Main goal: Visualize the "real slice" of elliptic curve from the complex torus perspective
Basic setup: for $\tau \in \mathbb H$ (the upper half plane $\{z\in \mathbb C: \Im(z)>0\}$), one can cook up a lattice $\Lambda_\tau := \mathbb Z + \tau \mathbb Z$, from which one can cook up the (Riemann surface) complex torus $\mathbb T_\tau := \mathbb C/\Lambda_\tau$, on which one can define the Weierstrass function and its derivative $$\wp_\tau(z) := \frac 1{z^2} + \sum_{\lambda \in \Lambda_\tau \setminus \{0\}} \left(\frac{1}{(z-\lambda)^2} - \frac 1{\lambda^2}\right), \quad \wp_\tau'(z) = 4 \wp_\tau(z)^3 - g_2(\tau) \wp_\tau(z) -g_3(\tau),$$
using which we can define the biholomorphic group isomorphism $\big[z \mapsto [1:\wp(z):\frac 12\wp'(z)] \big]$ from $\mathbb T_\tau$ to its image in $\mathbb C \mathbb P^2$ (the complex projective plane).
One can pick $\tau$ so that $g_2(\tau),g_3(\tau)$ land in $\mathbb R$ or $\mathbb Q$ or whatever (see https://mathoverflow.net/questions/338171/rationality-of-eisenstein-series-g2-and-g3-for-elliptic-curves-defined-over-numb).
We can then visualize the image of $\mathbb T_\tau$ intersected with the real (non-infinite) points of $\mathbb C \mathbb P^2$ by the graph of the equation $y^2=x^3+ax+b$ in $\mathbb R\times \mathbb R$.
I'm interested in what the preimage of this in $\mathbb T_\tau$ (under the isomorphism mentioned above) would look like. I imagine since the functions $\wp_\tau,\wp_\tau'$ are nice, it should look like some curve in $\mathbb T_\tau$ (not anything pathological). What would the rational points of the elliptic curve look like in $\mathbb T_\tau$? The torsion points are easy to see from the complex torus perspective --- how do they interact with the preimage curve? An ideal answer would include lots of pictures!