| Curve name |
$X_{120k}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 7 & 7 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{120}$ |
| Curves that $X_{120k}$ minimally covers |
|
| Curves that minimally cover $X_{120k}$ |
|
| Curves that minimally cover $X_{120k}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{12} + 2592t^{10} - 24192t^{8} + 110592t^{6} - 250560t^{4} +
235008t^{2} - 27648\]
\[B(t) = -432t^{18} + 15552t^{16} - 238464t^{14} + 2032128t^{12} -
10502784t^{10} + 33550848t^{8} - 64060416t^{6} + 65359872t^{4} - 25214976t^{2} -
1769472\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 993600x - 380962764$, with conductor $2070$ |
| Generic density of odd order reductions |
$193/1792$ |