| Curve name |
$X_{189b}$ |
| 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} 1 & 4 \\ 0 & 5 \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 $X_{189b}$ minimally covers |
|
| Curves that minimally cover $X_{189b}$ |
|
| Curves that minimally cover $X_{189b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{22} - 4752t^{21} - 95472t^{20} - 1157760t^{19} - 9821952t^{18} -
68345856t^{17} - 455196672t^{16} - 2924937216t^{15} - 16131391488t^{14} -
69713657856t^{13} - 227720429568t^{12} - 557709262848t^{11} -
1032409055232t^{10} - 1497567854592t^{9} - 1864485568512t^{8} -
2239557009408t^{7} - 2574765785088t^{6} - 2427998699520t^{5} -
1601754365952t^{4} - 637802643456t^{3} - 115964116992t^{2}\]
\[B(t) = 432t^{33} + 28512t^{32} + 886464t^{31} + 17248896t^{30} +
231704064t^{29} + 2179270656t^{28} + 12972662784t^{27} + 17820352512t^{26} -
593210179584t^{25} - 8041241640960t^{24} - 63458482323456t^{23} -
368682528669696t^{22} - 1703216683155456t^{21} - 6555356633235456t^{20} -
21848898033156096t^{19} - 65009179579908096t^{18} - 174791184265248768t^{17} -
419542824527069184t^{16} - 872046941775593472t^{15} - 1510123637431074816t^{14}
- 2079407548775006208t^{13} - 2107963248727818240t^{12} -
1244051914534944768t^{11} + 298975903289966592t^{10} + 1741161324978634752t^{9}
+ 2339974049163116544t^{8} + 1990322754460581888t^{7} + 1185335107393683456t^{6}
+ 487338737802412032t^{5} + 125397102124597248t^{4} + 15199648742375424t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 4348812x + 3412742384$, with conductor $20160$ |
| Generic density of odd order reductions |
$11/112$ |