Modular Parametrizations of Neumann-Setzer Elliptic Curves

Abstract

Suppose p is a prime of the form u2 + 64 for some integer u, which we take to be 3 mod 4. Then there are two Neumann-Setzer elliptic curves E0 and E1 of prime conductor p, and both have Mordell-Weil group ℤ/2ℤ. There is a surjective map X0(p)→πE0 that does not factor through any other elliptic curve (i.e., π is optimal), where X0(p) is the modular curve of level p. Our main result is that the degree of π is odd if and only if u≡3(mod8)⁠. We also prove the prime-conductor case of a conjecture of Glenn Stevens, namely that if E is an elliptic curve of prime conductor p, then the optimal quotient of X1(p) in the isogeny class of E is the curve with minimal Faltings height. Finally we discuss some conjectures and data about modular degrees and orders of Shafarevich-Tate groups of Neumann-Setzer curves.