| Curve name |
$X_{189i}$ |
| 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} 1 & 0 \\ 6 & 7 \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 $X_{189i}$ minimally covers |
|
| Curves that minimally cover $X_{189i}$ |
|
| Curves that minimally cover $X_{189i}$ 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 |
$81/896$ |