Let $X$ be a smooth projective algebraic curve defined over $\mathbb{Q}$ with genus $g$, we have an isomorphism $H^1_{dR}(X/\mathbb{Q})\otimes_\mathbb{Q}\mathbb{C}\cong H^1_{sing}(X,\mathbb{Z})\otimes_\mathbb{Z} \mathbb{C}$. Now, choose a $\mathbb{Q}$-basis $\{u_i\}$ for $H^1_{dR}(X/\mathbb{Q})$, a $\mathbb{Z}$-basis $\{v_j\}$ for $H^1_{sing}(X,\mathbb{Z})$, and considering the transfer matrix $A\in GL(2g,\mathbb{C})$ from $\{v_j\}$ to $\{u_i\}$, now different choice of basis make a differ by a right action of $GL(2g,\mathbb{Q})$.
Question:
(1) What is the geometric meaning of $A$, or at least $det(A)$?
(2) If (1) is too hard, what happens when X is an elliptic curve?
