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

Curve name $X_{102j}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$
Meaning/Special name
Chosen covering $X_{102}$
Curves that $X_{102j}$ minimally covers
Curves that minimally cover $X_{102j}$
Curves that minimally cover $X_{102j}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{12} - 864t^{11} + 864t^{10} + 13824t^{9} - 15552t^{8} - 76032t^{7} + 183168t^{6} - 76032t^{5} - 250560t^{4} + 483840t^{3} - 400896t^{2} + 165888t - 27648$ $B(t) = -432t^{18} - 5184t^{17} - 5184t^{16} + 117504t^{15} + 145152t^{14} - 1451520t^{13} - 145152t^{12} + 10077696t^{11} - 13488768t^{10} - 17266176t^{9} + 68719104t^{8} - 102021120t^{7} + 129862656t^{6} - 174182400t^{5} + 189444096t^{4} - 138018816t^{3} + 62373888t^{2} - 15925248t + 1769472$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - 48021x + 5112351$, with conductor $1470$
Generic density of odd order reductions $193/1792$