The equation $x^3+y^3 +z^3+c\, xyz= 0$ defines a non-singular elliptic curve $X$ in $\mathbb{C}P^2$ projective space.
In fact, how do we prove it has genus 1 both as an algebraic curve and as a Riemann surface?
Since $H^1(X,\mathbb{C})\simeq \mathbb{C}^2$ what are the explicit representatives of the de Rham cohomology? There should be two of them.