The modular curve $X_{227d}$

Curve name	$X_{227d}$
Index	$96$
Level	$32$
Genus	$0$
Does the subgroup contain $-I$?	No
Generating matrices	$\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 16 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 16 & 7 \end{matrix}\right]$
Images in lower levels	Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{84}$ $16$ $48$ $X_{227}$
Meaning/Special name
Chosen covering	$X_{227}$
Curves that $X_{227d}$ minimally covers
Curves that minimally cover $X_{227d}$
Curves that minimally cover $X_{227d}$ and have infinitely many rational points.
Model	$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $$y^2 = x^3 + A(t)x + B(t), \text{ where}$$ $$A(t) = -115964116992t^{32} + 3478923509760t^{30} - 3913788948480t^{28} + 1522029035520t^{26} - 494659436544t^{24} + 67947724800t^{22} + 15288238080t^{20} - 11041505280t^{18} + 3868065792t^{16} - 690094080t^{14} + 59719680t^{12} + 16588800t^{10} - 7547904t^{8} + 1451520t^{6} - 233280t^{4} + 12960t^{2} - 27$$ $$B(t) = 15199648742375424t^{48} + 957577870769651712t^{46} - 3950008716924813312t^{44} + 3650765632309297152t^{42} - 1977141809066803200t^{40} + 714442864519544832t^{38} - 169726937055363072t^{36} + 2571620258414592t^{34} + 15445724598632448t^{32} - 7568745987833856t^{30} + 2290349292650496t^{28} - 363460533682176t^{26} + 22716283355136t^{22} - 8946676924416t^{20} + 1847838375936t^{18} - 235683053568t^{16} - 2452488192t^{14} + 10116513792t^{12} - 2661507072t^{10} + 460339200t^{8} - 53125632t^{6} + 3592512t^{4} - 54432t^{2} - 54$$
Info about rational points
Comments on finding rational points	None
Elliptic curve whose $2$-adic image is the subgroup	$y^2 + xy + y = x^3 - x^2 + 425912395x + 15999991513647$, with conductor $130050$
Generic density of odd order reductions	$299/2688$

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