| Curve name |
$X_{181b}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 6 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 6 \\ 4 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{181}$ |
| Curves that $X_{181b}$ minimally covers |
|
| Curves that minimally cover $X_{181b}$ |
|
| Curves that minimally cover $X_{181b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -1855425871872t^{24} - 4638564679680t^{22} - 4232690270208t^{20} -
1637993152512t^{18} - 311200579584t^{16} - 147220070400t^{14} -
60671655936t^{12} - 2300313600t^{10} - 75976704t^{8} - 6248448t^{6} -
252288t^{4} - 4320t^{2} - 27\]
\[B(t) = -972777519512027136t^{36} - 3647915698170101760t^{34} -
5608670385936531456t^{32} - 4499096027743125504t^{30} -
1838207519781027840t^{28} - 84785540641062912t^{26} + 322413642903453696t^{24} +
163604031678185472t^{22} + 29631730994380800t^{20} + 1613379768680448t^{18} +
462995796787200t^{16} + 39942390546432t^{14} + 1229910441984t^{12} -
5053612032t^{10} - 1711964160t^{8} - 65470464t^{6} - 1275264t^{4} - 12960t^{2} -
54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 527232x + 121351860$, with conductor $13872$ |
| Generic density of odd order reductions |
$299/2688$ |