The modular curve $X_{189j}$

Curve name $X_{189j}$
Index $96$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 7 & 0 \\ 2 & 1 \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} 7 & 0 \\ 4 & 7 \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 $X_{189j}$ minimally covers
Curves that minimally cover $X_{189j}$
Curves that minimally cover $X_{189j}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -27t^{30} - 1620t^{29} - 44604t^{28} - 743904t^{27} - 8430912t^{26} - 70447104t^{25} - 478683648t^{24} - 2978463744t^{23} - 17811173376t^{22} - 93374595072t^{21} - 364222365696t^{20} - 781897826304t^{19} + 876844154880t^{18} + 14069482389504t^{17} + 55327001149440t^{16} + 112555859116032t^{15} + 56118025912320t^{14} - 400331687067648t^{13} - 1491854809890816t^{12} - 3059698731319296t^{11} - 4669092233478144t^{10} - 6246291197657088t^{9} - 8030978958163968t^{8} - 9455250243059712t^{7} - 9052622828863488t^{6} - 6390086702727168t^{5} - 3065163540332544t^{4} - 890604418498560t^{3} - 118747255799808t^{2}\] \[B(t) = 54t^{45} + 4860t^{44} + 206712t^{43} + 5517072t^{42} + 102949056t^{41} + 1404490752t^{40} + 13993827840t^{39} + 93983265792t^{38} + 250620604416t^{37} - 3157853257728t^{36} - 54769847500800t^{35} - 481368808488960t^{34} - 2916707599908864t^{33} - 12646363271528448t^{32} - 36148109755023360t^{31} - 36016947430686720t^{30} + 258453182097653760t^{29} + 1758124168257208320t^{28} + 6238476317830938624t^{27} + 14967448543546048512t^{26} + 23065384979923992576t^{25} - 184523079839391940608t^{23} - 957916706786947104768t^{22} - 3194099874729440575488t^{21} - 7201276593181525278720t^{20} - 8468993870975918407680t^{19} + 9441626667269939527680t^{18} + 75808080668966749470720t^{17} + 212170768220899422240768t^{16} + 391473867300100733140992t^{15} + 516865822443642594263040t^{14} + 470469407645686667673600t^{13} + 217006023480141108215808t^{12} - 137780134357820566929408t^{11} - 413342774218665511354368t^{10} - 492364045669616194682880t^{9} - 395329001709531731853312t^{8} - 231820665119872162725888t^{7} - 99386733613504788430848t^{6} - 29790338757536319012864t^{5} - 5603198512389276303360t^{4} - 498062089990157893632t^{3}\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 3329559x - 2285438387$, with conductor $4410$
Generic density of odd order reductions $271/2688$

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