| Curve name |
$X_{102f}$ |
| 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 \\ 8 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{102}$ |
| Curves that $X_{102f}$ minimally covers |
|
| Curves that minimally cover $X_{102f}$ |
|
| Curves that minimally cover $X_{102f}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{10} + 216t^{9} + 1620t^{8} - 6912t^{7} + 8640t^{6} + 3456t^{5} -
24192t^{4} + 34560t^{3} - 25920t^{2} + 10368t - 1728\]
\[B(t) = 432t^{15} - 1296t^{14} - 9072t^{13} + 51408t^{12} - 51840t^{11} -
259200t^{10} + 1064448t^{9} - 2052864t^{8} + 2809728t^{7} - 3411072t^{6} +
3846528t^{5} - 3535488t^{4} + 2322432t^{3} - 995328t^{2} + 248832t - 27648\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 3780x - 97200$, with conductor $2070$ |
| Generic density of odd order reductions |
$193/1792$ |