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

Curve name $X_{75c}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{75}$
Meaning/Special name
Chosen covering $X_{75}$
Curves that $X_{75c}$ minimally covers
Curves that minimally cover $X_{75c}$
Curves that minimally cover $X_{75c}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -7077888t^{12} + 51314688t^{10} - 1658880t^{8} - 2045952t^{6} - 25920t^{4} + 12528t^{2} - 27$ $B(t) = 7247757312t^{18} + 116870086656t^{16} - 74742497280t^{14} - 13872660480t^{12} + 2218917888t^{10} + 277364736t^{8} - 27095040t^{6} - 2280960t^{4} + 55728t^{2} + 54$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + 3484077x + 2577412978$, with conductor $16560$
Generic density of odd order reductions $635/5376$