Let $E$ be an elliptic curve over some subfield $k$ of $\mathbb{C}$, say given by an equation $$ y^2=4x^3+ax+b. $$ Then:

$E(\mathbb{C})$ is a complex torus, so $H_1(E(\mathbb{C}), \mathbb{Q})$ is spanned by two cycles $\gamma_1$ and $\gamma_2$. Assume the basis $\{\gamma_1, \gamma_2\}$ is oriented.

the algebraic de Rham cohomology $H^1_{dR}(E/k)$ is spanned by the differential forms $\frac{dx}{y}$ and $x \frac{dx}{y}$.

I've read several times that the following result is true

**Legendre relation**: The determinant of the period matrix with respect to these two basis is $2\pi i$.

Can someone indicate me how to prove this?

Is it really a result by Legendre? If so, where can I find it (in legendre's works)?