| Curve name |
$X_{115}$ |
| Index |
$24$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 6 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 9 \\ 12 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{32}$ |
| Curves that $X_{115}$ minimally covers |
$X_{32}$ |
| Curves that minimally cover $X_{115}$ |
$X_{314}$, $X_{334}$, $X_{115a}$, $X_{115b}$, $X_{115c}$, $X_{115d}$, $X_{115e}$, $X_{115f}$, $X_{115g}$, $X_{115h}$ |
| Curves that minimally cover $X_{115}$ and have infinitely many rational
points. |
$X_{115a}$, $X_{115b}$, $X_{115c}$, $X_{115d}$, $X_{115e}$, $X_{115f}$, $X_{115g}$, $X_{115h}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{115}) = \mathbb{Q}(f_{115}), f_{32} =
\frac{2}{f_{115}^{2}}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 18x - 27$, with conductor $360$ |
| Generic density of odd order reductions |
$13/84$ |