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

Curve name $X_{78d}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right]$
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 $X_{78d}$ minimally covers
Curves that minimally cover $X_{78d}$
Curves that minimally cover $X_{78d}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{16} - 3240t^{14} - 14256t^{12} + 12960t^{10} + 114912t^{8} + 51840t^{6} - 228096t^{4} - 207360t^{2} - 6912$ $B(t) = 54t^{24} - 13608t^{22} - 225504t^{20} - 598752t^{18} + 1801440t^{16} + 8273664t^{14} - 33094656t^{10} - 28823040t^{8} + 38320128t^{6} + 57729024t^{4} + 13934592t^{2} - 221184$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 280230232x - 1803784094132$, with conductor $142296$
Generic density of odd order reductions $307/2688$