| Curve name |
$X_{312}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$1$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 10 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 4 \\ 2 & 3 \end{matrix}\right],
\left[ \begin{matrix} 13 & 7 \\ 2 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 4 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{97}$ |
| Curves that $X_{312}$ minimally covers |
$X_{97}$, $X_{110}$, $X_{150}$ |
| Curves that minimally cover $X_{312}$ |
$X_{635}$, $X_{637}$, $X_{686}$, $X_{689}$ |
| Curves that minimally cover $X_{312}$ and have infinitely many rational
points. |
|
| Model |
\[y^2 = x^3 + x^2 - 3x + 1\] |
| Info about rational points |
$X_{312}(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}$ |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 13804780x + 191741059200$, with conductor $173600$ |
| Generic density of odd order reductions |
$42979/172032$ |