| Curve name |
$X_{226f}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 6 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 9 & 9 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 13 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{226}$ |
| Curves that $X_{226f}$ minimally covers |
|
| Curves that minimally cover $X_{226f}$ |
|
| Curves that minimally cover $X_{226f}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -891t^{32} - 22032t^{31} - 200016t^{30} - 557280t^{29} + 3524256t^{28}
+ 33659712t^{27} + 80961984t^{26} - 247197312t^{25} - 2137589568t^{24} -
4632913152t^{23} + 8099599104t^{22} + 78449859072t^{21} + 213957158400t^{20} +
131875734528t^{19} - 1149101614080t^{18} - 5345979402240t^{17} -
13764204274176t^{16} - 25101750251520t^{15} - 34381912043520t^{14} -
35562016382976t^{13} - 26727108157440t^{12} - 12575153995776t^{11} -
746820255744t^{10} + 4394642079744t^{9} + 4013768097792t^{8} +
1817111494656t^{7} + 303151251456t^{6} - 159008292864t^{5} - 133350948864t^{4} -
47669575680t^{3} - 9859497984t^{2} - 1146617856t - 58392576\]
\[B(t) = -10206t^{48} - 377136t^{47} - 5695920t^{46} - 40237344t^{45} -
40847328t^{44} + 1645645248t^{43} + 14006488896t^{42} + 43341194880t^{41} -
89857252800t^{40} - 1434710617344t^{39} - 6128131832064t^{38} -
9509456899584t^{37} + 34840642337280t^{36} + 286253610992640t^{35} +
1030168761154560t^{34} + 2152820386879488t^{33} + 398287080930816t^{32} -
19398629310455808t^{31} - 99865808111001600t^{30} - 303693208499994624t^{29} -
572188927474778112t^{28} - 267451815609925632t^{27} + 2611412598283075584t^{26}
+ 11912743745126203392t^{25} + 32185421015687331840t^{24} +
65042998170011172864t^{23} + 104675676442886799360t^{22} +
136926588726828859392t^{21} + 145473375836328689664t^{20} +
122666365844281884672t^{19} + 76507858972259647488t^{18} +
26682360121292488704t^{17} - 8061796785275928576t^{16} -
20801771267360292864t^{15} - 17307768354814033920t^{14} -
8345216215280517120t^{13} - 1564216307969163264t^{12} +
1103032859611889664t^{11} + 1181233857354006528t^{10} + 560345408757301248t^{9}
+ 133494881283735552t^{8} - 10449067708514304t^{7} - 23488306763268096t^{6} -
10775082251059200t^{5} - 2991219202326528t^{4} - 556238194606080t^{3} -
68344539512832t^{2} - 5055310725120t - 171228266496\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 62142366078251x - 188373226044392714094$, with conductor
$23762336$ |
| Generic density of odd order reductions |
$4769/28672$ |