| Curve name |
$X_{193i}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 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 $X_{193i}$ minimally covers |
|
| Curves that minimally cover $X_{193i}$ |
|
| Curves that minimally cover $X_{193i}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -1769472t^{16} + 3538944t^{14} - 1327104t^{12} - 221184t^{10} -
1589760t^{8} - 13824t^{6} - 5184t^{4} + 864t^{2} - 27\]
\[B(t) = -905969664t^{24} + 2717908992t^{22} - 2378170368t^{20} +
396361728t^{18} + 1985347584t^{16} - 1847328768t^{14} - 229146624t^{12} -
115458048t^{10} + 7755264t^{8} + 96768t^{6} - 36288t^{4} + 2592t^{2} - 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 17120x - 499968$, with conductor $1680$ |
| Generic density of odd order reductions |
$19/336$ |