| Curve name |
$X_{85}$ |
| Index |
$24$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \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 |
$X_{36}$ |
| Curves that $X_{85}$ minimally covers |
$X_{36}$ |
| Curves that minimally cover $X_{85}$ |
$X_{192}$, $X_{199}$, $X_{202}$, $X_{205}$, $X_{211}$, $X_{219}$, $X_{225}$, $X_{236}$, $X_{335}$, $X_{339}$, $X_{341}$, $X_{344}$, $X_{85a}$, $X_{85b}$, $X_{85c}$, $X_{85d}$, $X_{85e}$, $X_{85f}$, $X_{85g}$, $X_{85h}$, $X_{85i}$, $X_{85j}$, $X_{85k}$, $X_{85l}$, $X_{85m}$, $X_{85n}$, $X_{85o}$, $X_{85p}$ |
| Curves that minimally cover $X_{85}$ and have infinitely many rational
points. |
$X_{192}$, $X_{199}$, $X_{202}$, $X_{205}$, $X_{211}$, $X_{219}$, $X_{225}$, $X_{236}$, $X_{85a}$, $X_{85b}$, $X_{85c}$, $X_{85d}$, $X_{85e}$, $X_{85f}$, $X_{85g}$, $X_{85h}$, $X_{85i}$, $X_{85j}$, $X_{85k}$, $X_{85l}$, $X_{85m}$, $X_{85n}$, $X_{85o}$, $X_{85p}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{85}) = \mathbb{Q}(f_{85}), f_{36} =
\frac{8}{f_{85}^{2} - 2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 + x^2 - 850x - 27125$, with conductor $525$ |
| Generic density of odd order reductions |
$19/168$ |