| Curve name |
$X_{181g}$ |
| 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 & 2 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 4 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{181}$ |
| Curves that $X_{181g}$ minimally covers |
|
| Curves that minimally cover $X_{181g}$ |
|
| Curves that minimally cover $X_{181g}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -1811939328t^{16} - 1811939328t^{14} - 339738624t^{12} + 28311552t^{10}
- 101744640t^{8} + 442368t^{6} - 82944t^{4} - 6912t^{2} - 108\]
\[B(t) = -29686813949952t^{24} - 44530220924928t^{22} - 19481971654656t^{20} -
1623497637888t^{18} + 4065991852032t^{16} + 1891664658432t^{14} -
117323071488t^{12} + 29557260288t^{10} + 992673792t^{8} - 6193152t^{6} -
1161216t^{4} - 41472t^{2} - 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 7297x + 195455$, with conductor $3264$ |
| Generic density of odd order reductions |
$271/2688$ |