| Curve name |
$X_{113}$ |
| Index |
$24$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 3 \\ 12 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 12 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 3 \\ 14 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{50}$ |
| Curves that $X_{113}$ minimally covers |
$X_{50}$ |
| Curves that minimally cover $X_{113}$ |
$X_{284}$, $X_{290}$, $X_{317}$, $X_{324}$, $X_{328}$, $X_{348}$ |
| Curves that minimally cover $X_{113}$ and have infinitely many rational
points. |
$X_{284}$, $X_{324}$, $X_{328}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{113}) = \mathbb{Q}(f_{113}), f_{50} =
-f_{113}^{2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 785472x - 267747760$, with conductor $2093058$ |
| Generic density of odd order reductions |
$85091/344064$ |