| Curve name |
$X_{231d}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 3 \\ 4 & 13 \end{matrix}\right],
\left[ \begin{matrix} 3 & 6 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 0 & 13 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{231}$ |
| Curves that $X_{231d}$ minimally covers |
|
| Curves that minimally cover $X_{231d}$ |
|
| Curves that minimally cover $X_{231d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = 81t^{32} + 2592t^{31} + 29376t^{30} + 77760t^{29} - 1260576t^{28} -
12845952t^{27} - 36336384t^{26} + 138108672t^{25} + 1395223488t^{24} +
3825695232t^{23} - 2768200704t^{22} - 50173350912t^{21} - 150280220160t^{20} -
145445787648t^{19} + 379652935680t^{18} + 1708861501440t^{17} +
3220399351296t^{16} + 3417723002880t^{15} + 1518611742720t^{14} -
1163566301184t^{13} - 2404483522560t^{12} - 1605547229184t^{11} -
177164845056t^{10} + 489688989696t^{9} + 357177212928t^{8} + 70711640064t^{7} -
37208457216t^{6} - 26308509696t^{5} - 5163319296t^{4} + 637009920t^{3} +
481296384t^{2} + 84934656t + 5308416\]
\[B(t) = 3888t^{47} + 182736t^{46} + 3659040t^{45} + 38568960t^{44} +
187811136t^{43} - 344399040t^{42} - 10800065664t^{41} - 60940615680t^{40} -
48629687040t^{39} + 1245342428928t^{38} + 7189108452864t^{37} +
10251054268416t^{36} - 70032466916352t^{35} - 428112906716160t^{34} -
815251264763904t^{33} + 1529004209209344t^{32} + 13197677010739200t^{31} +
33634427462295552t^{30} + 19212936944762880t^{29} - 147683421237805056t^{28} -
579319473820631040t^{27} - 1127610012248801280t^{26} - 1181341148239429632t^{25}
+ 2362682296478859264t^{23} + 4510440048995205120t^{22} +
4634555790565048320t^{21} + 2362934739804880896t^{20} - 614813982232412160t^{19}
- 2152603357586915328t^{18} - 1689302657374617600t^{17} -
391425077557592064t^{16} + 417408647559118848t^{15} + 438387616477347840t^{14} +
143426492244688896t^{13} - 41988318283431936t^{12} - 58893176445861888t^{11} -
20403690355556352t^{10} + 1593497584926720t^{9} + 3993804189204480t^{8} +
1415586206711808t^{7} + 90282141941760t^{6} - 98467124871168t^{5} -
40442485800960t^{4} - 7673563054080t^{3} - 766450335744t^{2} - 32614907904t\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 15527434426559x - 23572637476693292960$, with conductor
$47524672$ |
| Generic density of odd order reductions |
$13411/86016$ |