| Curve name |
$X_{158}$ |
| Index |
$24$ |
| Level |
$16$ |
| Genus |
$1$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 3 \\ 12 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 4 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 9 \\ 12 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{32}$ |
| Curves that $X_{158}$ minimally covers |
$X_{32}$ |
| Curves that minimally cover $X_{158}$ |
$X_{314}$, $X_{333}$ |
| Curves that minimally cover $X_{158}$ and have infinitely many rational
points. |
|
| Model |
\[y^2 = x^3 + 4x\] |
| Info about rational points |
| Rational point | Image on the $j$-line |
| $(0 : 1 : 0)$ |
\[ \infty \]
|
| $(2 : 4 : 1)$ |
\[287496 \,\,(\text{CM by }-16)\]
|
| $(0 : 0 : 1)$ |
\[ \infty \]
|
| $(2 : -4 : 1)$ |
\[287496 \,\,(\text{CM by }-16)\]
|
|
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
None |
| Generic density of odd order reductions |
N/A |