| Curve name |
$X_{234f}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 0 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{234}$ |
| Curves that $X_{234f}$ minimally covers |
|
| Curves that minimally cover $X_{234f}$ |
|
| Curves that minimally cover $X_{234f}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -280179t^{24} - 5218992t^{23} - 49333968t^{22} - 304613568t^{21} -
1340147808t^{20} - 4438409472t^{19} - 11416156416t^{18} - 23670973440t^{17} -
41993821440t^{16} - 62320140288t^{15} - 100312915968t^{14} - 81966366720t^{13} -
303563096064t^{12} + 327865466880t^{11} - 1605006655488t^{10} +
3988488978432t^{9} - 10750418288640t^{8} + 24239076802560t^{7} -
46760576679936t^{6} + 72718900789248t^{5} - 87827926745088t^{4} +
79852619169792t^{3} - 51730414829568t^{2} + 21890039021568t - 4700623601664\]
\[B(t) = 57082158t^{36} + 1594947024t^{35} + 22503701232t^{34} +
210454932672t^{33} + 1447355918304t^{32} + 7738282543104t^{31} +
33263197267968t^{30} + 117802193412096t^{29} + 350412448978944t^{28} +
892277877620736t^{27} + 1980589216776192t^{26} + 3911009424703488t^{25} +
7000732924084224t^{24} + 11527629250756608t^{23} + 17940542312153088t^{22} +
24898922990272512t^{21} + 37516577293467648t^{20} + 28837109362065408t^{19} +
97727264235454464t^{18} - 115348437448261632t^{17} + 600265236695482368t^{16} -
1593531071377440768t^{15} + 4592778831911190528t^{14} -
11804292352774766592t^{13} + 28675002057048981504t^{12} -
64077978414341947392t^{11} + 129799894910644518912t^{10} -
233905291951010217984t^{9} + 367434084100545183744t^{8} -
494098211037127901184t^{7} + 558063845415309017088t^{6} -
519307350778740473856t^{5} + 388521645924232986624t^{4} -
225974263277030473728t^{3} + 96652660830394908672t^{2} - 27400981227730108416t +
3922656028721676288\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 3459x + 78302$, with conductor $72$ |
| Generic density of odd order reductions |
$299/2688$ |