| Curve name |
$X_{122n}$ |
| 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} 3 & 0 \\ 8 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 0 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{122}$ |
| Curves that $X_{122n}$ minimally covers |
|
| Curves that minimally cover $X_{122n}$ |
|
| Curves that minimally cover $X_{122n}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27648t^{16} + 138240t^{14} - 290304t^{12} + 331776t^{10} - 223020t^{8}
+ 88452t^{6} - 19359t^{4} + 1890t^{2} - 27\]
\[B(t) = 1769472t^{24} - 13271040t^{22} + 44457984t^{20} - 87699456t^{18} +
113021568t^{16} - 99734976t^{14} + 61347888t^{12} - 26214840t^{10} +
7585812t^{8} - 1402326t^{6} + 145962t^{4} - 5994t^{2} - 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 5532051x + 5008150546$, with conductor $7056$ |
| Generic density of odd order reductions |
$41/336$ |