Modular parametrizations of Neumann-Setzer elliptic curves

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 $X_0{(p)}\xrightarrow\pi E_0$ 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 $u\equiv3{(\operatorname{mod}8)}$ . 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.