| Curve name |
$X_{190f}$ |
| 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} 7 & 0 \\ 4 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{190}$ |
| Curves that $X_{190f}$ minimally covers |
|
| Curves that minimally cover $X_{190f}$ |
|
| Curves that minimally cover $X_{190f}$ 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 |
$109/896$ |