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

Curve name $X_{102n}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right]$
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 $X_{102n}$ minimally covers
Curves that minimally cover $X_{102n}$
Curves that minimally cover $X_{102n}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{8} + 1728t^{6} - 3456t^{5} + 6912t^{3} - 10368t^{2} + 6912t - 1728$ $B(t) = -432t^{12} + 10368t^{10} - 20736t^{9} - 41472t^{8} + 207360t^{7} - 338688t^{6} + 373248t^{5} - 466560t^{4} + 552960t^{3} - 414720t^{2} + 165888t - 27648$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 2497x - 48577$, with conductor $1344$
Generic density of odd order reductions $193/1792$