| Curve name |
$X_{70}$ |
| Index |
$24$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 6 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 4 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{45}$ |
| Curves that $X_{70}$ minimally covers |
$X_{45}$, $X_{47}$, $X_{49}$ |
| Curves that minimally cover $X_{70}$ |
$X_{257}$, $X_{260}$, $X_{325}$, $X_{326}$ |
| Curves that minimally cover $X_{70}$ and have infinitely many rational
points. |
$X_{326}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{70}) = \mathbb{Q}(f_{70}), f_{45} =
\frac{8f_{70}}{f_{70}^{2} + 2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 513x - 5103$, with conductor $2400$ |
| Generic density of odd order reductions |
$1343/5376$ |