2004/04/19 by William Stein, Mark Watkins
Mathematics · #math.NT
published as IMRN 2004, no. 27, 1395-1405
arxiv created 2004/04/19 · arxiv updated 2009/12/01
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/2\Z. There is a surjective map X0(p)\xrightarrowπ E0 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 \con 3\pmod8. We also prove the prime-conductor case of a conjecture of Glenn Stevens, namely that 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.