| Curve name |
$X_{188c}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 2 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 4 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{188}$ |
| Curves that $X_{188c}$ minimally covers |
|
| Curves that minimally cover $X_{188c}$ |
|
| Curves that minimally cover $X_{188c}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{16} + 432t^{14} - 1296t^{12} - 1728t^{10} - 99360t^{8} -
6912t^{6} - 20736t^{4} + 27648t^{2} - 6912\]
\[B(t) = 54t^{24} - 1296t^{22} + 9072t^{20} - 12096t^{18} - 484704t^{16} +
3608064t^{14} + 3580416t^{12} + 14432256t^{10} - 7755264t^{8} - 774144t^{6} +
2322432t^{4} - 1327104t^{2} + 221184\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 + x^2 - 84x + 261$, with conductor $42$ |
| Generic density of odd order reductions |
$53/896$ |