The modular curve $X_{189h}$

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
LevelIndex of imageCorresponding curve
$2$ $6$ $X_{8}$
$4$ $24$ $X_{58}$
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 $271/2688$

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