## The modular curve $X_{204h}$

Curve name $X_{204h}$
Index $96$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 5 & 2 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 4 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $24$ $X_{25n}$
Meaning/Special name
Chosen covering $X_{204}$
Curves that $X_{204h}$ minimally covers
Curves that minimally cover $X_{204h}$
Curves that minimally cover $X_{204h}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -1971t^{16} - 25056t^{15} - 170208t^{14} - 556416t^{13} - 1850688t^{12} - 3386880t^{11} - 3580416t^{10} - 6248448t^{9} - 18510336t^{8} + 24993792t^{7} - 57286656t^{6} + 216760320t^{5} - 473776128t^{4} + 569769984t^{3} - 697171968t^{2} + 410517504t - 129171456$ $B(t) = 32130t^{24} + 681696t^{23} + 6052320t^{22} + 38586240t^{21} + 168049728t^{20} + 539675136t^{19} + 1376331264t^{18} + 3399542784t^{17} + 5064187392t^{16} + 8345272320t^{15} + 6496174080t^{14} + 4347592704t^{13} + 50158338048t^{12} - 17390370816t^{11} + 103938785280t^{10} - 534097428480t^{9} + 1296431972352t^{8} - 3481131810816t^{7} + 5637452857344t^{6} - 8842037428224t^{5} + 11013306974208t^{4} - 10115151298560t^{3} + 6346317496320t^{2} - 2859240259584t + 539051950080$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 24x + 36$, with conductor $48$
Generic density of odd order reductions $53/896$