Consider the plane algebraic curve
$$f(x, y) = y^4 - (2x - 1)y^2 - (4x - 1) y + x^2 + x + 1 = 0.\tag{1}$$
Its compactification results in a Riemann surface $C_1$ of genus $1$.
Hence, it can be transformed into an ellptic curve
$$
w^2 + z^3 - \frac{640}{3}z + \frac{5888}{27} = 0\tag{2}
$$
through the birational transformation
$$x = \frac{1}{256}z^2 - \frac{5}{48}z - \frac{29}{36} - \frac{1}{32}w, \ \ y = -\frac{1}{16}z - \frac{1}{6}.\label{3}\tag{3}$$

I want to perform integration on $C_1$, especially the integration of Abelian differentials along the canonical cycles, that is, periods of Abelian differentials.
However, I don't know how to proceed.
Is it possible to calculate elliptic integrals and then obtain the integral of the corresponding differential on $C_1$ through transformation \eqref{3}?