| Curve name |
$X_{204h}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 2 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 4 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{204}$ |
| Curves that $X_{204h}$ minimally covers |
|
| Curves that minimally cover $X_{204h}$ |
|
| Curves that minimally cover $X_{204h}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -1971t^{16} - 25056t^{15} - 170208t^{14} - 556416t^{13} - 1850688t^{12}
- 3386880t^{11} - 3580416t^{10} - 6248448t^{9} - 18510336t^{8} + 24993792t^{7} -
57286656t^{6} + 216760320t^{5} - 473776128t^{4} + 569769984t^{3} -
697171968t^{2} + 410517504t - 129171456\]
\[B(t) = 32130t^{24} + 681696t^{23} + 6052320t^{22} + 38586240t^{21} +
168049728t^{20} + 539675136t^{19} + 1376331264t^{18} + 3399542784t^{17} +
5064187392t^{16} + 8345272320t^{15} + 6496174080t^{14} + 4347592704t^{13} +
50158338048t^{12} - 17390370816t^{11} + 103938785280t^{10} - 534097428480t^{9} +
1296431972352t^{8} - 3481131810816t^{7} + 5637452857344t^{6} -
8842037428224t^{5} + 11013306974208t^{4} - 10115151298560t^{3} +
6346317496320t^{2} - 2859240259584t + 539051950080\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 24x + 36$, with conductor $48$ |
| Generic density of odd order reductions |
$53/896$ |