| Curve name |
$X_{243}$ |
| 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} 5 & 0 \\ 16 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 16 & 3 \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 $X_{243}$ minimally covers |
$X_{118}$ |
| Curves that minimally cover $X_{243}$ |
$X_{490}$, $X_{492}$, $X_{494}$, $X_{495}$, $X_{498}$, $X_{499}$, $X_{243a}$, $X_{243b}$, $X_{243c}$, $X_{243d}$, $X_{243e}$, $X_{243f}$, $X_{243g}$, $X_{243h}$, $X_{243i}$, $X_{243j}$, $X_{243k}$, $X_{243l}$, $X_{243m}$, $X_{243n}$, $X_{243o}$, $X_{243p}$ |
| Curves that minimally cover $X_{243}$ and have infinitely many rational
points. |
$X_{243a}$, $X_{243b}$, $X_{243c}$, $X_{243d}$, $X_{243e}$, $X_{243f}$, $X_{243g}$, $X_{243h}$, $X_{243i}$, $X_{243j}$, $X_{243k}$, $X_{243l}$, $X_{243m}$, $X_{243n}$, $X_{243o}$, $X_{243p}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{243}) = \mathbb{Q}(f_{243}), f_{118} =
\frac{2}{f_{243}^{2}}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 720x - 8960$, with conductor $1530$ |
| Generic density of odd order reductions |
$25/224$ |