| Curve name |
$X_{57}$ |
| Index |
$16$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 7 \\ 1 & 2 \end{matrix}\right],
\left[ \begin{matrix} 15 & 11 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{22}$ |
| Curves that $X_{57}$ minimally covers |
$X_{22}$ |
| Curves that minimally cover $X_{57}$ |
$X_{370}$, $X_{440}$, $X_{57a}$, $X_{57b}$ |
| Curves that minimally cover $X_{57}$ and have infinitely many rational
points. |
$X_{57a}$, $X_{57b}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{57}) = \mathbb{Q}(f_{57}), f_{22} =
\frac{3f_{57}^{2} + 6f_{57} - 3}{f_{57}^{2} + 1}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 13x + 43$, with conductor $6400$ |
| Generic density of odd order reductions |
$91681/172032$ |