  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 $E_0$
and $E_1$ of prime conductor $p$, and both have Mordell--Weil group $\Z/2\Z$.
There is a surjective map $X_0(p)\xrightarrow{\pi} E_0$ that does not factor
through any other elliptic curve (i.e., $\pi$ is optimal), where $X_0(p)$ is
the modular curve of level $p$. Our main result is that the degree of $\pi$ is
odd if and only if $u \con 3\pmod{8}$. 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 $X_1(p)$ 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.
