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

Curve name $X_{188e}$
Index $96$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 6 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 2 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 4 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$
Meaning/Special name
Chosen covering $X_{188}$
Curves that $X_{188e}$ minimally covers
Curves that minimally cover $X_{188e}$
Curves that minimally cover $X_{188e}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{24} + 4320t^{22} - 63072t^{20} + 390528t^{18} - 1187136t^{16} + 8985600t^{14} - 59249664t^{12} + 35942400t^{10} - 18994176t^{8} + 24993792t^{6} - 16146432t^{4} + 4423680t^{2} - 442368$ $B(t) = 432t^{36} - 25920t^{34} + 637632t^{32} - 8183808t^{30} + 53498880t^{28} - 39481344t^{26} - 2402168832t^{24} + 19503120384t^{22} - 56518041600t^{20} + 49236443136t^{18} - 226072166400t^{16} + 312049926144t^{14} - 153738805248t^{12} - 10107224064t^{10} + 54782853120t^{8} - 33520877568t^{6} + 10446962688t^{4} - 1698693120t^{2} + 113246208$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 263489x - 51927231$, with conductor $9408$
Generic density of odd order reductions $109/896$