| Curve name |
$X_{190e}$ |
| 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 & 0 \\ 4 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 2 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{190}$ |
| Curves that $X_{190e}$ minimally covers |
|
| Curves that minimally cover $X_{190e}$ |
|
| Curves that minimally cover $X_{190e}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -12636t^{24} - 642816t^{23} - 16274304t^{22} - 273881088t^{21} -
3454631424t^{20} - 34821881856t^{19} - 290593529856t^{18} - 2043304869888t^{17}
- 12186818592768t^{16} - 61714710134784t^{15} - 265037942882304t^{14} -
963801784516608t^{13} - 2964589469761536t^{12} - 7710414276132864t^{11} -
16962428344467456t^{10} - 31597931589009408t^{9} - 49917208955977728t^{8} -
66955013976489984t^{7} - 76177350290571264t^{6} - 73026779178074112t^{5} -
57959097600835584t^{4} - 36759697373528064t^{3} - 17474400861290496t^{2} -
5521747394691072t - 868339308036096\]
\[B(t) = 408240t^{36} + 26500608t^{35} + 781519104t^{34} + 12800692224t^{33} +
90776153088t^{32} - 1103964733440t^{31} - 45355934416896t^{30} -
804958477221888t^{29} - 10099087302721536t^{28} - 99930964041400320t^{27} -
817466125213237248t^{26} - 5676411356681601024t^{25} -
34041681707676991488t^{24} - 178513103298169405440t^{23} -
826351701931793055744t^{22} - 3401933683149596786688t^{21} -
12527514578175664324608t^{20} - 41439469243504986685440t^{19} -
123452570890783105744896t^{18} - 331515753948039893483520t^{17} -
801760933003242516774912t^{16} - 1741790045772593554784256t^{15} -
3384736571112624356327424t^{14} - 5849517368874415077457920t^{13} -
8923822609577277256630272t^{12} - 11904297429487532950683648t^{11} -
13714805755385527368941568t^{10} - 13412506950486448888872960t^{9} -
10843812421159462228721664t^{8} - 6914540668611939790749696t^{7} -
3116836080001426397331456t^{6} - 606911030535956170014720t^{5} +
399237743380120995889152t^{4} + 450384318203834419642368t^{3} +
219978071597332744372224t^{2} + 59674064156945792630784t +
7354198047510925148160\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 156x + 560$, with conductor $576$ |
| Generic density of odd order reductions |
$109/896$ |