The modular curve $X_{84g}$

Curve name	$X_{84g}$
Index	$48$
Level	$16$
Genus	$0$
Does the subgroup contain $-I$?	No
Generating matrices	$\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels	Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{84}$
Meaning/Special name
Chosen covering	$X_{84}$
Curves that $X_{84g}$ minimally covers
Curves that minimally cover $X_{84g}$
Curves that minimally cover $X_{84g}$ and have infinitely many rational points.
Model	$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $$y^2 = x^3 + A(t)x + B(t), \text{ where}$$ $$A(t) = -27t^{16} - 270t^{14} - 729t^{12} + 648t^{10} + 6615t^{8} + 12474t^{6} + 9801t^{4} + 2700t^{2} - 108$$ $$B(t) = -54t^{24} - 810t^{22} - 7128t^{20} - 43524t^{18} - 177552t^{16} - 470448t^{14} - 793044t^{12} - 807732t^{10} - 418122t^{8} - 14310t^{6} + 89100t^{4} + 30456t^{2} + 432$$
Info about rational points
Comments on finding rational points	None
Elliptic curve whose $2$-adic image is the subgroup	$y^2 = x^3 - 72300x - 64078000$, with conductor $14400$
Generic density of odd order reductions	$89/672$

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