| Curve name |
$X_{238d}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 31 & 31 \\ 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 $X_{238d}$ minimally covers |
|
| Curves that minimally cover $X_{238d}$ |
|
| Curves that minimally cover $X_{238d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = 231928233984t^{32} - 5566277615616t^{31} + 30614526885888t^{30} +
37572373905408t^{29} - 517779782369280t^{28} - 7305739370496t^{27} +
3287988591132672t^{26} - 47400332820480t^{25} - 9525962987274240t^{24} -
16285710680064t^{23} + 14613511736918016t^{22} + 9126738395136t^{21} -
12784184586141696t^{20} + 6782541889536t^{19} + 6673860636180480t^{18} +
1039260450816t^{17} - 2127098747289600t^{16} - 259815112704t^{15} +
417116289761280t^{14} - 105977217024t^{13} - 49938221039616t^{12} -
8912830464t^{11} + 3567751888896t^{10} + 994000896t^{9} - 145354659840t^{8} +
180817920t^{7} + 3135670272t^{6} + 1741824t^{5} - 30862080t^{4} - 559872t^{3} +
114048t^{2} + 5184t + 54\]
\[B(t) = 151996487423754240t^{48} - 2188749418902061056t^{47} +
3830311483078606848t^{46} + 80436541144650743808t^{45} -
369465461805290618880t^{44} - 695064737340085764096t^{43} +
5849558219117882769408t^{42} + 2021959873339789934592t^{41} -
44782252713077140094976t^{40} - 1732660208935926497280t^{39} +
194180518355667026706432t^{38} - 805124206411068211200t^{37} -
517071234623214194786304t^{36} + 1237789629566272143360t^{35} +
894895700817073636638720t^{34} + 221919339504180658176t^{33} -
1045933513852795748352000t^{32} - 335602522098260508672t^{31} +
847677266770429269245952t^{30} - 61961770725707612160t^{29} -
485184263414926641463296t^{28} + 41797859093099053056t^{27} +
198484929953037721534464t^{26} + 9477421538551529472t^{25} -
58430586996037936742400t^{24} - 2369355384637882368t^{23} +
12405308122064857595904t^{22} - 653091548329672704t^{21} -
1895251028964557193216t^{20} + 60509541724323840t^{19} +
206952457707624333312t^{18} + 20483552374161408t^{17} -
15959678861279232000t^{16} - 846555097595904t^{15} + 853439045731614720t^{14} -
295112044707840t^{13} - 30819847263289344t^{12} + 11997285580800t^{11} +
723378801180672t^{10} + 1613665566720t^{9} - 10426680723456t^{8} -
117693554688t^{7} + 85122275328t^{6} + 2528630784t^{5} - 336026880t^{4} -
18289152t^{3} + 217728t^{2} + 31104t + 540\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 + x + 1$, with conductor $128$ |
| Generic density of odd order reductions |
$410657/1835008$ |