| Curve name |
$X_{118}$ |
| 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} 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 |
$X_0(16)$ |
| Chosen covering |
$X_{36}$ |
| Curves that $X_{118}$ minimally covers |
$X_{36}$ |
| Curves that minimally cover $X_{118}$ |
$X_{211}$, $X_{212}$, $X_{213}$, $X_{225}$, $X_{227}$, $X_{235}$, $X_{240}$, $X_{243}$, $X_{305}$, $X_{313}$, $X_{329}$, $X_{331}$, $X_{353}$, $X_{354}$, $X_{355}$, $X_{356}$, $X_{405}$, $X_{406}$, $X_{118a}$, $X_{118b}$, $X_{118c}$, $X_{118d}$, $X_{118e}$, $X_{118f}$, $X_{118g}$, $X_{118h}$, $X_{118i}$, $X_{118j}$, $X_{118k}$, $X_{118l}$, $X_{118m}$, $X_{118n}$, $X_{118o}$, $X_{118p}$, $X_{118q}$, $X_{118r}$, $X_{118s}$, $X_{118t}$, $X_{118u}$, $X_{118v}$, $X_{118w}$, $X_{118x}$ |
| Curves that minimally cover $X_{118}$ and have infinitely many rational
points. |
$X_{211}$, $X_{212}$, $X_{213}$, $X_{225}$, $X_{227}$, $X_{235}$, $X_{240}$, $X_{243}$, $X_{118a}$, $X_{118b}$, $X_{118c}$, $X_{118d}$, $X_{118e}$, $X_{118f}$, $X_{118g}$, $X_{118h}$, $X_{118i}$, $X_{118j}$, $X_{118k}$, $X_{118l}$, $X_{118m}$, $X_{118n}$, $X_{118o}$, $X_{118p}$, $X_{118q}$, $X_{118r}$, $X_{118s}$, $X_{118t}$, $X_{118u}$, $X_{118v}$, $X_{118w}$, $X_{118x}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{118}) = \mathbb{Q}(f_{118}), f_{36} =
-f_{118}^{2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 40148370x - 97905135425$, with conductor $6435$ |
| Generic density of odd order reductions |
$19/168$ |