The modular curve $X_{189d}$

Curve name	$X_{189d}$
Index	$96$
Level	$8$
Genus	$0$
Does the subgroup contain $-I$?	No
Generating matrices	$\left[ \begin{matrix} 5 & 0 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 4 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 6 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right]$
Images in lower levels	Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $48$ $X_{58i}$
Meaning/Special name
Chosen covering	$X_{189}$
Curves that $X_{189d}$ minimally covers
Curves that minimally cover $X_{189d}$
Curves that minimally cover $X_{189d}$ and have infinitely many rational points.
Model	$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $$y^2 = x^3 + A(t)x + B(t), \text{ where}$$ $$A(t) = -27t^{16} - 864t^{15} - 12096t^{14} - 96768t^{13} - 580608t^{12} - 3870720t^{11} - 27095040t^{10} - 142884864t^{9} - 500539392t^{8} - 1143078912t^{7} - 1734082560t^{6} - 1981808640t^{5} - 2378170368t^{4} - 3170893824t^{3} - 3170893824t^{2} - 1811939328t - 452984832$$ $$B(t) = 54t^{24} + 2592t^{23} + 57024t^{22} + 760320t^{21} + 6386688t^{20} + 25546752t^{19} - 127733760t^{18} - 2934226944t^{17} - 24999321600t^{16} - 138195763200t^{15} - 563838713856t^{14} - 1862107398144t^{13} - 5461864611840t^{12} - 14896859185152t^{11} - 36085677686784t^{10} - 70756230758400t^{9} - 102397221273600t^{8} - 96148748500992t^{7} - 33484638781440t^{6} + 53575422050304t^{5} + 107150844100608t^{4} + 102048422952960t^{3} + 61229053771776t^{2} + 22265110462464t + 3710851743744$$
Info about rational points
Comments on finding rational points	None
Elliptic curve whose $2$-adic image is the subgroup	$y^2 + xy = x^3 - 7550x - 247500$, with conductor $210$
Generic density of odd order reductions	$1/28$

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