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

Curve name $X_{194h}$
Index $96$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 6 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $24$ $X_{25i}$
Meaning/Special name
Chosen covering $X_{194}$
Curves that $X_{194h}$ minimally covers
Curves that minimally cover $X_{194h}$
Curves that minimally cover $X_{194h}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -452984832t^{24} - 2264924160t^{22} - 4133486592t^{20} - 3199205376t^{18} - 1215627264t^{16} - 1150156800t^{14} - 947994624t^{12} - 71884800t^{10} - 4748544t^{8} - 781056t^{6} - 63072t^{4} - 2160t^{2} - 27$ $B(t) = -3710851743744t^{36} - 27831388078080t^{34} - 85581518340096t^{32} - 137301514518528t^{30} - 112195283189760t^{28} - 10349797441536t^{26} + 78714268286976t^{24} + 79884781092864t^{22} + 28937237299200t^{20} + 3151132360704t^{18} + 1808577331200t^{16} + 312049926144t^{14} + 19217350656t^{12} - 157925376t^{10} - 106997760t^{8} - 8183808t^{6} - 318816t^{4} - 6480t^{2} - 54$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 133541505x - 556209541503$, with conductor $277440$
Generic density of odd order reductions $5/42$