 56�
�31, ±22, 58�
�19, ±14, 91�
�33, ±12, 52�
�37, ±32, 53�
�29, ±2, 59�
�39, ±18, 47�
�31, ±23, 61�
�29, ±2, 61�
�23, ±6, 78�
�38, ±2, 47�
�31, ±1, 59�
�4, 4, 463�
�11, 0, 168�
�24, 24, 83�
�44, 44, 53�
�31, ±13, 61�
�38, ±24, 53�
�34, ±22, 59�
�21, ±15, 95�
�29, ±6, 69�
�29, ±3, 69�
�34, ±4, 59�
17
INTEGERS: 13 (2013)
Table 1: Representatives for 2779 SL2 (Z)-equivalence Classes of Regular Forms
|∆|
8008
8052
8160
8160
8320
8352
8512
8547
8580
8580
8680
8715
8835
8932
9108
9120
9120
9240
9240
9568
9867
10080
10080
10528
10560
10560
10920
10920
10948
11040
11040
11067
11328
11715
11872
12160
12180
12180
12768
12768
12915
13195
13440
13440
13728
13728
13860
13860
13920
13920
14280
14280
14560
14560
14763
14820
14820
16192
16555
17220
17220
17472
17472
17760
�A, B, C�
�37, ±24, 58�
�41, ±36, 57�
�28, ±4, 73�
�43, ±28, 52�
�32, ±16, 67�
�37, ±26, 61�
�41, ±4, 52�
�43, ±15, 51�
�29, ±2, 74�
�42, ±18, 53�
�43, ±36, 58�
�43, ±33, 57�
�43, ±25, 55�
�38, ±6, 59�
�34, ±2, 67�
�28, ±20, 85�
�51, ±48, 56�
�26, ±4, 89�
�46, ±12, 51�
�53, ±48, 56�
�47, ±35, 59�
�36, ±12, 71�
�47, ±42, 63�
�41, ±38, 73�
�32, ±24, 87�
�52, ±36, 57�
�29, ±10, 95�
�57, ±48, 58�
�47, ±12, 59�
�33, ±24, 88�
�52, ±20, 55�
�47, ±5, 59�
�43, ±14, 67�
�51, ±27, 61�
�52, ±20, 59�
�43, ±40, 80�
�34, ±14, 91�
�53, ±40, 65�
�33, ±6, 97�
�51, ±30, 67�
�53, ±21, 63�
�55, ±15, 61�
�32, ±16, 107�
�48, ±24, 73�
�31, ±6, 111�
�57, ±54, 73�
�31, ±20, 115�
�62, ±42, 63�
�39, ±30, 95�
�61, ±54, 69�
�33, ±30, 115�
�59, ±36, 66�
�41, ±6, 89�
�55, ±20, 68�
�59, ±39, 69�
�43, ±12, 87�
�59, ±44, 71�
�61, ±20, 68�
�47, ±41, 97�
�34, ±18, 129�
�62, ±58, 83�
�32, ±8, 137�
�68, ±36, 69�
�44, ±4, 101�
|∆|
8052
8160
8160
8320
8352
8512
8547
8580
8580
8680
8715
8835
8932
9108
9120
9120
9240
9240
9568
9867
10080
10080
10528
10560
10560
10920
10920
10948
11040
11040
11067
11328
11715
11872
12160
12180
12180
12768
12768
12915
13195
13440
13440
13728
13728
13860
13860
13920
13920
14280
14280
14560
14560
14763
14820
14820
16192
16555
17220
17220
17472
17472
17760
17760
�A, B, C�
�19, ±2, 106�
�7, ±4, 292�
�35, ±10, 59�
�16, ±8, 131�
�9, ±6, 233�
�13, ±4, 164�
�17, ±15, 129�
�7, ±4, 307�
�31, ±10, 70�
�13, ±2, 167�
�19, ±5, 115�
�11, ±3, 201�
�13, ±8, 173�
�9, ±6, 254�
�7, ±6, 327�
�31, ±26, 79�
�13, ±4, 178�
�34, ±12, 69�
�7, ±6, 343�
�29, ±15, 87�
�9, ±6, 281�
�37, ±24, 72�
�19, ±6, 139�
�13, ±10, 205�
�32, ±8, 83�
�11, ±6, 249�
�33, ±6, 83�
�37, ±2, 74�
�11, ±2, 251�
�39, ±6, 71�
�13, ±3, 213�
�31, ±24, 96�
�17, ±7, 173�
�13, ±6, 229�
�16, ±8, 191�
�13, ±12, 237�
�37, ±20, 85�
�11, ±6, 291�
�37, ±16, 88�
�9, ±3, 359�
�11, ±7, 301�
�16, ±8, 211�
�37, ±18, 93�
�17, ±12, 204�
�37, ±6, 93�
�9, ±6, 386�
�41, ±30, 90�
�13, ±4, 268�
�41, ±26, 89�
�11, ±8, 326�
�46, ±16, 79�
�11, ±2, 331�
�43, ±24, 88�
�23, ±7, 161�
�17, ±2, 218�
�47, ±28, 83�
�17, ±14, 241�
�29, ±27, 149�
�17, ±16, 257�
�43, ±18, 102�
�17, ±2, 257�
�47, ±24, 96�
�11, ±4, 404�
�47, ±10, 95�
|∆|
8052
8160
8160
8320
8352
8512
8547
8580
8580
8680
8715
8835
8932
9108
9120
9120
9240
9240
9568
9867
10080
10080
10528
10560
10560
10920
10920
10948
11040
11040
11067
11328
11715
11872
12160
12180
12180
12768
12768
12915
13195
13440
13440
13728
13728
13860
13860
13920
13920
14280
14280
14560
14560
14763
14820
14820
16192
16555
17220
17220
17472
17472
17760
17760
�A, B, C�
�31, ±16, 67�
�13, ±2, 157�
�39, ±24, 56�
�23, ±12, 92�
�31, ±24, 72�
�32, ±24, 71�
�23, ±3, 93�
�14, ±10, 155�
�35, ±10, 62�
�26, ±24, 89�
�23, ±5, 95�
�33, ±3, 67�
�19, ±6, 118�
�17, ±2, 134�
�17, ±14, 137�
�35, ±20, 68�
�17, ±12, 138�
�37, ±26, 67�
�28, ±20, 89�
�37, ±7, 67�
�17, ±16, 152�
�43, ±38, 67�
�23, ±12, 116�
�19, ±2, 139�
�39, ±36, 76�
�19, ±10, 145�
�38, ±28, 77�
�41, ±32, 73�
�13, ±6, 213�
�43, ±22, 67�
�37, ±25, 79�
�32, ±24, 93�
�29, ±1, 101�
�31, ±30, 103�
�29, ±22, 109�
�17, ±14, 182�
�39, ±12, 79�
�17, ±4, 188�
�44, ±28, 77�
�19, ±9, 171�
�43, ±7, 77�
�29, ±4, 116�
�41, ±34, 89�
�19, ±16, 184�
�51, ±12, 68�
�18, ±6, 193�
�45, ±30, 82�
�19, ±8, 184�
�52, ±4, 67�
�22, ±8, 163�
�47, ±14, 77�
�17, ±14, 217�
�44, ±20, 85�
�47, ±29, 83�
�29, ±12, 129�
�51, ±36, 79�
�32, ±24, 131�
�37, ±13, 113�
�29, ±8, 149�
�51, ±18, 86�
�23, ±10, 191�
�51, ±36, 92�
�19, ±10, 235�
�55, ±40, 88�
|∆|
8052
8160
8160
8320
8352
8512
8547
8580
8580
8680
8715
8835
8932
9108
9120
9120
9240
9240
9568
9867
10080
10080
10528
10560
10560
10920
10920
10948
11040
11040
11067
11328
11715
11872
12160
12180
12180
12768
12768
12915
13195
13440
13440
13728
13728
13860
13860
13920
13920
14280
14280
14560
14560
14763
14820
14820
16192
16555
17220
17220
17472
17472
17760
17760
�A, B, C�
�38, ±2, 53�
�21, ±18, 101�
�41, ±32, 56�
�31, ±22, 71�
�36, ±12, 59�
�32, ±8, 67�
�31, ±3, 69�
�21, ±18, 106�
�37, ±2, 58�
�29, ±22, 79�
�41, ±31, 59�
�41, ±29, 59�
�26, ±18, 89�
�18, ±6, 127�
�21, ±6, 109�
�41, ±8, 56�
�23, ±12, 102�
�39, ±30, 65�
�43, ±8, 56�
�43, ±25, 61�
�19, ±16, 136�
�45, ±30, 61�
�29, ±12, 92�
�29, ±24, 96�
�41, ±10, 65�
�22, ±16, 127�
�55, ±50, 61�
�43, ±24, 67�
�29, ±26, 101�
�44, ±20, 65�
�39, ±3, 71�
�32, ±8, 89�
�43, ±29, 73�
�41, ±10, 73�
�32, ±16, 97�
�26, ±14, 119�
�51, ±48, 71�
�31, ±2, 103�
�47, ±4, 68�
�45, ±15, 73�
�47, ±23, 73�
�31, ±18, 111�
�47, ±40, 80�
�23, ±16, 152�
�53, ±30, 69�
�23, ±20, 155�
�46, ±26, 79�
�23, ±8, 152�
�57, ±30, 65�
�23, ±16, 158�
�55, ±30, 69�
�31, ±14, 119�
�53, ±42, 77�
�53, ±17, 71�
�34, ±2, 109�
�58, ±46, 73�
�32, ±8, 127�
�41, ±3, 101�
�31, ±4, 139�
�58, ±50, 85�
�32, ±24, 141�
�59, ±46, 83�
�33, ±18, 137�
�57, ±48, 88�
18
INTEGERS: 13 (2013)
Table 1: Representatives for 2779 SL2 (Z)-equivalence Classes of Regular Forms
|∆|
17760
17952
17952
18720
18720
19320
19320
19380
19380
19635
19635
20020
20020
20640
20640
20832
20832
21120
21120
21840
21840
22080
22080
22848
22848
24640
24640
27360
27360
29568
29568
29920
29920
31395
31395
32032
32032
33915
33915
34720
34720
36960
36960
36960
36960
40755
40755
43680
43680
43680
43680
57120
57120
57120
57120
77280
77280
77280
77280
87360
87360
87360
87360
�A, B, C�
�61, ±28, 76�
�37, ±20, 124�
�53, ±42, 93�
�36, ±12, 131�
�72, ±48, 73�
�41, ±14, 119�
�73, ±68, 82�
�39, ±30, 130�
�69, ±66, 86�
�41, ±39, 129�
�59, ±37, 89�
�38, ±14, 133�
�74, ±58, 79�
�51, ±24, 104�
�68, ±44, 83�
�41, ±18, 129�
�69, ±12, 76�
�41, ±6, 129�
�71, ±40, 80�
�40, ±20, 139�
�59, ±52, 104�
�37, ±34, 157�
�76, ±44, 79�
�32, ±8, 179�
�76, ±60, 87�
�32, ±8, 193�
�67, ±4, 92�
�43, ±26, 163�
�72, ±24, 97�
�47, ±18, 159�
�83, ±48, 96�
�53, ±48, 152�
�92, ±76, 97�
�47, ±1, 167�
�85, ±15, 93�
�59, ±8, 136�
�89, ±50, 97�
�55, ±25, 157�
�79, ±23, 109�
�52, ±4, 167�
�91, ±56, 104�
�37, ±22, 253�
�65, ±60, 156�
�85, ±10, 109�
�104, ±96, 111�
�43, ±3, 237�
�93, ±45, 115�
�33, ±12, 332�
�61, ±22, 181�
�83, ±12, 132�
�95, ±20, 116�
�44, ±28, 329�
�69, ±60, 220�
�88, ±16, 163�
�115, ±60, 132�
�51, ±6, 379�
�85, ±40, 232�
�116, ±76, 179�
�136, ±40, 145�
�43, ±4, 508�
�96, ±24, 229�
�127, ±4, 172�
�159, ±120, 160�
|∆|
17952
17952
18720
18720
19320
19320
19380
19380
19635
19635
20020
20020
20640
20640
20832
20832
21120
21120
21840
21840
22080
22080
22848
22848
24640
24640
27360
27360
29568
29568
29920
29920
31395
31395
32032
32032
33915
33915
34720
34720
36960
36960
36960
36960
40755
40755
43680
43680
43680
43680
57120
57120
57120
57120
77280
77280
77280
77280
87360
87360
87360
87360
�A, B, C�
�13, ±12, 348�
�39, ±12, 116�
�9, ±6, 521�
�45, ±30, 109�
�17, ±14, 287�
�51, ±48, 106�
�13, ±4, 373�
�43, ±20, 115�
�19, ±7, 259�
�43, ±39, 123�
�19, ±14, 266�
�46, ±6, 109�
�13, ±2, 397�
�52, ±28, 103�
�19, ±12, 276�
�43, ±18, 123�
�16, ±8, 331�
�43, ±6, 123�
�8, ±4, 683�
�43, ±2, 127�
�19, ±6, 291�
�57, ±6, 97�
�19, ±16, 304�
�57, ±54, 113�
�23, ±4, 268�
�41, ±40, 160�
�9, ±6, 761�
�45, ±30, 157�
�16, ±8, 463�
�48, ±24, 157�
�19, ±10, 395�
�67, ±30, 115�
�17, ±15, 465�
�51, ±15, 155�
�17, ±8, 472�
�68, ±60, 131�
�11, ±3, 771�
�61, ±1, 139�
�13, ±4, 668�
�65, ±30, 137�
�13, ±8, 712�
�39, ±18, 239�
�67, ±52, 148�
�89, ±8, 104�
�23, ±1, 443�
�69, ±45, 155�
�11, ±10, 995�
�44, ±12, 249�
�67, ±2, 163�
�87, ±78, 143�
�11, ±6, 1299�
�47, ±28, 308�
�77, ±28, 188�
�89, ±14, 161�
�17, ±6, 1137�
�53, ±10, 365�
�87, ±18, 223�
�119, ±28, 164�
�32, ±24, 687�
�53, ±14, 413�
�101, ±56, 224�
�129, ±90, 185�
|∆|
17952
17952
18720
18720
19320
19320
19380
19380
19635
19635
20020
20020
20640
20640
20832
20832
21120
21120
21840
21840
22080
22080
22848
22848
24640
24640
27360
27360
29568
29568
29920
29920
31395
31395
32032
32032
33915
33915
34720
34720
36960
36960
36960
36960
40755
40755
43680
43680
43680
43680
57120
57120
57120
57120
77280
77280
77280
77280
87360
87360
87360
87360
�A, B, C�
�29, ±12, 156�
�47, ±40, 104�
�23, ±18, 207�
�53, ±28, 92�
�29, ±20, 170�
�53, ±48, 102�
�23, ±20, 215�
�46, ±26, 109�
�31, ±9, 159�
�53, ±9, 93�
�23, ±6, 218�
�47, ±40, 115�
�17, ±10, 305�
�61, ±10, 85�
�23, ±12, 228�
�47, ±6, 111�
�32, ±16, 167�
�48, ±24, 113�
�24, ±12, 229�
�53, ±46, 113�
�32, ±24, 177�
�59, ±24, 96�
�29, ±2, 197�
�61, ±24, 96�
�31, ±6, 199�
�59, ±50, 115�
�29, ±4, 236�
�59, ±4, 116�
�32, ±16, 233�
�53, ±18, 141�
�23, ±16, 328�
�76, ±28, 101�
�31, ±15, 255�
�61, ±9, 129�
�29, ±10, 277�
�71, ±42, 119�
�33, ±3, 257�
�67, ±11, 127�
�29, ±14, 301�
�79, ±44, 116�
�17, ±10, 545�
�51, ±24, 184�
�68, ±44, 143�
�91, ±70, 115�
�31, ±17, 331�
�79, ±3, 129�
�19, ±18, 579�
�55, ±10, 199�
�76, ±20, 145�
�88, ±56, 133�
�23, ±14, 623�
�55, ±50, 271�
�79, ±32, 184�
�92, ±60, 165�
�29, ±18, 669�
�68, ±28, 287�
�107, ±98, 203�
�123, ±54, 163�
�32, ±8, 683�
�59, ±14, 371�
�111, ±90, 215�
�139, ±40, 160�
|∆|
17952
17952
18720
18720
19320
19320
19380
19380
19635
19635
20020
20020
20640
20640
20832
20832
21120
21120
21840
21840
22080
22080
22848
22848
24640
24640
27360
27360
29568
29568
29920
29920
31395
31395
32032
32032
33915
33915
34720
34720
36960
36960
36960
36960
40755
40755
43680
43680
43680
43680
57120
57120
57120
57120
77280
77280
77280
77280
87360
87360
87360
87360
�A, B, C�
�31, ±20, 148�
�52, ±12, 87�
�31, ±2, 151�
�67, ±24, 72�
�34, ±20, 145�
�58, ±20, 85�
�26, ±22, 191�
�65, ±30, 78�
�37, ±7, 133�
�57, ±45, 95�
�37, ±16, 137�
�61, ±54, 94�
�39, ±24, 136�
�65, ±50, 89�
�37, ±6, 141�
�57, ±12, 92�
�37, ±28, 148�
�61, ±48, 96�
�37, ±8, 148�
�56, ±28, 101�
�32, ±8, 173�
�71, ±70, 95�
�32, ±24, 183�
�73, ±72, 96�
�32, ±24, 197�
�61, ±2, 101�
�36, ±12, 191�
�72, ±48, 103�
�43, ±4, 172�
�73, ±56, 112�
�41, ±16, 184�
�79, ±10, 95�
�43, ±9, 183�
�71, ±49, 119�
�37, ±26, 221�
�79, ±68, 116�
�41, ±19, 209�
�77, ±63, 123�
�43, ±14, 203�
�89, ±48, 104�
�23, ±22, 407�
�52, ±44, 187�
�69, ±24, 136�
�92, ±68, 113�
�41, ±9, 249�
�83, ±9, 123�
�29, ±20, 380�
�57, ±18, 193�
�77, ±56, 152�
�88, ±32, 127�
�33, ±6, 433�
�59, ±46, 251�
�88, ±72, 177�
�109, ±66, 141�
�41, ±28, 476�
�73, ±10, 265�
�109, ±108, 204�
�136, ±96, 159�
�37, ±16, 592�
�96, ±72, 241�
�113, ±92, 212�
�148, ±132, 177�