## The modular curve $X_{75b}$

Curve name $X_{75b}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 0 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{75}$
Meaning/Special name
Chosen covering $X_{75}$
Curves that $X_{75b}$ minimally covers
Curves that minimally cover $X_{75b}$
Curves that minimally cover $X_{75b}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -1811939328t^{16} + 13589544960t^{14} - 3737124864t^{12} - 212336640t^{10} + 117669888t^{8} - 3317760t^{6} - 912384t^{4} + 51840t^{2} - 108$ $B(t) = 29686813949952t^{24} + 467567319711744t^{22} - 484266152558592t^{20} + 80363133075456t^{18} + 15111573995520t^{16} - 4337782751232t^{14} + 67777855488t^{10} - 3689349120t^{8} - 306561024t^{6} + 28864512t^{4} - 435456t^{2} - 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 + 330498271x + 793472964033$, with conductor $1138368$
Generic density of odd order reductions $335/2688$