TITLE: Mazur's Question on Mod $N$ Galois representations
QUESTION [8 upvotes]: In Rational Isogenies of Prime Degree, Mazur poses:
"the problem of determining all elliptic curves $E'/\mathbb{Q}$ with symplectic $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ isomorphisms $E'[N]\cong V$. 
He then claims:
"This can be reduced to (or rephrased as) the problem of determining all $\mathbb{Q}$-rational points of a certain twisted form $X(V)$ of the modular curve $X(N)$."
My question is, how does one define $X(V)$ and how can one see that it is a twist of $X(N)$. Over which field is this twist defined?

REPLY [3 votes]: In addition to the previous comments, let me also say that this question has been considered by Kraus in joint works with Freitas, Halberstadt, and Oesterlé.
The properties of the modular curve associated to the problem you mention are given in Proposition 1 of the paper by Kraus and Oesterlé, Sur une question de B. Mazur, Math. Ann. 293 (1992). The proof of this proposition is given in the reference [6] therein. Email me if you want an electronic copy of it.