| Curve name |
$X_{189l}$ |
| 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} 1 & 0 \\ 4 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{189}$ |
| Curves that $X_{189l}$ minimally covers |
|
| Curves that minimally cover $X_{189l}$ |
|
| Curves that minimally cover $X_{189l}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{16} - 3456t^{15} - 48384t^{14} - 387072t^{13} - 2322432t^{12} -
15482880t^{11} - 108380160t^{10} - 571539456t^{9} - 2002157568t^{8} -
4572315648t^{7} - 6936330240t^{6} - 7927234560t^{5} - 9512681472t^{4} -
12683575296t^{3} - 12683575296t^{2} - 7247757312t - 1811939328\]
\[B(t) = 432t^{24} + 20736t^{23} + 456192t^{22} + 6082560t^{21} + 51093504t^{20}
+ 204374016t^{19} - 1021870080t^{18} - 23473815552t^{17} - 199994572800t^{16} -
1105566105600t^{15} - 4510709710848t^{14} - 14896859185152t^{13} -
43694916894720t^{12} - 119174873481216t^{11} - 288685421494272t^{10} -
566049846067200t^{9} - 819177770188800t^{8} - 769189988007936t^{7} -
267877110251520t^{6} + 428603376402432t^{5} + 857206752804864t^{4} +
816387383623680t^{3} + 489832430174208t^{2} + 178120883699712t +
29686813949952\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 483201x - 126236799$, with conductor $6720$ |
| Generic density of odd order reductions |
$299/2688$ |