Real geometric transcendence for Weierstrass elliptic functions

Given a lattice $\mathrm{\Lambda }\subset \mathbb{C}\simeq {\mathbb{R}}^2$ with associated Weierstrass function ${\mathrm{\wp }}_{\mathrm{\Lambda }}$, we determine the algebraic curves in ${\mathbb{R}}^2$ whose image via ${\mathrm{\wp }}_{\mathrm{\Lambda }}$ is contained in an algebraic curve.