| Curve name |
$X_{199d}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{199}$ |
| Curves that $X_{199d}$ minimally covers |
|
| Curves that minimally cover $X_{199d}$ |
|
| Curves that minimally cover $X_{199d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{24} + 47520t^{22} + 870048t^{20} - 1012608t^{18} -
69201216t^{16} - 219663360t^{14} - 394343424t^{12} - 878653440t^{10} -
1107219456t^{8} - 64806912t^{6} + 222732288t^{4} + 48660480t^{2} - 442368\]
\[B(t) = -432t^{36} - 461376t^{34} + 3716928t^{32} + 510879744t^{30} +
7607043072t^{28} + 35131428864t^{26} + 38478827520t^{24} + 466915000320t^{22} +
3442233729024t^{20} + 9889733836800t^{18} + 13768934916096t^{16} +
7470640005120t^{14} + 2462644961280t^{12} + 8993645789184t^{10} +
7789612105728t^{8} + 2092563431424t^{6} + 60898148352t^{4} - 30236737536t^{2} -
113246208\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 30407809x + 77498755105$, with conductor $55488$ |
| Generic density of odd order reductions |
$109/896$ |