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

Curve name $X_{192j}$
Index $96$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 6 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $24$ $X_{25i}$
Meaning/Special name
Chosen covering $X_{192}$
Curves that $X_{192j}$ minimally covers
Curves that minimally cover $X_{192j}$
Curves that minimally cover $X_{192j}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{22} - 99360t^{20} - 464832t^{18} + 16754688t^{16} - 105781248t^{14} + 420691968t^{12} - 1692499968t^{10} + 4289200128t^{8} - 1903951872t^{6} - 6511656960t^{4} - 113246208t^{2}$ $B(t) = 432t^{33} - 896832t^{31} - 69113088t^{29} + 844729344t^{27} - 2322763776t^{25} + 25245499392t^{23} - 542808539136t^{21} + 4414542446592t^{19} - 17658169786368t^{17} + 34739746504704t^{15} - 25851391377408t^{13} + 38056161705984t^{11} - 221440729153536t^{9} + 289881301450752t^{7} + 60185376718848t^{5} - 463856467968t^{3}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 69148812x + 221322182384$, with conductor $20160$
Generic density of odd order reductions $11/112$