| Curve name |
$X_{188g}$ |
| 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 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 4 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{188}$ |
| Curves that $X_{188g}$ minimally covers |
|
| Curves that minimally cover $X_{188g}$ |
|
| Curves that minimally cover $X_{188g}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{16} + 1728t^{14} - 5184t^{12} - 6912t^{10} - 397440t^{8} -
27648t^{6} - 82944t^{4} + 110592t^{2} - 27648\]
\[B(t) = 432t^{24} - 10368t^{22} + 72576t^{20} - 96768t^{18} - 3877632t^{16} +
28864512t^{14} + 28643328t^{12} + 115458048t^{10} - 62042112t^{8} - 6193152t^{6}
+ 18579456t^{4} - 10616832t^{2} + 1769472\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 5377x + 149855$, with conductor $1344$ |
| Generic density of odd order reductions |
$271/2688$ |