| Curve name |
$X_{26}$ |
| Index |
$12$ |
| Level |
$4$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}\right]$ |
| Images in lower levels |
| Level | Index of image | Corresponding curve |
| $2$ |
$3$ |
$X_{6}$ |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{11}$ |
| Curves that $X_{26}$ minimally covers |
$X_{7}$, $X_{11}$ |
| Curves that minimally cover $X_{26}$ |
$X_{59}$, $X_{60}$, $X_{73}$, $X_{74}$, $X_{83}$, $X_{139}$, $X_{140}$ |
| Curves that minimally cover $X_{26}$ and have infinitely many rational
points. |
$X_{60}$, $X_{73}$, $X_{74}$, $X_{83}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{26}) = \mathbb{Q}(f_{26}), f_{11} =
\frac{8}{f_{26}^{2} + 1}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 168x - 855$, with conductor $468$ |
| Generic density of odd order reductions |
$89/336$ |