| Curve name |
$X_{202a}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{202}$ |
| Curves that $X_{202a}$ minimally covers |
|
| Curves that minimally cover $X_{202a}$ |
|
| Curves that minimally cover $X_{202a}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -442368t^{24} - 48660480t^{22} + 222732288t^{20} + 64806912t^{18} -
1107219456t^{16} + 878653440t^{14} - 394343424t^{12} + 219663360t^{10} -
69201216t^{8} + 1012608t^{6} + 870048t^{4} - 47520t^{2} - 108\]
\[B(t) = 113246208t^{36} - 30236737536t^{34} - 60898148352t^{32} +
2092563431424t^{30} - 7789612105728t^{28} + 8993645789184t^{26} -
2462644961280t^{24} + 7470640005120t^{22} - 13768934916096t^{20} +
9889733836800t^{18} - 3442233729024t^{16} + 466915000320t^{14} -
38478827520t^{12} + 35131428864t^{10} - 7607043072t^{8} + 510879744t^{6} -
3716928t^{4} - 461376t^{2} + 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 2866369x + 1850890945$, with conductor $9408$ |
| Generic density of odd order reductions |
$109/896$ |