## The modular curve $X_{194c}$

Curve name $X_{194c}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$ $8$ $48$ $X_{194}$
Meaning/Special name
Chosen covering $X_{194}$
Curves that $X_{194c}$ minimally covers
Curves that minimally cover $X_{194c}$
Curves that minimally cover $X_{194c}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -28311552t^{22} - 70778880t^{20} - 51314688t^{18} - 10616832t^{16} - 24993792t^{14} - 12275712t^{12} - 1562112t^{10} - 41472t^{8} - 12528t^{6} - 1080t^{4} - 27t^{2}$ $B(t) = 57982058496t^{33} + 217432719360t^{31} + 293534171136t^{29} + 173040205824t^{27} - 76780929024t^{25} - 206391214080t^{23} - 97434206208t^{21} - 20543569920t^{19} - 5135892480t^{17} - 1522409472t^{15} - 201553920t^{13} - 4686336t^{11} + 660096t^{9} + 69984t^{7} + 3240t^{5} + 54t^{3}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - 154251x + 7145398$, with conductor $6150$
Generic density of odd order reductions $73/672$