| Curve name |
$X_{236}$ |
| Index |
$48$ |
| Level |
$16$ |
| 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} 1 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{85}$ |
| Curves that $X_{236}$ minimally covers |
$X_{85}$, $X_{117}$, $X_{120}$ |
| Curves that minimally cover $X_{236}$ |
$X_{466}$, $X_{474}$, $X_{476}$, $X_{483}$, $X_{236a}$, $X_{236b}$, $X_{236c}$, $X_{236d}$, $X_{236e}$, $X_{236f}$, $X_{236g}$, $X_{236h}$ |
| Curves that minimally cover $X_{236}$ and have infinitely many rational
points. |
$X_{236a}$, $X_{236b}$, $X_{236c}$, $X_{236d}$, $X_{236e}$, $X_{236f}$, $X_{236g}$, $X_{236h}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{236}) = \mathbb{Q}(f_{236}), f_{85} =
\frac{f_{236}^{2} + \frac{1}{2}}{f_{236}}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 12096x - 509036$, with conductor $126$ |
| Generic density of odd order reductions |
$193/1792$ |