| Curve name |
$X_{13}$ |
| Index |
$6$ |
| Level |
$4$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
| Level | Index of image | Corresponding curve |
| $2$ |
$3$ |
$X_{6}$ |
|
| Meaning/Special name |
Elliptic curves with a cyclic $4$-isogeny defined over $\mathbb{Q}$ |
| Chosen covering |
$X_{6}$ |
| Curves that $X_{13}$ minimally covers |
$X_{6}$ |
| Curves that minimally cover $X_{13}$ |
$X_{25}$, $X_{27}$, $X_{32}$, $X_{33}$, $X_{34}$, $X_{36}$, $X_{44}$, $X_{48}$, $X_{13a}$, $X_{13b}$, $X_{13c}$, $X_{13d}$, $X_{13e}$, $X_{13f}$, $X_{13g}$, $X_{13h}$ |
| Curves that minimally cover $X_{13}$ and have infinitely many rational
points. |
$X_{25}$, $X_{27}$, $X_{32}$, $X_{33}$, $X_{34}$, $X_{36}$, $X_{44}$, $X_{48}$, $X_{13a}$, $X_{13b}$, $X_{13c}$, $X_{13d}$, $X_{13e}$, $X_{13f}$, $X_{13g}$, $X_{13h}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{13}) = \mathbb{Q}(f_{13}), f_{6} =
-f_{13}^{2} + 48\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - x^2 - 23x - 34$, with conductor $315$ |
| Generic density of odd order reductions |
$1/7$ |