| Curve name |
$X_{11}$ |
| Index |
$6$ |
| 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 & 2 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 1 \\ 2 & 1 \end{matrix}\right]$ |
| Images in lower levels |
| Level | Index of image | Corresponding curve |
| $2$ |
$3$ |
$X_{6}$ |
|
| Meaning/Special name |
Elliptic curves with discriminant $\Delta$ whose $2$-isogenous curve has
discriminant in the square class of $-\Delta$ |
| Chosen covering |
$X_{6}$ |
| Curves that $X_{11}$ minimally covers |
$X_{6}$ |
| Curves that minimally cover $X_{11}$ |
$X_{23}$, $X_{24}$, $X_{26}$, $X_{27}$, $X_{28}$, $X_{29}$, $X_{35}$, $X_{39}$, $X_{41}$, $X_{43}$, $X_{45}$, $X_{47}$, $X_{49}$, $X_{50}$, $X_{53}$, $X_{54}$ |
| Curves that minimally cover $X_{11}$ and have infinitely many rational
points. |
$X_{23}$, $X_{24}$, $X_{26}$, $X_{27}$, $X_{28}$, $X_{29}$, $X_{35}$, $X_{39}$, $X_{41}$, $X_{43}$, $X_{45}$, $X_{47}$, $X_{49}$, $X_{50}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{11}) = \mathbb{Q}(f_{11}), f_{6} =
f_{11}^{2} - 16\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + 72x + 485$, with conductor $2772$ |
| Generic density of odd order reductions |
$83/336$ |