| Curve name |
$X_{238c}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 31 & 29 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 17 & 19 \\ 2 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{238}$ |
| Curves that $X_{238c}$ minimally covers |
|
| Curves that minimally cover $X_{238c}$ |
|
| Curves that minimally cover $X_{238c}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = 927712935936t^{32} - 22265110462464t^{31} + 122458107543552t^{30} +
150289495621632t^{29} - 2071119129477120t^{28} - 29222957481984t^{27} +
13151954364530688t^{26} - 189601331281920t^{25} - 38103851949096960t^{24} -
65142842720256t^{23} + 58454046947672064t^{22} + 36506953580544t^{21} -
51136738344566784t^{20} + 27130167558144t^{19} + 26695442544721920t^{18} +
4157041803264t^{17} - 8508394989158400t^{16} - 1039260450816t^{15} +
1668465159045120t^{14} - 423908868096t^{13} - 199752884158464t^{12} -
35651321856t^{11} + 14271007555584t^{10} + 3976003584t^{9} - 581418639360t^{8} +
723271680t^{7} + 12542681088t^{6} + 6967296t^{5} - 123448320t^{4} - 2239488t^{3}
+ 456192t^{2} + 20736t + 216\]
\[B(t) = -1215971899390033920t^{48} + 17509995351216488448t^{47} -
30642491864628854784t^{46} - 643492329157205950464t^{45} +
2955723694442324951040t^{44} + 5560517898720686112768t^{43} -
46796465752943062155264t^{42} - 16175678986718319476736t^{41} +
358258021704617120759808t^{40} + 13861281671487411978240t^{39} -
1553444146845336213651456t^{38} + 6440993651288545689600t^{37} +
4136569876985713558290432t^{36} - 9902317036530177146880t^{35} -
7159165606536589093109760t^{34} - 1775354716033445265408t^{33} +
8367468110822365986816000t^{32} + 2684820176786084069376t^{31} -
6781418134163434153967616t^{30} + 495694165805660897280t^{29} +
3881474107319413131706368t^{28} - 334382872744792424448t^{27} -
1587879439624301772275712t^{26} - 75819372308412235776t^{25} +
467444695968303493939200t^{24} + 18954843077103058944t^{23} -
99242464976518860767232t^{22} + 5224732386637381632t^{21} +
15162008231716457545728t^{20} - 484076333794590720t^{19} -
1655619661660994666496t^{18} - 163868418993291264t^{17} +
127677430890233856000t^{16} + 6772440780767232t^{15} - 6827512365852917760t^{14}
+ 2360896357662720t^{13} + 246558778106314752t^{12} - 95978284646400t^{11} -
5787030409445376t^{10} - 12909324533760t^{9} + 83413445787648t^{8} +
941548437504t^{7} - 680978202624t^{6} - 20229046272t^{5} + 2688215040t^{4} +
146313216t^{3} - 1741824t^{2} - 248832t - 4320\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 + 3x - 5$, with conductor $128$ |
| Generic density of odd order reductions |
$1461347/5505024$ |