| Curve name |
$X_{192n}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 6 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{192}$ |
| Curves that $X_{192n}$ minimally covers |
|
| Curves that minimally cover $X_{192n}$ |
|
| Curves that minimally cover $X_{192n}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{16} - 100224t^{14} - 1264896t^{12} + 8239104t^{10} -
19630080t^{8} + 131825664t^{6} - 323813376t^{4} - 410517504t^{2} - 7077888\]
\[B(t) = 432t^{24} - 891648t^{22} - 79833600t^{20} - 70447104t^{18} +
606818304t^{16} + 30799429632t^{14} - 206851276800t^{12} + 492790874112t^{10} +
155345485824t^{8} - 288551337984t^{6} - 5231974809600t^{4} - 934960693248t^{2} +
7247757312\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 7683201x - 8194556799$, with conductor $6720$ |
| Generic density of odd order reductions |
$299/2688$ |