| Curve name |
$X_{181a}$ |
| 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} 5 & 2 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{181}$ |
| Curves that $X_{181a}$ minimally covers |
|
| Curves that minimally cover $X_{181a}$ |
|
| Curves that minimally cover $X_{181a}$ 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 |
$y^2 + xy + y = x^3 + x^2 - 32952x - 1912599$, with conductor $1734$ |
| Generic density of odd order reductions |
$299/2688$ |