## The modular curve $X_{78c}$

Curve name $X_{78c}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{78}$
Meaning/Special name
Chosen covering $X_{78}$
Curves that $X_{78c}$ minimally covers
Curves that minimally cover $X_{78c}$
Curves that minimally cover $X_{78c}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{16} - 12960t^{14} - 57024t^{12} + 51840t^{10} + 459648t^{8} + 207360t^{6} - 912384t^{4} - 829440t^{2} - 27648$ $B(t) = 432t^{24} - 108864t^{22} - 1804032t^{20} - 4790016t^{18} + 14411520t^{16} + 66189312t^{14} - 264757248t^{10} - 230584320t^{8} + 306561024t^{6} + 461832192t^{4} + 111476736t^{2} - 1769472$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 1120920929x - 14431393673985$, with conductor $1138368$
Generic density of odd order reductions $335/2688$