| Curve name |
$X_{178}$ |
| Index |
$32$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 9 & 31 \\ 1 & 2 \end{matrix}\right],
\left[ \begin{matrix} 23 & 19 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{56}$ |
| Curves that $X_{178}$ minimally covers |
$X_{56}$ |
| Curves that minimally cover $X_{178}$ |
$X_{718}$ |
| Curves that minimally cover $X_{178}$ and have infinitely many rational
points. |
|
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{178}) = \mathbb{Q}(f_{178}), f_{56} =
\frac{3f_{178}^{2} + 3}{f_{178}^{2} + 2f_{178} - 1}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 120x - 512$, with conductor $20736$ |
| Generic density of odd order reductions |
$977931/1835008$ |