| Curve name |
$X_{231b}$ |
| 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 & 0 \\ 12 & 15 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{231}$ |
| Curves that $X_{231b}$ minimally covers |
|
| Curves that minimally cover $X_{231b}$ |
|
| Curves that minimally cover $X_{231b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = 324t^{32} + 10368t^{31} + 117504t^{30} + 311040t^{29} - 5042304t^{28} -
51383808t^{27} - 145345536t^{26} + 552434688t^{25} + 5580893952t^{24} +
15302780928t^{23} - 11072802816t^{22} - 200693403648t^{21} - 601120880640t^{20}
- 581783150592t^{19} + 1518611742720t^{18} + 6835446005760t^{17} +
12881597405184t^{16} + 13670892011520t^{15} + 6074446970880t^{14} -
4654265204736t^{13} - 9617934090240t^{12} - 6422188916736t^{11} -
708659380224t^{10} + 1958755958784t^{9} + 1428708851712t^{8} + 282846560256t^{7}
- 148833828864t^{6} - 105234038784t^{5} - 20653277184t^{4} + 2548039680t^{3} +
1925185536t^{2} + 339738624t + 21233664\]
\[B(t) = 31104t^{47} + 1461888t^{46} + 29272320t^{45} + 308551680t^{44} +
1502489088t^{43} - 2755192320t^{42} - 86400525312t^{41} - 487524925440t^{40} -
389037496320t^{39} + 9962739431424t^{38} + 57512867622912t^{37} +
82008434147328t^{36} - 560259735330816t^{35} - 3424903253729280t^{34} -
6522010118111232t^{33} + 12232033673674752t^{32} + 105581416085913600t^{31} +
269075419698364416t^{30} + 153703495558103040t^{29} - 1181467369902440448t^{28}
- 4634555790565048320t^{27} - 9020880097990410240t^{26} -
9450729185915437056t^{25} + 18901458371830874112t^{23} +
36083520391961640960t^{22} + 37076446324520386560t^{21} +
18903477918439047168t^{20} - 4918511857859297280t^{19} -
17220826860695322624t^{18} - 13514421258996940800t^{17} -
3131400620460736512t^{16} + 3339269180472950784t^{15} +
3507100931818782720t^{14} + 1147411937957511168t^{13} - 335906546267455488t^{12}
- 471145411566895104t^{11} - 163229522844450816t^{10} + 12747980679413760t^{9} +
31950433513635840t^{8} + 11324689653694464t^{7} + 722257135534080t^{6} -
787736998969344t^{5} - 323539886407680t^{4} - 61388504432640t^{3} -
6131602685952t^{2} - 260919263232t\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 62109737706236x - 188581099813546343680$, with conductor
$23762336$ |
| Generic density of odd order reductions |
$4769/28672$ |