| Curve name |
$X_{193a}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 6 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 6 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 2 \\ 0 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{193}$ |
| Curves that $X_{193a}$ minimally covers |
|
| Curves that minimally cover $X_{193a}$ |
|
| Curves that minimally cover $X_{193a}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -7077888t^{16} + 14155776t^{14} - 5308416t^{12} - 884736t^{10} -
6359040t^{8} - 55296t^{6} - 20736t^{4} + 3456t^{2} - 108\]
\[B(t) = 7247757312t^{24} - 21743271936t^{22} + 19025362944t^{20} -
3170893824t^{18} - 15882780672t^{16} + 14778630144t^{14} + 1833172992t^{12} +
923664384t^{10} - 62042112t^{8} - 774144t^{6} + 290304t^{4} - 20736t^{2} + 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 68481x + 4068225$, with conductor $6720$ |
| Generic density of odd order reductions |
$299/2688$ |