It's known that for a elliptic curve like $E:y^2=x(x-1)(x-t)$ we have a basis $\frac{dx}{y}, \frac{xdx}{y}$ for $H_{dR}^1(E)$. But find such a basis is not an easy thing. I wonder for a general abelian variety or a jacobian variety over a smooth curve is there a way to write out an basis for $H_{dR}^1$ (algebraic de Rham) explicitly?
In my settings, suppose $K$ is an finite unramified extension of $\mathbb Q_p$ and $C$ is a curve defined over $K$ (smooth projective). Let $J$ be its jacobian. I need to calculate the matrix of Frobenius on $H_{dR}^1(J)$ by the comparison theorem (or using rigid cohomology).
For rigid cohomology I need to find a basis for $H_{dR}^1(J)$ and lift Frobenius to the weak completion of $J$. But then I'm stuck. Firstly I don't know how to find a canonical basis and I don't know how to find the weak completion since I know nothing about the equation of $J$.
Anything about calculating the frobenius would be of help. Thanks.
