| Curve name |
$X_{239a}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 15 & 15 \\ 6 & 3 \end{matrix}\right],
\left[ \begin{matrix} 31 & 0 \\ 2 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{239}$ |
| Curves that $X_{239a}$ minimally covers |
|
| Curves that minimally cover $X_{239a}$ |
|
| Curves that minimally cover $X_{239a}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -189t^{32} - 3456t^{31} - 9504t^{30} + 314496t^{29} + 4009824t^{28} +
16740864t^{27} - 57891456t^{26} - 1159156224t^{25} - 7549260480t^{24} -
30445811712t^{23} - 86535959040t^{22} - 190095316992t^{21} - 375246079488t^{20}
- 787829317632t^{19} - 1697577854976t^{18} - 3094579814400t^{17} -
4053122330112t^{16} - 3030202220544t^{15} + 355184123904t^{14} +
4076722814976t^{13} + 5345880809472t^{12} + 3577847611392t^{11} +
770395373568t^{10} - 974770274304t^{9} - 1266722979840t^{8} - 866659074048t^{7}
- 440531288064t^{6} - 183614570496t^{5} - 62395121664t^{4} - 16024338432t^{3} -
2788687872t^{2} - 283115520t - 12386304\]
\[B(t) = 918t^{48} + 28512t^{47} + 222912t^{46} - 2403648t^{45} - 57788640t^{44}
- 371952000t^{43} + 833317632t^{42} + 31569585408t^{41} + 248514630336t^{40} +
843453319680t^{39} - 2101305203712t^{38} - 47805188041728t^{37} -
359010401637888t^{36} - 1833245944621056t^{35} - 7066909877760000t^{34} -
20756908241178624t^{33} - 43739276560233984t^{32} - 49053453009960960t^{31} +
63180001798225920t^{30} + 489459682266808320t^{29} + 1422505823720325120t^{28} +
2772097602494201856t^{27} + 3905829301768814592t^{26} +
3838006907502723072t^{25} + 2001035405994393600t^{24} -
1052720510068850688t^{23} - 3914126315685937152t^{22} -
5412319688935342080t^{21} - 5306570353426563072t^{20} -
3922969766893977600t^{19} - 1612418743976067072t^{18} +
1077987637582626816t^{17} + 3121731360134922240t^{16} +
3569569567315329024t^{15} + 2450904811265064960t^{14} + 778430446093467648t^{13}
- 395078547957350400t^{12} - 740449871814721536t^{11} - 564186735307653120t^{10}
- 282303556013260800t^{9} - 98734905835388928t^{8} - 22914840196546560t^{7} -
2500883958988800t^{6} + 423303453868032t^{5} + 252507334901760t^{4} +
55883832754176t^{3} + 7066563379200t^{2} + 500095254528t + 15401484288\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 2x + 2$, with conductor $128$ |
| Generic density of odd order reductions |
$1461347/5505024$ |