## The modular curve $X_{188a}$

Curve name $X_{188a}$
Index $96$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 6 \\ 4 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $24$ $X_{25i}$
Meaning/Special name
Chosen covering $X_{188}$
Curves that $X_{188a}$ minimally covers
Curves that minimally cover $X_{188a}$
Curves that minimally cover $X_{188a}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{24} + 1080t^{22} - 15768t^{20} + 97632t^{18} - 296784t^{16} + 2246400t^{14} - 14812416t^{12} + 8985600t^{10} - 4748544t^{8} + 6248448t^{6} - 4036608t^{4} + 1105920t^{2} - 110592$ $B(t) = -54t^{36} + 3240t^{34} - 79704t^{32} + 1022976t^{30} - 6687360t^{28} + 4935168t^{26} + 300271104t^{24} - 2437890048t^{22} + 7064755200t^{20} - 6154555392t^{18} + 28259020800t^{16} - 39006240768t^{14} + 19217350656t^{12} + 1263403008t^{10} - 6847856640t^{8} + 4190109696t^{6} - 1305870336t^{4} + 212336640t^{2} - 14155776$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 65872x + 6523840$, with conductor $2352$
Generic density of odd order reductions $299/2688$