| Curve name |
$X_{83}$ |
| Index |
$24$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 6 \\ 2 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{26}$ |
| Curves that $X_{83}$ minimally covers |
$X_{22}$, $X_{26}$, $X_{29}$, $X_{39}$ |
| Curves that minimally cover $X_{83}$ |
$X_{232}$, $X_{237}$, $X_{275}$, $X_{276}$, $X_{370}$, $X_{402}$ |
| Curves that minimally cover $X_{83}$ and have infinitely many rational
points. |
$X_{232}$, $X_{237}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{83}) = \mathbb{Q}(f_{83}), f_{26} =
\frac{f_{83}}{f_{83}^{2} - \frac{1}{8}}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 2110x + 682176$, with conductor $202496$ |
| Generic density of odd order reductions |
$1427/5376$ |