| Curve name |
$X_{71}$ |
| 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 \\ 2 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 0 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{43}$ |
| Curves that $X_{71}$ minimally covers |
$X_{43}$, $X_{49}$, $X_{50}$ |
| Curves that minimally cover $X_{71}$ |
$X_{258}$, $X_{259}$, $X_{327}$, $X_{328}$ |
| Curves that minimally cover $X_{71}$ and have infinitely many rational
points. |
$X_{328}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{71}) = \mathbb{Q}(f_{71}), f_{43} =
\frac{2f_{71}^{2} - 4}{f_{71}^{2} + 2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 + x^2 + 62x + 224$, with conductor $867$ |
| Generic density of odd order reductions |
$401/1792$ |