| Curve name |
$X_{56}$ |
| Index |
$16$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 13 & 11 \\ 1 & 6 \end{matrix}\right],
\left[ \begin{matrix} 7 & 3 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{22}$ |
| Curves that $X_{56}$ minimally covers |
$X_{22}$ |
| Curves that minimally cover $X_{56}$ |
$X_{177}$, $X_{178}$, $X_{402}$, $X_{439}$ |
| Curves that minimally cover $X_{56}$ and have infinitely many rational
points. |
$X_{177}$, $X_{178}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{56}) = \mathbb{Q}(f_{56}), f_{22} =
\frac{f_{56}^{2} + \frac{9}{2}}{f_{56}}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 17400x + 4787200$, with conductor $3628800$ |
| Generic density of odd order reductions |
$91681/172032$ |