Math 346/646 - Homework 5
Assigned: 9/24.
Due: 10/1 at the start of class.
Notation: Exercise a.b stands for exercise b in Chapter a of Silverman and Tate. Note that the
exercises are all together at the end of the chapter. A problem marked with a ∗ is a challenge
problem.
A problem is prefaced (Sage) if the use of Sage on that problem is permissible.
Problems:
1. Exercise 1.19 from the textbook.
2. (Sage) Find four rational solutions to the system of equations
y 2 − 3x2 = w2
2z 2 − y 2 = w2 .
Do this by mapping this curve to an elliptic curve in Weierstrass form, finding rational points
there, and mapping back to the original curve.
(For such a system, if we let n = (y 2 − w2 )/w2 , then n is three times a square, n + 1 is a square,
and n + 2 is twice a square.)
3. (Sage) ∗ Construct an elliptic curve with as high rank as possible. Full credit will be earned
for an elliptic curve of rank 4. If you find a curve of rank r, then you will earn r − 4 extra
credit points (or extra extra-credit points for students in 346). There will be a special bonus
for students who find the highest rank curve/curves.
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