| Curve name |
$X_{189j}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 0 \\ 2 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 4 \\ 4 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 4 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 4 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{189}$ |
| Curves that $X_{189j}$ minimally covers |
|
| Curves that minimally cover $X_{189j}$ |
|
| Curves that minimally cover $X_{189j}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{30} - 1620t^{29} - 44604t^{28} - 743904t^{27} - 8430912t^{26} -
70447104t^{25} - 478683648t^{24} - 2978463744t^{23} - 17811173376t^{22} -
93374595072t^{21} - 364222365696t^{20} - 781897826304t^{19} + 876844154880t^{18}
+ 14069482389504t^{17} + 55327001149440t^{16} + 112555859116032t^{15} +
56118025912320t^{14} - 400331687067648t^{13} - 1491854809890816t^{12} -
3059698731319296t^{11} - 4669092233478144t^{10} - 6246291197657088t^{9} -
8030978958163968t^{8} - 9455250243059712t^{7} - 9052622828863488t^{6} -
6390086702727168t^{5} - 3065163540332544t^{4} - 890604418498560t^{3} -
118747255799808t^{2}\]
\[B(t) = 54t^{45} + 4860t^{44} + 206712t^{43} + 5517072t^{42} + 102949056t^{41}
+ 1404490752t^{40} + 13993827840t^{39} + 93983265792t^{38} + 250620604416t^{37}
- 3157853257728t^{36} - 54769847500800t^{35} - 481368808488960t^{34} -
2916707599908864t^{33} - 12646363271528448t^{32} - 36148109755023360t^{31} -
36016947430686720t^{30} + 258453182097653760t^{29} + 1758124168257208320t^{28} +
6238476317830938624t^{27} + 14967448543546048512t^{26} +
23065384979923992576t^{25} - 184523079839391940608t^{23} -
957916706786947104768t^{22} - 3194099874729440575488t^{21} -
7201276593181525278720t^{20} - 8468993870975918407680t^{19} +
9441626667269939527680t^{18} + 75808080668966749470720t^{17} +
212170768220899422240768t^{16} + 391473867300100733140992t^{15} +
516865822443642594263040t^{14} + 470469407645686667673600t^{13} +
217006023480141108215808t^{12} - 137780134357820566929408t^{11} -
413342774218665511354368t^{10} - 492364045669616194682880t^{9} -
395329001709531731853312t^{8} - 231820665119872162725888t^{7} -
99386733613504788430848t^{6} - 29790338757536319012864t^{5} -
5603198512389276303360t^{4} - 498062089990157893632t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 3329559x - 2285438387$, with conductor $4410$ |
| Generic density of odd order reductions |
$271/2688$ |