| Curve name |
$X_{185}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 6 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{87}$ |
| Curves that $X_{185}$ minimally covers |
$X_{87}$, $X_{96}$, $X_{101}$ |
| Curves that minimally cover $X_{185}$ |
$X_{442}$, $X_{452}$, $X_{456}$, $X_{458}$, $X_{485}$, $X_{486}$, $X_{185a}$, $X_{185b}$, $X_{185c}$, $X_{185d}$, $X_{185e}$, $X_{185f}$, $X_{185g}$, $X_{185h}$, $X_{185i}$, $X_{185j}$, $X_{185k}$, $X_{185l}$ |
| Curves that minimally cover $X_{185}$ and have infinitely many rational
points. |
$X_{185a}$, $X_{185b}$, $X_{185c}$, $X_{185d}$, $X_{185e}$, $X_{185f}$, $X_{185g}$, $X_{185h}$, $X_{185i}$, $X_{185j}$, $X_{185k}$, $X_{185l}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{185}) = \mathbb{Q}(f_{185}), f_{87} =
\frac{f_{185}}{f_{185}^{2} - \frac{1}{2}}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - 6616x + 206471$, with conductor $735$ |
| Generic density of odd order reductions |
$25/224$ |