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

Curve name $X_{75d}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 0 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right]$
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 $X_{75d}$ minimally covers
Curves that minimally cover $X_{75d}$
Curves that minimally cover $X_{75d}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -452984832t^{16} + 3397386240t^{14} - 934281216t^{12} - 53084160t^{10} + 29417472t^{8} - 829440t^{6} - 228096t^{4} + 12960t^{2} - 27$ $B(t) = 3710851743744t^{24} + 58445914963968t^{22} - 60533269069824t^{20} + 10045391634432t^{18} + 1888946749440t^{16} - 542222843904t^{14} + 8472231936t^{10} - 461168640t^{8} - 38320128t^{6} + 3608064t^{4} - 54432t^{2} - 54$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 + 82624568x - 99225432788$, with conductor $142296$
Generic density of odd order reductions $307/2688$