| Curve name |
$X_{120e}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 8 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 8 & 5 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 8 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{120}$ |
| Curves that $X_{120e}$ minimally covers |
|
| Curves that minimally cover $X_{120e}$ |
|
| Curves that minimally cover $X_{120e}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{12} + 3456t^{10} - 43200t^{8} + 262656t^{6} - 775872t^{4} +
912384t^{2} - 110592\]
\[B(t) = 432t^{18} - 20736t^{16} + 425088t^{14} - 4838400t^{12} + 33229440t^{10}
- 139677696t^{8} + 345461760t^{6} - 445243392t^{4} + 207028224t^{2} + 14155776\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - 823201x + 287410955$, with conductor $1470$ |
| Generic density of odd order reductions |
$193/1792$ |