## The modular curve $X_{102f}$

Curve name $X_{102f}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 8 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{102}$
Meaning/Special name
Chosen covering $X_{102}$
Curves that $X_{102f}$ minimally covers
Curves that minimally cover $X_{102f}$
Curves that minimally cover $X_{102f}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{10} + 216t^{9} + 1620t^{8} - 6912t^{7} + 8640t^{6} + 3456t^{5} - 24192t^{4} + 34560t^{3} - 25920t^{2} + 10368t - 1728$ $B(t) = 432t^{15} - 1296t^{14} - 9072t^{13} + 51408t^{12} - 51840t^{11} - 259200t^{10} + 1064448t^{9} - 2052864t^{8} + 2809728t^{7} - 3411072t^{6} + 3846528t^{5} - 3535488t^{4} + 2322432t^{3} - 995328t^{2} + 248832t - 27648$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 3780x - 97200$, with conductor $2070$
Generic density of odd order reductions $193/1792$