| Curve name |
$X_{117}$ |
| Index |
$24$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 8 & 7 \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} 3 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{36}$ |
| Curves that $X_{117}$ minimally covers |
$X_{36}$ |
| Curves that minimally cover $X_{117}$ |
$X_{219}$, $X_{223}$, $X_{228}$, $X_{229}$, $X_{230}$, $X_{236}$, $X_{306}$, $X_{315}$, $X_{330}$, $X_{332}$, $X_{117a}$, $X_{117b}$, $X_{117c}$, $X_{117d}$, $X_{117e}$, $X_{117f}$, $X_{117g}$, $X_{117h}$, $X_{117i}$, $X_{117j}$, $X_{117k}$, $X_{117l}$, $X_{117m}$, $X_{117n}$, $X_{117o}$, $X_{117p}$ |
| Curves that minimally cover $X_{117}$ and have infinitely many rational
points. |
$X_{219}$, $X_{223}$, $X_{228}$, $X_{229}$, $X_{230}$, $X_{236}$, $X_{117a}$, $X_{117b}$, $X_{117c}$, $X_{117d}$, $X_{117e}$, $X_{117f}$, $X_{117g}$, $X_{117h}$, $X_{117i}$, $X_{117j}$, $X_{117k}$, $X_{117l}$, $X_{117m}$, $X_{117n}$, $X_{117o}$, $X_{117p}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{117}) = \mathbb{Q}(f_{117}), f_{36} =
\frac{2}{f_{117}^{2}}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 18714x - 985367$, with conductor $5544$ |
| Generic density of odd order reductions |
$643/5376$ |