| Curve name |
$X_{101a}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 2 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 6 \\ 4 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 6 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 2 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{101}$ |
| Curves that $X_{101a}$ minimally covers |
|
| Curves that minimally cover $X_{101a}$ |
|
| Curves that minimally cover $X_{101a}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -452984832t^{16} + 566231040t^{14} - 297271296t^{12} + 84934656t^{10} -
14376960t^{8} + 1492992t^{6} - 98496t^{4} + 4320t^{2} - 108\]
\[B(t) = 3710851743744t^{24} - 6957847019520t^{22} + 5827196878848t^{20} -
2873735774208t^{18} + 924089057280t^{16} - 201804742656t^{14} +
29974855680t^{12} - 2890432512t^{10} + 151953408t^{8} - 359424t^{6} -
476928t^{4} + 25920t^{2} - 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - x^2 - 771980x - 203140353$, with conductor
$22050$ |
| Generic density of odd order reductions |
$193/1792$ |