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?