 Genus one tangent incidence curve of two conics

If two smooth plane conics over $\mathbb{C}$ meet at four distinct points, their tangent incidence curve is a smooth projective genus one curve. Projection to $C_2$ is a double cover branched at those four intersections. The tangent equation becomes a quadratic whose discriminant cuts out $C_1\cap C_2$; its four simple zeros make the cover smooth and connected. The Riemann-Hurwitz formula then gives genus one.