| Curve name |
$X_{23}$ |
| Index |
$12$ |
| Level |
$4$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 2 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 2 & 3 \end{matrix}\right]$ |
| Images in lower levels |
| Level | Index of image | Corresponding curve |
| $2$ |
$3$ |
$X_{6}$ |
|
| Meaning/Special name |
$X_{s}^{+}(4)$ |
| Chosen covering |
$X_{11}$ |
| Curves that $X_{23}$ minimally covers |
$X_{11}$ |
| Curves that minimally cover $X_{23}$ |
$X_{58}$, $X_{60}$, $X_{64}$, $X_{65}$, $X_{68}$, $X_{69}$, $X_{76}$, $X_{80}$, $X_{127}$, $X_{136}$, $X_{146}$, $X_{148}$ |
| Curves that minimally cover $X_{23}$ and have infinitely many rational
points. |
$X_{58}$, $X_{60}$, $X_{64}$, $X_{65}$, $X_{68}$, $X_{69}$, $X_{76}$, $X_{80}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{23}) = \mathbb{Q}(f_{23}), f_{11} =
\frac{8}{f_{23}^{2} - 1}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 128x + 540$, with conductor $1176$ |
| Generic density of odd order reductions |
$25/112$ |