| Curve name |
$X_{211k}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 5 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 9 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 24 & 7 \end{matrix}\right],
\left[ \begin{matrix} 15 & 0 \\ 24 & 1 \end{matrix}\right],
\left[ \begin{matrix} 11 & 11 \\ 8 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{211}$ |
| Curves that $X_{211k}$ minimally covers |
|
| Curves that minimally cover $X_{211k}$ |
|
| Curves that minimally cover $X_{211k}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{30} - 51840t^{29} - 923616t^{28} - 1451520t^{27} +
44051904t^{26} + 197406720t^{25} - 472476672t^{24} - 4293181440t^{23} -
3027373056t^{22} + 27510865920t^{21} + 46805409792t^{20} + 3344302080t^{19} -
78802993152t^{18} - 753370398720t^{17} - 320585859072t^{16} +
3013481594880t^{15} - 1260847890432t^{14} - 214035333120t^{13} +
11982184906752t^{12} - 28171126702080t^{11} - 12400120037376t^{10} +
70339484712960t^{9} - 30964231176192t^{8} - 51748987207680t^{7} +
46191769288704t^{6} + 6088116142080t^{5} - 15495705133056t^{4} +
3478923509760t^{3} - 28991029248t^{2}\]
\[B(t) = 432t^{45} - 435456t^{44} - 28797120t^{43} - 369266688t^{42} -
115250688t^{41} + 26210967552t^{40} + 129520982016t^{39} - 490163208192t^{38} -
5035364683776t^{37} - 3008645627904t^{36} + 70673895014400t^{35} +
187202436857856t^{34} - 256411313897472t^{33} - 1945681061216256t^{32} -
2328773561155584t^{31} + 6744453112922112t^{30} + 20817234647580672t^{29} +
10041225079947264t^{28} - 1641881800802304t^{27} - 153803164879945728t^{26} -
571120875365990400t^{25} + 696081715046645760t^{24} + 2284483501463961600t^{23}
- 2460850638079131648t^{22} + 105080435251347456t^{21} +
2570553620466499584t^{20} - 21316848279122608128t^{19} +
27625279950528970752t^{18} + 38154626025973088256t^{17} -
127512154027868553216t^{16} + 67216687470338899968t^{15} +
196295982430663213056t^{14} - 296427800554477977600t^{13} -
50476697566801035264t^{12} + 337917603753926590464t^{11} -
131577184305442455552t^{10} - 139072095476131037184t^{9} +
112575248432357179392t^{8} + 1979991743205998592t^{7} -
25375813575395770368t^{6} + 7915692071615201280t^{5} - 478788935384825856t^{4} -
1899956092796928t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 54212659788x + 4858466207610512$, with conductor $141120$ |
| Generic density of odd order reductions |
$139/1344$ |