| Curve name |
$X_{189h}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 6 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 4 \\ 4 & 3 \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 $X_{189h}$ minimally covers |
|
| Curves that minimally cover $X_{189h}$ |
|
| Curves that minimally cover $X_{189h}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{24} - 5184t^{23} - 110592t^{22} - 1368576t^{21} - 11045376t^{20}
- 67129344t^{19} - 396361728t^{18} - 2518843392t^{17} - 13256441856t^{16} -
41094217728t^{15} - 7247757312t^{14} + 530105499648t^{13} + 2316451184640t^{12}
+ 4240843997184t^{11} - 463856467968t^{10} - 21040239476736t^{9} -
54298385842176t^{8} - 82537460269056t^{7} - 103903848824832t^{6} -
140780438028288t^{5} - 185310658953216t^{4} - 183687161315328t^{3} -
118747255799808t^{2} - 44530220924928t - 7421703487488\]
\[B(t) = 432t^{36} + 31104t^{35} + 1036800t^{34} + 21150720t^{33} +
289889280t^{32} + 2670133248t^{31} + 13831962624t^{30} - 19216465920t^{29} -
1151724552192t^{28} - 12130962112512t^{27} - 75919691612160t^{26} -
299648786890752t^{25} - 573301906735104t^{24} + 1112614096601088t^{23} +
12222654169743360t^{22} + 44035402037723136t^{21} + 90313463124983808t^{20} +
105403728714006528t^{19} - 843229829712052224t^{17} - 5780061639998963712t^{16}
- 22546125843314245632t^{15} - 50063991479268802560t^{14} -
36458138717424451584t^{13} + 150287655039167102976t^{12} +
628409052725514338304t^{11} + 1273721064830596546560t^{10} +
1628190173195441012736t^{9} + 1236654821416221278208t^{8} +
165068185342197104640t^{7} - 950525233753189515264t^{6} -
1467921276943648948224t^{5} - 1274946536510450565120t^{4} -
744174802426700759040t^{3} - 291833255853608140800t^{2} - 70039981404865953792t
- 7782220156096217088\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 23676865x + 43346575775$, with conductor $47040$ |
| Generic density of odd order reductions |
$271/2688$ |