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# Suggested Problems

1. Consider the rational function field Q(d) in one variable d. a. Create in Sage the elliptic curve with a-invariants (a1, a2, a3, a4, a6) = (1+d-d^2, d^2-d^3, d^2-d^3, 0, 0) that appears on page 2 of Elkies' slides. b. Put it in short Weierstrass form y^2 = x^3 + A*x + B. ----------------- 2. a. Find a quadratic imaginary number field with class number 5. b. Find a cubic field with class number 3. ----------------- 3. For a given integer a, let E = EllipticCurve([0,(a-1),1,-a,0]) For r = 0, 1, 2, 3, 4, 5, find the smallest positive integer a such that E has rank r.