| Curve name |
$X_{214c}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 12 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 4 \\ 0 & 15 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 2 & 9 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{214}$ |
| Curves that $X_{214c}$ minimally covers |
|
| Curves that minimally cover $X_{214c}$ |
|
| Curves that minimally cover $X_{214c}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -81t^{32} - 2592t^{31} - 38016t^{30} - 336960t^{29} - 1893024t^{28} -
5298048t^{27} + 12870144t^{26} + 214126848t^{25} + 1062338112t^{24} +
2500167168t^{23} - 813459456t^{22} - 28623172608t^{21} - 114311554560t^{20} -
271953303552t^{19} - 456998215680t^{18} - 614522695680t^{17} -
813711730176t^{16} - 1229045391360t^{15} - 1827992862720t^{14} -
2175626428416t^{13} - 1828984872960t^{12} - 915941523456t^{11} -
52061405184t^{10} + 320021397504t^{9} + 271958556672t^{8} + 109632946176t^{7} +
13179027456t^{6} - 10850402304t^{5} - 7753826304t^{4} - 2760376320t^{3} -
622854144t^{2} - 84934656t - 5308416\]
\[B(t) = 3888t^{47} + 182736t^{46} + 3852576t^{45} + 47278080t^{44} +
359090496t^{43} + 1528061760t^{42} + 555465600t^{41} - 39611400192t^{40} -
286729286400t^{39} - 1034275537152t^{38} - 1312790229504t^{37} +
7556633690112t^{36} + 57587976981504t^{35} + 219808833960960t^{34} +
549382875383808t^{33} + 668510824955904t^{32} - 1852623510183936t^{31} -
15548874012942336t^{30} - 60353237054177280t^{29} - 164329428593147904t^{28} -
333603502985281536t^{27} - 493611918339833856t^{26} - 455446735239315456t^{25} +
910893470478630912t^{23} + 1974447673359335424t^{22} + 2668828023882252288t^{21}
+ 2629270857490366464t^{20} + 1931303585733672960t^{19} +
995127936828309504t^{18} + 237135809303543808t^{17} - 171138771188711424t^{16} -
281284032196509696t^{15} - 225084245976023040t^{14} - 117940176858120192t^{13} -
30951971594698752t^{12} + 10754377560096768t^{11} + 16945570400698368t^{10} +
9395545256755200t^{9} + 2595972722982912t^{8} - 72805987123200t^{7} -
400572222013440t^{6} - 188266837966848t^{5} - 49574660014080t^{4} -
8079437463552t^{3} - 766450335744t^{2} - 32614907904t\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 62126051749441x - 188477166568120620960$, with conductor
$23762336$ |
| Generic density of odd order reductions |
$4769/28672$ |