| Curve name |
$X_{122b}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 8 & 1 \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} 3 & 0 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{122}$ |
| Curves that $X_{122b}$ minimally covers |
|
| Curves that minimally cover $X_{122b}$ |
|
| Curves that minimally cover $X_{122b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27648t^{8} + 55296t^{6} - 34560t^{4} + 6912t^{2} - 108\]
\[B(t) = -1769472t^{12} + 5308416t^{10} - 5971968t^{8} + 3096576t^{6} -
715392t^{4} + 51840t^{2} + 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 50177x - 4309503$, with conductor $1344$ |
| Generic density of odd order reductions |
$635/5376$ |