Let $\Lambda = \mathbb{Z} + \mathbb{Z}\tau \subset \mathbb{C}$ be a lattice generated by $1$ and $\tau$. Weierstrass $\wp$-function $\wp_\Lambda: \mathbb{C} \to \mathbb{CP}^1$ gives an isomorphism $$ \mathbb{C} / \Lambda \to E = E_\Lambda: \{y^2z = 4x^3 - g_2 xz^2 - g_3 z^3\} \cup \{\infty\}, \quad z\mapsto [\wp_\Lambda(z): \wp'_\Lambda(z): 1] $$ where $$ g_2 = 60 \sum_{0 \neq \lambda \in \Lambda} \frac{1}{\lambda^4}, \quad g_3 = 140 \sum_{0 \neq \lambda \in \Lambda} \frac{1}{\lambda^6}. $$ Now I want to consider the cases when $E_\Lambda$ is defined over $\mathbb{R}$, i.e. when $g_2, g_3 \in \mathbb{R}$. This happens when $\Lambda = \overline{\Lambda}$, i.e. $\bar{\tau} = a\tau + b$ for some $a, b \in \mathbb{Z}$. In this case, we can check that $\wp_\Lambda(\mathbb{R}) \subset \mathbb{R}\cup\{\infty\}$ and $\wp'_\Lambda(\mathbb{R}) \subset \mathbb{R}\cup\{\infty\}$. Hence the real points $\mathbb{R}/\mathbb{Z} \subset \mathbb{C}/ \Lambda$ maps to the real points $E_\Lambda(\mathbb{R}) \subset E_\Lambda(\mathbb{C})$.
Q1. Is it true that $\Lambda = \overline{\Lambda}$ if and only if $g_2, g_3 \in \mathbb{R}$?
I'm interested in dermining the precise preimage of $E_\Lambda(\mathbb{R})$ under the isomorphism. Especially, when $E_\Lambda(\mathbb{R})$ has two connected components, its preimage should also have two connected components, which should properly contain $\mathbb{R}/ \mathbb{Z}$. I guess the answer is that we need to add the cycle in the middle:
Q2. When $E_\Lambda(\mathbb{R})$ is not connected, does its preimage coincides with $(\mathbb{R} \cup (\mathbb{R} + \tau/2)) / \mathbb{Z}$?
In this case, the group structure is isomorphic to $\mathbb{S}^{1} \times (\mathbb{Z}/2)$. To show this, we may need to show that
$$ \wp_\Lambda(t + \tau/2) \in \mathbb{R}, \quad \wp'_\Lambda(t + \tau/2) \in \mathbb{R} $$ for $t \in \mathbb{R}$, when $E_\Lambda(\mathbb{R})$ is not connected. I wonder if this easily follows from the definition of $\varphi_\Lambda$ and topology of $E_\Lambda$. Thanks in advance.