| Curve name |
$X_{189a}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 0 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 4 \\ 4 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 6 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{189}$ |
| Curves that $X_{189a}$ minimally covers |
|
| Curves that minimally cover $X_{189a}$ |
|
| Curves that minimally cover $X_{189a}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{30} - 6480t^{29} - 178416t^{28} - 2975616t^{27} - 33723648t^{26}
- 281788416t^{25} - 1914734592t^{24} - 11913854976t^{23} - 71244693504t^{22} -
373498380288t^{21} - 1456889462784t^{20} - 3127591305216t^{19} +
3507376619520t^{18} + 56277929558016t^{17} + 221308004597760t^{16} +
450223436464128t^{15} + 224472103649280t^{14} - 1601326748270592t^{13} -
5967419239563264t^{12} - 12238794925277184t^{11} - 18676368933912576t^{10} -
24985164790628352t^{9} - 32123915832655872t^{8} - 37821000972238848t^{7} -
36210491315453952t^{6} - 25560346810908672t^{5} - 12260654161330176t^{4} -
3562417673994240t^{3} - 474989023199232t^{2}\]
\[B(t) = 432t^{45} + 38880t^{44} + 1653696t^{43} + 44136576t^{42} +
823592448t^{41} + 11235926016t^{40} + 111950622720t^{39} + 751866126336t^{38} +
2004964835328t^{37} - 25262826061824t^{36} - 438158780006400t^{35} -
3850950467911680t^{34} - 23333660799270912t^{33} - 101170906172227584t^{32} -
289184878040186880t^{31} - 288135579445493760t^{30} + 2067625456781230080t^{29}
+ 14064993346057666560t^{28} + 49907810542647508992t^{27} +
119739588348368388096t^{26} + 184523079839391940608t^{25} -
1476184638715135524864t^{23} - 7663333654295576838144t^{22} -
25552798997835524603904t^{21} - 57610212745452202229760t^{20} -
67751950967807347261440t^{19} + 75533013338159516221440t^{18} +
606464645351733995765760t^{17} + 1697366145767195377926144t^{16} +
3131790938400805865127936t^{15} + 4134926579549140754104320t^{14} +
3763755261165493341388800t^{13} + 1736048187841128865726464t^{12} -
1102241074862564535435264t^{11} - 3306742193749324090834944t^{10} -
3938912365356929557463040t^{9} - 3162632013676253854826496t^{8} -
1854565320958977301807104t^{7} - 795093868908038307446784t^{6} -
238322710060290552102912t^{5} - 44825588099114210426880t^{4} -
3984496719921263149056t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 213091788x + 1170570637712$, with conductor $141120$ |
| Generic density of odd order reductions |
$139/1344$ |