Modular Degrees of Neumann-Setzer Curves

William Stein and Mark Watkins



Suppose p is a prime of the form u^2+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 pi: X0(p) --> E0 that does not factor through any other elliptic curve (i.e., pi is optimal), where X0(p) is the modular curve of level p. Our main result is that the degree of pi is odd if and only if u=3(mod 8). 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.

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This is a relevant excerpt from a paper by Mestre-Oesterle.