| Curve name |
$X_{189k}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 4 \\ 4 & 5 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 2 & 1 \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 $X_{189k}$ minimally covers |
|
| Curves that minimally cover $X_{189k}$ |
|
| Curves that minimally cover $X_{189k}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{22} - 1188t^{21} - 23868t^{20} - 289440t^{19} - 2455488t^{18} -
17086464t^{17} - 113799168t^{16} - 731234304t^{15} - 4032847872t^{14} -
17428414464t^{13} - 56930107392t^{12} - 139427315712t^{11} - 258102263808t^{10}
- 374391963648t^{9} - 466121392128t^{8} - 559889252352t^{7} - 643691446272t^{6}
- 606999674880t^{5} - 400438591488t^{4} - 159450660864t^{3} - 28991029248t^{2}\]
\[B(t) = 54t^{33} + 3564t^{32} + 110808t^{31} + 2156112t^{30} + 28963008t^{29} +
272408832t^{28} + 1621582848t^{27} + 2227544064t^{26} - 74151272448t^{25} -
1005155205120t^{24} - 7932310290432t^{23} - 46085316083712t^{22} -
212902085394432t^{21} - 819419579154432t^{20} - 2731112254144512t^{19} -
8126147447488512t^{18} - 21848898033156096t^{17} - 52442853065883648t^{16} -
109005867721949184t^{15} - 188765454678884352t^{14} - 259925943596875776t^{13} -
263495406090977280t^{12} - 155506489316868096t^{11} + 37371987911245824t^{10} +
217645165622329344t^{9} + 292496756145389568t^{8} + 248790344307572736t^{7} +
148166888424210432t^{6} + 60917342225301504t^{5} + 15674637765574656t^{4} +
1899956092796928t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 67950x + 6682500$, with conductor $630$ |
| Generic density of odd order reductions |
$271/2688$ |