## The modular curve $X_{192n}$

Curve name $X_{192n}$
Index $96$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 6 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$
Meaning/Special name
Chosen covering $X_{192}$
Curves that $X_{192n}$ minimally covers
Curves that minimally cover $X_{192n}$
Curves that minimally cover $X_{192n}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{16} - 100224t^{14} - 1264896t^{12} + 8239104t^{10} - 19630080t^{8} + 131825664t^{6} - 323813376t^{4} - 410517504t^{2} - 7077888$ $B(t) = 432t^{24} - 891648t^{22} - 79833600t^{20} - 70447104t^{18} + 606818304t^{16} + 30799429632t^{14} - 206851276800t^{12} + 492790874112t^{10} + 155345485824t^{8} - 288551337984t^{6} - 5231974809600t^{4} - 934960693248t^{2} + 7247757312$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 7683201x - 8194556799$, with conductor $6720$
Generic density of odd order reductions $299/2688$