| Curve name |
$X_{228e}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 8 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{228}$ |
| Curves that $X_{228e}$ minimally covers |
|
| Curves that minimally cover $X_{228e}$ |
|
| Curves that minimally cover $X_{228e}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = 486t^{24} + 33696t^{23} + 1050624t^{22} + 19761408t^{21} +
250953984t^{20} + 2258896896t^{19} + 14508343296t^{18} + 62900748288t^{17} +
133924036608t^{16} - 402781372416t^{15} - 5273960841216t^{14} -
27155478085632t^{13} - 91803661369344t^{12} - 217243824685056t^{11} -
337533493837824t^{10} - 206224062676992t^{9} + 548552853946368t^{8} +
2061131719901184t^{7} + 3803275144986624t^{6} + 4737250143240192t^{5} +
4210309195628544t^{4} + 2652331283841024t^{3} + 1128098930098176t^{2} +
289446436012032t + 33397665693696\]
\[B(t) = 5103t^{36} + 373248t^{35} + 12204432t^{34} + 219691008t^{33} +
1792787904t^{32} - 15984304128t^{31} - 770855657472t^{30} - 13858803744768t^{29}
- 169061744001024t^{28} - 1570120425013248t^{27} - 11594918710738944t^{26} -
69335416427249664t^{25} - 336873109192704000t^{24} - 1314710158332395520t^{23} -
3972084924558606336t^{22} - 8325423280919937024t^{21} -
6618150161627480064t^{20} + 30199927916074106880t^{19} +
144044548694514597888t^{18} + 241599423328592855040t^{17} -
423561610344158724096t^{16} - 4262616719831007756288t^{15} -
16269659850992051552256t^{14} - 43080422468235936399360t^{13} -
88309264336212197376000t^{12} - 145406907231239487356928t^{11} -
194530455712508783099904t^{10} - 210737996131672516460544t^{9} -
181528665372280567627776t^{8} - 119046217690921782214656t^{7} -
52972797420461088571392t^{6} - 8787464125321958129664t^{5} +
7884764586336652886016t^{4} + 7729690170042567622656t^{3} +
3435242212966784827392t^{2} + 840479776858391445504t + 91927475593886564352\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + 6x + 7$, with conductor $144$ |
| Generic density of odd order reductions |
$299/2688$ |