| Curve name |
$X_{102g}$ |
| 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 \\ 8 & 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 $X_{102g}$ minimally covers |
|
| Curves that minimally cover $X_{102g}$ |
|
| Curves that minimally cover $X_{102g}$ 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 |
$y^2 + xy = x^3 - x^2 - 351x - 2430$, with conductor $63$ |
| Generic density of odd order reductions |
$17/168$ |