| Curve name |
$X_{120o}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{120}$ |
| Curves that $X_{120o}$ minimally covers |
|
| Curves that minimally cover $X_{120o}$ |
|
| Curves that minimally cover $X_{120o}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{16} + 1080t^{14} - 18144t^{12} + 165888t^{10} - 892080t^{8} +
2830464t^{6} - 4955904t^{4} + 3870720t^{2} - 442368\]
\[B(t) = -54t^{24} + 3240t^{22} - 86832t^{20} + 1370304t^{18} - 14127696t^{16} +
99734976t^{14} - 490783104t^{12} + 1677749760t^{10} - 3883935744t^{8} +
5743927296t^{6} - 4782882816t^{4} + 1571291136t^{2} + 113246208\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + 6909x + 48706$, with conductor $7056$ |
| Generic density of odd order reductions |
$41/336$ |