| Curve name |
$X_{122o}$ |
| 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} 3 & 0 \\ 0 & 1 \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 $X_{122o}$ minimally covers |
|
| Curves that minimally cover $X_{122o}$ |
|
| Curves that minimally cover $X_{122o}$ 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 |
$y^2 + xy = x^3 - x^2 - 345753x - 78165914$, with conductor $441$ |
| Generic density of odd order reductions |
$25/224$ |