Internat. J. Math. & Math. Sci.
Vol. 9 No. 4 (1986) 797-800
797
ON THE COMPUTATION OF THE CLASS
NUMBERS OF SOME CUBIC FIELDS
MANNY SCAROWSKY and ABRAHAM BOYARSKY
Department of Mathematics
Loyola Campus
Concordia University
Montreal, Canada H4B IR6
(Received December I0, 1985)
ABSTRACT.
x3+12Ax-12
Class numbers are calculated for cubic fields of the form
A > 0, for
!
and for some other values of
! 17
a
A.
0,
These fields have a known
unit, which under certain conditions is the fundamental unit, and are important in
studying the Diophantine Equation
+ y3 +
x
Class numbers, cubic fields, Diophantine equation.
KEY WORDS AND PHRASES.
12A04, 12A50.
1980 AMS SUBJECT CLASSIFICATION CODES.
I.
3.
z
INTRODUCTION AND SOME THEOREMS.
We consider the cubic fields defined by an equation of the form
(I.I)
0,
12
to
related
is
it
because
is
important
equation
The field defined by this
f(x)
A > 0.
where
the Diophantine equation
+ y3 +
x
when
3
z
9a 2 [i].
A
Equation (I.I) is
f(x) is increasing, it defines a real cubic field
(with two complex conjugates) with exactly one fundamental unit. Let @ be the real
and as
clearly irreducible,
K
+ 12Ax
x
root of (1.1).
D
33
-2
have
son
q3.
(3)
p3,
(2)
+ 9).
(16A
Also as
and as
< i, we have
0 <
As
K.
fines a unit of
6
6
Thus the descriminant, D, of K
THEOREM I.
In
K
square, prime to
then
D
- 02
we see that
divides
the discriminant
3, dividing
D
D
-22-33
3.
The primes
Pi
02
(16A
O, we see that for the same
OK,
+ 9).
A
D
I.
tea-
K.
the ring of integers of
We now state:
where
is never a cube,
The unit
dividing
18
-22.33(16A +9)
q2
is the fundamental unit except when
divisible by
A@ de-
B
The discriminant of f(x) is
0 <
x 3- 36Ax
satisfies
6A +
Also
is an Eisenstein polynomial with respect to 3, we
f(x)
As
0 < @ < I.
Note that
Q(@).
K
We write
q2
is the largest
and if
The class-number
(except for 2 and 3)
q=l or
h, of
ramify as
K
q=5
is
M. SCAROWSKY AND A. BOYARSKY
798
2
(pi)
(B
A basis for
piqi
i
O
is given by
K
0o
02
0
(3,A)).
02
16A2+30+2A02
i
3 q
As the proof is similar to the proof of the corresponding theorem in [i], we omit
it, as well as the proof of the following two theorems, also in [i].
THEOREM 2.
If the 3-component of the class-group of
K
is a direct product of cyclic
groups of order 3, then
+ 12Ax
x
4z
12
(1.2)
has no solutions.
Corollary:
If
3
THEOREM 3.
If
(h,2)
to solving
-AG 4 -2
A
e.g.
2.
then (1.2) has no solutions.
h
i, and
G3H +
i, then solving
q
3 H4= -I
+12Ax
x
12
y2
is equivalent
(This has no solutions (mod p) for small primes
p,
14, p=5).
NUMERICAL COMPUTATIONS.
k (s)
wher4
4 Io$ e.h
2/22.33 (16A3+9)/q 2
lim
s+l+ (s)
We note that
is the fundamental unit of
e
=
(2.1)
As in [2], the left-hand side of (2.1) can
K.
P
be expressed as
f
lim f
p+ P
f(p)
lim
po P 5
- P- p-I
p2
p2 +p+
where
if
p
ramifies
if
p
remains inert
if
(p)
if
p splits completely
((pi)
pqi
f(p)
p2
p2
(p
p
i)2
pq
Hence (2.1) implies that approximately,
h
for
P
/27 (16A + 9)
"q
?
lo
fe
(2.2)
sufficiently large.
For Table i, the product in (2.2) was calculated for
where
P(i)
indicates the
i th
A
prime, and
P
P(2027), (at intervals of 50),
36:
TABLE 1
,DI2 2- 3 3
__h
_.A
-D/2 2- 3 3
1
52
6
2
137
3
19
20
7-15679
7-18287
21
22
23
24
32- 5- 37- 89
_.A
3
32. 72
6
4
5
1033
6
72.41
32. 5- 7-11
3
6"
21
347-/91
194681
7- 3511
32-
h
39
72
54
36
72
54
COMPUTATION OF THE CLASS NUMBERS OF SOME CUBIC FIELDS
A
-D/22.33
h
A
-D/22.33
h
7
23-239
9
29"37"233
72
8
59"139
18
25
26
9
32"1297
12
27
52-7"1607
32-7"4999
54
10
7-2287
27
28
11-37"863
78
11
5-4261
24
29
359.1087
48
12
32. 7-439
24
30
32-23-2087
72
7- 5023
43913
48
31
5- 7" 13619
162
21
32
17- 3084
78
13
14
15
15
32.17. 353
36
33
7- 82143
114
16
5-13109
36
34
17
18
7-11-1021
48
35
87
75
32.10369
36
36
7 89839
686009
5- 53" 313
In all the cases above except when
K.
unit of
A
or
A
When
-2.
1,5,
32.
A
K
A
or
799
156
5,
is
the fundamental
is a pure cubic field if and only if
A
3.
Also because of the equivalence of (1.2) with the Diophantine equation
+ y3 +
x
z
3
when
9a 2, the class-numbers of
A
! a ! 17 (Actually Cassels has shown that for
for
this case, one must have
31a
[3]).
K
were calculated using
(2.2)
solutions of (1.2) to exist in
While most of the values obtained in this way
were approximate, perhaps congruence conditions may be used to find them exactly, or
perhaps they may be of use in regards to Brauer-Siegel Theorem, so we list them in
3 log A
Table 2. (The Brauer-Siegel Theorem applied here states log h
log q)
i
TABLE 2
a
-D/22"35
1
1297
2
5-53-313
3
5-188957
4
5380417
5
6
3557-5693
I (P(I) is the I
th
prlme)
h
7
37-241-6781
5" 30494621
8303
14903
10803
4303
2201
3003
1002
8
53.17.29- 37-149
1002
9
17. 40514561
1002
3549
10
181-1361- 5261
1002
6999
12
156
216
420
789 (*)
1410
3285
873
11
89- 25797113
212
6753
12
5- 23761- 32573
212
13
5- 8821-141833
212
15999
21864
M. SCAROWSKY AND A. BOYARSKY
800
5
_a
-D/2 2 -3
14
37- 263737261
212
10062
15
1193- 2381. 5197
212
22653
16
35801.607337
212
16764
17
52.1251291577
212
4644
I (P(I) is the I th prime)
The second column gives the factorization of
(*)
For
a
5
the values of
bly accurate within
h
h
-D/22-35.
should be considered as estimates, but are proba-
%.
All computations were done at Concordia University, Loyola Campus, on the Cyber
170, Model 835.
ACKNOWLEDGEMNT.
#A9072
The research of Abraham Boyarsky was supported by an NSERC Grant
and an FCAC grant from the Quebec Department of Education.
REFERENCES
I.
SCAROWSKY, M.,
x
+ y3 + z
On units of Certain Cubic Fields and the Diophantine Equation
3, Proc. Amer. Soc., 91(3), 1984, 351-356.
2.
SHANKS, D., The Simplest Cubic Fields, Math. of Comp., 28(1974), 1137-1152.
3.
CASSELS, J.W.S.,
Personal communication.