| Curve name |
$X_{102h}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{102}$ |
| Curves that $X_{102h}$ minimally covers |
|
| Curves that minimally cover $X_{102h}$ |
|
| Curves that minimally cover $X_{102h}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{10} + 54t^{9} + 405t^{8} - 1728t^{7} + 2160t^{6} + 864t^{5} -
6048t^{4} + 8640t^{3} - 6480t^{2} + 2592t - 432\]
\[B(t) = 54t^{15} - 162t^{14} - 1134t^{13} + 6426t^{12} - 6480t^{11} -
32400t^{10} + 133056t^{9} - 256608t^{8} + 351216t^{7} - 426384t^{6} +
480816t^{5} - 441936t^{4} + 290304t^{3} - 124416t^{2} + 31104t - 3456\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 5619x + 161138$, with conductor $1008$ |
| Generic density of odd order reductions |
$25/224$ |