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

Curve name $X_{78b}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 8 & 5 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right]$
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 $X_{78b}$ minimally covers
Curves that minimally cover $X_{78b}$
Curves that minimally cover $X_{78b}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{12} - 3132t^{10} - 1620t^{8} + 31968t^{6} - 6480t^{4} - 50112t^{2} - 1728$ $B(t) = 54t^{18} - 13932t^{16} - 142560t^{14} + 423360t^{12} + 1083456t^{10} - 2166912t^{8} - 3386880t^{6} + 4561920t^{4} + 1783296t^{2} - 27648$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 2315952x - 1356051852$, with conductor $25872$
Generic density of odd order reductions $193/1792$