The modular curve $X_{118r}$

Curve name	$X_{118r}$
Index	$48$
Level	$32$
Genus	$0$
Does the subgroup contain $-I$?	No
Generating matrices	$\left[ \begin{matrix} 7 & 7 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 16 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels	Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $12$ $X_{36}$ $16$ $24$ $X_{118}$
Meaning/Special name
Chosen covering	$X_{118}$
Curves that $X_{118r}$ minimally covers
Curves that minimally cover $X_{118r}$
Curves that minimally cover $X_{118r}$ and have infinitely many rational points.
Model	$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $$y^2 = x^3 + A(t)x + B(t), \text{ where}$$ $$A(t) = -27t^{18} + 1296t^{14} - 21168t^{10} + 124416t^{6} - 110592t^{2}$$ $$B(t) = 54t^{27} - 3888t^{23} + 110160t^{19} - 1524096t^{15} + 10119168t^{11} - 23887872t^{7} - 14155776t^{3}$$
Info about rational points
Comments on finding rational points	None
Elliptic curve whose $2$-adic image is the subgroup	$y^2 + xy = x^3 - x^2 - 4183542x - 3292403009$, with conductor $38025$
Generic density of odd order reductions	$307/2688$

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