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

Curve name $X_{102h}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{102}$
Meaning/Special name
Chosen covering $X_{102}$
Curves that $X_{102h}$ minimally covers
Curves that minimally cover $X_{102h}$
Curves that minimally cover $X_{102h}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{10} + 54t^{9} + 405t^{8} - 1728t^{7} + 2160t^{6} + 864t^{5} - 6048t^{4} + 8640t^{3} - 6480t^{2} + 2592t - 432$ $B(t) = 54t^{15} - 162t^{14} - 1134t^{13} + 6426t^{12} - 6480t^{11} - 32400t^{10} + 133056t^{9} - 256608t^{8} + 351216t^{7} - 426384t^{6} + 480816t^{5} - 441936t^{4} + 290304t^{3} - 124416t^{2} + 31104t - 3456$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 5619x + 161138$, with conductor $1008$
Generic density of odd order reductions $25/224$