cv.complex variables - conformally embedding complex tori into R^3

Let $L$ be a lattice in $mathbb{C}$ with two fundamental periods, so that $mathbb{C}/L$ is topologically a torus. Let $p:mathbb{C}/L mapsto mathbb{R}^3$ be an embedding ($C^1$, say). Call $p$ conformal if pulling back the standard metric on $mathbb{R}^3$ along $p$ yields a metric in the equivalence class of metrics on $mathbb{C}/L$ (i.e. a multiple of the identity matrix).

Is there an explicit formula for such a p in the case of L an oblique lattice?

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Backgorund

The existence of such $C^1$ embeddings is implied by the nash embedding theorem (fix a metric on $mathbb{C}/L$, pick any short embedding, apply nash iteration to make it isometric and hence conformal).

For orthogonal lattices, the solution is simple: Parametrise the standard torus of radii $r_1$, $r_2$ in the usual way. Make the ansatz $pi(theta, phi) = (f(theta), h(phi))$, pull back the standard metric on $mathbb{C}/L$ and solve the resulting system of ODEs. This relates $r_1/r_2$ to the ratio of the magnitudes of the periods. This shows that no standard torus can be the image of $p$ in the original question (oblique lattice), although that is geometrically clear anyway.