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

Curve name $X_{192e}$
Index $96$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 6 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 7 & 6 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 5 \end{matrix}\right]$
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 $X_{192e}$ minimally covers
Curves that minimally cover $X_{192e}$
Curves that minimally cover $X_{192e}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{22} - 24840t^{20} - 116208t^{18} + 4188672t^{16} - 26445312t^{14} + 105172992t^{12} - 423124992t^{10} + 1072300032t^{8} - 475987968t^{6} - 1627914240t^{4} - 28311552t^{2}$ $B(t) = 54t^{33} - 112104t^{31} - 8639136t^{29} + 105591168t^{27} - 290345472t^{25} + 3155687424t^{23} - 67851067392t^{21} + 551817805824t^{19} - 2207271223296t^{17} + 4342468313088t^{15} - 3231423922176t^{13} + 4757020213248t^{11} - 27680091144192t^{9} + 36235162681344t^{7} + 7523172089856t^{5} - 57982058496t^{3}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 1080450x + 432540000$, with conductor $630$
Generic density of odd order reductions $271/2688$