| Curve name |
$X_{156}$ |
| Index |
$24$ |
| Level |
$16$ |
| Genus |
$1$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 3 \\ 12 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 2 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 0 & 13 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{50}$ |
| Curves that $X_{156}$ minimally covers |
$X_{50}$ |
| Curves that minimally cover $X_{156}$ |
$X_{291}$, $X_{324}$, $X_{412}$, $X_{413}$, $X_{418}$, $X_{426}$ |
| Curves that minimally cover $X_{156}$ and have infinitely many rational
points. |
$X_{291}$, $X_{324}$ |
| Model |
\[y^2 = x^3 + 8x\] |
| Info about rational points |
$X_{156}(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}$ |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
None. All the rational points lift to covering modular curves. |
| Generic density of odd order reductions |
N/A |