Explicit basis for homology of elliptic curve

Question

Let $X=\{y^2=x^3+ax+b\}$ be an elliptic curve in $\mathbb{C}^2$ (so no point at infinity). Is it possible to write down explicitly a basis for the homology of $X$. By explicit I mean giving explicit maps $\sigma_i:[0,1]\to X$ such that the $[\sigma_i]$ in $H_1(X,\mathbb{Z})$ are a basis.