| Curve name |
$X_{204b}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 6 \\ 4 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 6 \\ 4 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{204}$ |
| Curves that $X_{204b}$ minimally covers |
|
| Curves that minimally cover $X_{204b}$ |
|
| Curves that minimally cover $X_{204b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -7884t^{16} - 100224t^{15} - 680832t^{14} - 2225664t^{13} -
7402752t^{12} - 13547520t^{11} - 14321664t^{10} - 24993792t^{9} - 74041344t^{8}
+ 99975168t^{7} - 229146624t^{6} + 867041280t^{5} - 1895104512t^{4} +
2279079936t^{3} - 2788687872t^{2} + 1642070016t - 516685824\]
\[B(t) = -257040t^{24} - 5453568t^{23} - 48418560t^{22} - 308689920t^{21} -
1344397824t^{20} - 4317401088t^{19} - 11010650112t^{18} - 27196342272t^{17} -
40513499136t^{16} - 66762178560t^{15} - 51969392640t^{14} - 34780741632t^{13} -
401266704384t^{12} + 139122966528t^{11} - 831510282240t^{10} +
4272779427840t^{9} - 10371455778816t^{8} + 27849054486528t^{7} -
45099622858752t^{6} + 70736299425792t^{5} - 88106455793664t^{4} +
80921210388480t^{3} - 50770539970560t^{2} + 22873922076672t - 4312415600640\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 97x - 385$, with conductor $192$ |
| Generic density of odd order reductions |
$271/2688$ |