| Curve name |
$X_{36}$ |
| Index |
$12$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \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} 5 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
$X_{0}(8)$ |
| Chosen covering |
$X_{13}$ |
| Curves that $X_{36}$ minimally covers |
$X_{13}$ |
| Curves that minimally cover $X_{36}$ |
$X_{75}$, $X_{78}$, $X_{84}$, $X_{85}$, $X_{86}$, $X_{92}$, $X_{96}$, $X_{102}$, $X_{117}$, $X_{118}$, $X_{119}$, $X_{120}$, $X_{121}$, $X_{122}$, $X_{159}$, $X_{168}$, $X_{36a}$, $X_{36b}$, $X_{36c}$, $X_{36d}$, $X_{36e}$, $X_{36f}$, $X_{36g}$, $X_{36h}$, $X_{36i}$, $X_{36j}$, $X_{36k}$, $X_{36l}$, $X_{36m}$, $X_{36n}$, $X_{36o}$, $X_{36p}$, $X_{36q}$, $X_{36r}$, $X_{36s}$, $X_{36t}$ |
| Curves that minimally cover $X_{36}$ and have infinitely many rational
points. |
$X_{75}$, $X_{78}$, $X_{84}$, $X_{85}$, $X_{86}$, $X_{92}$, $X_{96}$, $X_{102}$, $X_{117}$, $X_{118}$, $X_{119}$, $X_{120}$, $X_{121}$, $X_{122}$, $X_{36a}$, $X_{36b}$, $X_{36c}$, $X_{36d}$, $X_{36e}$, $X_{36f}$, $X_{36g}$, $X_{36h}$, $X_{36i}$, $X_{36j}$, $X_{36k}$, $X_{36l}$, $X_{36m}$, $X_{36n}$, $X_{36o}$, $X_{36p}$, $X_{36q}$, $X_{36r}$, $X_{36s}$, $X_{36t}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{36}) = \mathbb{Q}(f_{36}), f_{13} =
-f_{36}^{2} + 8\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 2385x - 44240$, with conductor $3465$ |
| Generic density of odd order reductions |
$5/42$ |