| Curve name |
$X_{58}$ |
| Index |
$24$ |
| Level |
$4$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 2 & 1 \end{matrix}\right]$ |
| Images in lower levels |
| Level | Index of image | Corresponding curve |
| $2$ |
$6$ |
$X_{8}$ |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{23}$ |
| Curves that $X_{58}$ minimally covers |
$X_{23}$, $X_{24}$, $X_{25}$ |
| Curves that minimally cover $X_{58}$ |
$X_{183}$, $X_{186}$, $X_{187}$, $X_{189}$, $X_{248}$, $X_{249}$, $X_{251}$, $X_{255}$, $X_{357}$, $X_{58a}$, $X_{58b}$, $X_{58c}$, $X_{58d}$, $X_{58e}$, $X_{58f}$, $X_{58g}$, $X_{58h}$, $X_{58i}$, $X_{58j}$, $X_{58k}$ |
| Curves that minimally cover $X_{58}$ and have infinitely many rational
points. |
$X_{183}$, $X_{187}$, $X_{189}$, $X_{58a}$, $X_{58b}$, $X_{58c}$, $X_{58d}$, $X_{58e}$, $X_{58f}$, $X_{58g}$, $X_{58h}$, $X_{58i}$, $X_{58j}$, $X_{58k}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{58}) = \mathbb{Q}(f_{58}), f_{23} =
\frac{f_{58}^{2} + 2f_{58} - 1}{f_{58}^{2} + 1}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 40807260x - 94524898325$, with conductor $6435$ |
| Generic density of odd order reductions |
$19/168$ |