| Curve name |
$X_{240}$ |
| Index |
$48$ |
| Level |
$32$ |
| 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} 3 & 0 \\ 16 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 16 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{118}$ |
| Curves that $X_{240}$ minimally covers |
$X_{118}$ |
| Curves that minimally cover $X_{240}$ |
$X_{488}$, $X_{489}$, $X_{491}$, $X_{493}$, $X_{496}$, $X_{497}$, $X_{240a}$, $X_{240b}$, $X_{240c}$, $X_{240d}$, $X_{240e}$, $X_{240f}$, $X_{240g}$, $X_{240h}$, $X_{240i}$, $X_{240j}$, $X_{240k}$, $X_{240l}$, $X_{240m}$, $X_{240n}$, $X_{240o}$, $X_{240p}$ |
| Curves that minimally cover $X_{240}$ and have infinitely many rational
points. |
$X_{240a}$, $X_{240b}$, $X_{240c}$, $X_{240d}$, $X_{240e}$, $X_{240f}$, $X_{240g}$, $X_{240h}$, $X_{240i}$, $X_{240j}$, $X_{240k}$, $X_{240l}$, $X_{240m}$, $X_{240n}$, $X_{240o}$, $X_{240p}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{240}) = \mathbb{Q}(f_{240}), f_{118} =
f_{240}^{2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x - 64$, with conductor $735$ |
| Generic density of odd order reductions |
$25/224$ |