| Curve name |
$X_{350}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$1$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 3 \\ 10 & 15 \end{matrix}\right],
\left[ \begin{matrix} 1 & 6 \\ 2 & 11 \end{matrix}\right],
\left[ \begin{matrix} 1 & 5 \\ 12 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 2 & 11 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{73}$ |
| Curves that $X_{350}$ minimally covers |
$X_{73}$, $X_{114}$, $X_{155}$ |
| Curves that minimally cover $X_{350}$ |
$X_{595}$, $X_{597}$, $X_{626}$, $X_{676}$, $X_{679}$, $X_{714}$ |
| Curves that minimally cover $X_{350}$ and have infinitely many rational
points. |
|
| Model |
\[y^2 = x^3 + 8x\] |
| Info about rational points |
$X_{350}(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}$ |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 1725814293280x + 872648220992027997$, with conductor
$2094789268$ |
| Generic density of odd order reductions |
$45667/172032$ |