| Curve name |
$X_{204e}$ |
| 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 \\ 4 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{204}$ |
| Curves that $X_{204e}$ minimally covers |
|
| Curves that minimally cover $X_{204e}$ |
|
| Curves that minimally cover $X_{204e}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -70956t^{24} - 1280448t^{23} - 12578112t^{22} - 75071232t^{21} -
337087872t^{20} - 1100141568t^{19} - 2935194624t^{18} - 5721477120t^{17} -
10318648320t^{16} - 15232499712t^{15} - 31467405312t^{14} - 21888368640t^{13} -
33873002496t^{12} + 87553474560t^{11} - 503478484992t^{10} + 974879981568t^{9} -
2641573969920t^{8} + 5858792570880t^{7} - 12022557179904t^{6} +
18024719450112t^{5} - 22091390779392t^{4} + 19679473041408t^{3} -
13189106368512t^{2} + 5370588168192t - 1190444138496\]
\[B(t) = 6940080t^{36} + 202766976t^{35} + 2799888768t^{34} + 26266056192t^{33}
+ 180833575680t^{32} + 970623959040t^{31} + 4155648786432t^{30} +
14694773096448t^{29} + 43586085961728t^{28} + 112372304707584t^{27} +
249245897195520t^{26} + 493863131676672t^{25} + 828976065675264t^{24} +
1422508381175808t^{23} + 2191642652049408t^{22} + 4079057293541376t^{21} +
4644612423548928t^{20} + 4084939754569728t^{19} + 6302806944251904t^{18} -
16339759018278912t^{17} + 74313798776782848t^{16} - 261059666786648064t^{15} +
561060518924648448t^{14} - 1456648582324027392t^{13} + 3395485965005881344t^{12}
- 8091453549390594048t^{11} + 16334579118605598720t^{10} -
29457725445264900096t^{9} + 45703323673404899328t^{8} -
61634345577524232192t^{7} + 69720217310107533312t^{6} -
65137471262356930560t^{5} + 48542143347771310080t^{4} -
28202963084884574208t^{3} + 12025430690997731328t^{2} - 3483510122515267584t +
476918666105978880\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 876x + 9520$, with conductor $576$ |
| Generic density of odd order reductions |
$109/896$ |