| Curve name |
$X_{116f}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 3 \\ 4 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{116}$ |
| Curves that $X_{116f}$ minimally covers |
|
| Curves that minimally cover $X_{116f}$ |
|
| Curves that minimally cover $X_{116f}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{16} - 1296t^{12} - 21168t^{8} - 124416t^{4} - 110592\]
\[B(t) = -54t^{24} - 3888t^{20} - 110160t^{16} - 1524096t^{12} - 10119168t^{8} -
23887872t^{4} + 14155776\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 + x^2 - 6153682x - 5147588002$, with conductor
$28227$ |
| Generic density of odd order reductions |
$419/2688$ |