| Curve name |
$X_{207h}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 3 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 7 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 8 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{207}$ |
| Curves that $X_{207h}$ minimally covers |
|
| Curves that minimally cover $X_{207h}$ |
|
| Curves that minimally cover $X_{207h}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{30} - 1620t^{29} - 18684t^{28} + 707616t^{27} + 27857088t^{26} +
457906176t^{25} + 4372296192t^{24} + 24565579776t^{23} + 56918052864t^{22} -
256873807872t^{21} - 2776578932736t^{20} - 10229179613184t^{19} -
8457474539520t^{18} + 83797437579264t^{17} + 394804026408960t^{16} +
670379500634112t^{15} - 541278370529280t^{14} - 5237339961950208t^{13} -
11372867308486656t^{12} - 8417240936349696t^{11} + 14920726049980416t^{10} +
51517754758397952t^{9} + 73354957629161472t^{8} + 61459126579888128t^{7} +
29911320480448512t^{6} + 6078375156252672t^{5} - 1283954703335424t^{4} -
890604418498560t^{3} - 118747255799808t^{2}\]
\[B(t) = 54t^{45} + 4860t^{44} + 315576t^{43} + 14879376t^{42} + 468732096t^{41}
+ 9902850048t^{40} + 141713072640t^{39} + 1321770617856t^{38} +
6255914176512t^{37} - 22568405483520t^{36} - 693399667802112t^{35} -
6442637430620160t^{34} - 34250733920452608t^{33} - 87403012024172544t^{32} +
192424738629353472t^{31} + 2859595714969731072t^{30} +
12252095728645570560t^{29} + 18326371229025435648t^{28} -
75898766525051437056t^{27} - 529474459814414254080t^{26} -
1302108677984841891840t^{25} + 10416869423878735134720t^{23} +
33886365428122512261120t^{22} + 38860168460826335772672t^{21} -
75064816554088184414208t^{20} - 401476672836258056110080t^{19} -
749625859105025182138368t^{18} - 403543925466025892511744t^{17} +
1466379211780139991957504t^{16} + 4597055689135681777434624t^{15} +
6917729266124764049571840t^{14} + 5956257792534670479458304t^{13} +
1550889015593367471390720t^{12} - 3439225189721831852998656t^{11} -
5813208654341359244673024t^{10} - 4986085477617437748756480t^{9} -
2787404486629918651711488t^{8} - 1055490846441095779319808t^{7} -
268043008836422005161984t^{6} - 45479294592226292662272t^{5} -
5603198512389276303360t^{4} - 498062089990157893632t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 + 560061x - 7308493655$, with conductor $4410$ |
| Generic density of odd order reductions |
$271/2688$ |