| Curve name |
$X_{241}$ |
| Index |
$48$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 28 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 15 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 20 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{116}$ |
| Curves that $X_{241}$ minimally covers |
$X_{116}$ |
| Curves that minimally cover $X_{241}$ |
$X_{241a}$, $X_{241b}$, $X_{241c}$, $X_{241d}$, $X_{241e}$, $X_{241f}$, $X_{241g}$, $X_{241h}$ |
| Curves that minimally cover $X_{241}$ and have infinitely many rational
points. |
$X_{241a}$, $X_{241b}$, $X_{241c}$, $X_{241d}$, $X_{241e}$, $X_{241f}$, $X_{241g}$, $X_{241h}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{241}) = \mathbb{Q}(f_{241}), f_{116} =
\frac{2}{f_{241}^{2}}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 816x + 7424$, with conductor $4626$ |
| Generic density of odd order reductions |
$9249/57344$ |