## The modular curve $X_{200e}$

Curve name $X_{200e}$
Index $96$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 6 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 4 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 2 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$
Meaning/Special name
Chosen covering $X_{200}$
Curves that $X_{200e}$ minimally covers
Curves that minimally cover $X_{200e}$
Curves that minimally cover $X_{200e}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -110592t^{24} + 1105920t^{22} - 10672128t^{20} + 59332608t^{18} - 157365504t^{16} + 208051200t^{14} - 150011136t^{12} + 52012800t^{10} - 9835344t^{8} + 927072t^{6} - 41688t^{4} + 1080t^{2} - 27$ $B(t) = 14155776t^{36} - 212336640t^{34} - 477757440t^{32} + 18997051392t^{30} - 126478319616t^{28} + 456237121536t^{26} - 1060075634688t^{24} + 1623202136064t^{22} - 1586118537216t^{20} + 982579286016t^{18} - 396529634304t^{16} + 101450133504t^{14} - 16563681792t^{12} + 1782176256t^{10} - 123513984t^{8} + 4637952t^{6} - 29160t^{4} - 3240t^{2} + 54$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - 5097x - 49995$, with conductor $294$
Generic density of odd order reductions $81/896$