| Curve name |
$X_{214a}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 6 \\ 6 & 11 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 14 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 6 \\ 2 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{214}$ |
| Curves that $X_{214a}$ minimally covers |
|
| Curves that minimally cover $X_{214a}$ |
|
| Curves that minimally cover $X_{214a}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -324t^{32} - 10368t^{31} - 152064t^{30} - 1347840t^{29} - 7572096t^{28}
- 21192192t^{27} + 51480576t^{26} + 856507392t^{25} + 4249352448t^{24} +
10000668672t^{23} - 3253837824t^{22} - 114492690432t^{21} - 457246218240t^{20} -
1087813214208t^{19} - 1827992862720t^{18} - 2458090782720t^{17} -
3254846920704t^{16} - 4916181565440t^{15} - 7311971450880t^{14} -
8702505713664t^{13} - 7315939491840t^{12} - 3663766093824t^{11} -
208245620736t^{10} + 1280085590016t^{9} + 1087834226688t^{8} + 438531784704t^{7}
+ 52716109824t^{6} - 43401609216t^{5} - 31015305216t^{4} - 11041505280t^{3} -
2491416576t^{2} - 339738624t - 21233664\]
\[B(t) = 31104t^{47} + 1461888t^{46} + 30820608t^{45} + 378224640t^{44} +
2872723968t^{43} + 12224494080t^{42} + 4443724800t^{41} - 316891201536t^{40} -
2293834291200t^{39} - 8274204297216t^{38} - 10502321836032t^{37} +
60453069520896t^{36} + 460703815852032t^{35} + 1758470671687680t^{34} +
4395063003070464t^{33} + 5348086599647232t^{32} - 14820988081471488t^{31} -
124390992103538688t^{30} - 482825896433418240t^{29} - 1314635428745183232t^{28}
- 2668828023882252288t^{27} - 3948895346718670848t^{26} -
3643573881914523648t^{25} + 7287147763829047296t^{23} +
15795581386874683392t^{22} + 21350624191058018304t^{21} +
21034166859922931712t^{20} + 15450428685869383680t^{19} +
7961023494626476032t^{18} + 1897086474428350464t^{17} -
1369110169509691392t^{16} - 2250272257572077568t^{15} -
1800673967808184320t^{14} - 943521414864961536t^{13} - 247615772757590016t^{12}
+ 86035020480774144t^{11} + 135564563205586944t^{10} + 75164362054041600t^{9} +
20767781783863296t^{8} - 582447896985600t^{7} - 3204577776107520t^{6} -
1506134703734784t^{5} - 396597280112640t^{4} - 64635499708416t^{3} -
6131602685952t^{2} - 260919263232t\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 248504206997764x - 1507817332544964967680$, with conductor
$47524672$ |
| Generic density of odd order reductions |
$13411/86016$ |