# Real elliptic curves in the fundamental domain of $\Gamma(2)$

An elliptic curve (over $\mathbf{C}$) is real if its j-invariant is real.

The set of real elliptic curves in the standard fundamental domain of $\mathrm{SL}_2(\mathbf{Z})$ can be explicitly described.

In the standard fundamental domain $$F(\mathbf{SL}_2(\mathbf{Z}))=\left\{\tau \in \mathbf{H} : -\frac{1}{2} \leq \Re(\tau) \leq 0, \vert \tau \vert \geq 1 \right\} \cup \left\{ \tau \in \mathbf{H} : 0 < \Re(\tau) < \frac{1}{2}, \vert \tau \vert > 1\right\},$$ the set of real elliptic curves is the boundary of this fundamental domain together with the imaginary axis lying in this fundamental domain.

Let $F$ be the standard fundamental domain for $\Gamma(2)$. Can one describe how the set of real elliptic curves in this fundamental domain looks like? Of course, the set of real elliptic curves in $F(\mathbf{SL}_2(\mathbf{Z}))$ is contained in the set of real elliptic curves in $F$. But there should be more.

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I'm not sure which fundamental domain for $\Gamma(2)$ you consider to be "standard"? But whichever one you go for, it'll just be the points in your bigger domain whose $SL_2(\mathbf{Z})$ orbit contains a point of the set you just wrote down. –  David Loeffler Oct 15 '11 at 10:38

I'm not sure which fundamental domain for $\Gamma(2)$ you consider to be "standard"? But whichever one you go for, it'll just be the points in your bigger domain whose $SL_2(\Bbb{Z})$ orbit contains a point of the set you just wrote down.