| Curve name |
$X_{181d}$ |
| 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} 1 & 0 \\ 0 & 7 \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 $X_{181d}$ minimally covers |
|
| Curves that minimally cover $X_{181d}$ |
|
| Curves that minimally cover $X_{181d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -7421703487488t^{24} - 18554258718720t^{22} - 16930761080832t^{20} -
6551972610048t^{18} - 1244802318336t^{16} - 588880281600t^{14} -
242686623744t^{12} - 9201254400t^{10} - 303906816t^{8} - 24993792t^{6} -
1009152t^{4} - 17280t^{2} - 108\]
\[B(t) = -7782220156096217088t^{36} - 29183325585360814080t^{34} -
44869363087492251648t^{32} - 35992768221945004032t^{30} -
14705660158248222720t^{28} - 678284325128503296t^{26} +
2579309143227629568t^{24} + 1308832253425483776t^{22} + 237053847955046400t^{20}
+ 12907038149443584t^{18} + 3703966374297600t^{16} + 319539124371456t^{14} +
9839283535872t^{12} - 40428896256t^{10} - 13695713280t^{8} - 523763712t^{6} -
10202112t^{4} - 103680t^{2} - 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 2108929x + 972923809$, with conductor $55488$ |
| Generic density of odd order reductions |
$109/896$ |