| Curve name |
$X_{354}$ |
| Index |
$48$ |
| Level |
$32$ |
| Genus |
$1$ |
| 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 |
$X_{0}(32)$ |
| Chosen covering |
$X_{118}$ |
| Curves that $X_{354}$ minimally covers |
$X_{118}$ |
| Curves that minimally cover $X_{354}$ |
$X_{490}$, $X_{496}$, $X_{617}$, $X_{621}$, $X_{629}$, $X_{638}$, $X_{663}$, $X_{665}$, $X_{667}$, $X_{668}$, $X_{697}$, $X_{698}$, $X_{699}$, $X_{700}$ |
| Curves that minimally cover $X_{354}$ and have infinitely many rational
points. |
|
| Model |
\[y^2 = x^3 + 4x\] |
| Info about rational points |
| Rational point | Image on the $j$-line |
| $(0 : 1 : 0)$ |
\[ \infty \]
|
| $(2 : 4 : 1)$ |
\[ \infty \]
|
| $(0 : 0 : 1)$ |
\[ \infty \]
|
| $(2 : -4 : 1)$ |
\[ \infty \]
|
|
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
None |
| Generic density of odd order reductions |
N/A |