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

Curve name $X_{102p}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13h}$
Meaning/Special name $X_1(8)$
Chosen covering $X_{102}$
Curves that $X_{102p}$ minimally covers
Curves that minimally cover $X_{102p}$
Curves that minimally cover $X_{102p}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{8} + 432t^{6} - 864t^{5} + 1728t^{3} - 2592t^{2} + 1728t - 432$ $B(t) = 54t^{12} - 1296t^{10} + 2592t^{9} + 5184t^{8} - 25920t^{7} + 42336t^{6} - 46656t^{5} + 58320t^{4} - 69120t^{3} + 51840t^{2} - 20736t + 3456$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - 39x + 90$, with conductor $21$
Generic density of odd order reductions $5/84$