| Curve name |
$X_{122e}$ |
| 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 & 7 \\ 8 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 8 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 8 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{122}$ |
| Curves that $X_{122e}$ minimally covers |
|
| Curves that minimally cover $X_{122e}$ |
|
| Curves that minimally cover $X_{122e}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -6912t^{12} + 27648t^{10} - 43200t^{8} + 32832t^{6} - 12123t^{4} +
1782t^{2} - 27\]
\[B(t) = -221184t^{18} + 1327104t^{16} - 3400704t^{14} + 4838400t^{12} -
4153680t^{10} + 2182464t^{8} - 674730t^{6} + 108702t^{4} - 6318t^{2} - 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 112899x - 14601022$, with conductor $1008$ |
| Generic density of odd order reductions |
$25/224$ |