## The modular curve $X_{118p}$

Curve name $X_{118p}$
Index $48$
Level $32$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 16 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $12$ $X_{36}$ $16$ $24$ $X_{118}$
Meaning/Special name
Chosen covering $X_{118}$
Curves that $X_{118p}$ minimally covers
Curves that minimally cover $X_{118p}$
Curves that minimally cover $X_{118p}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{18} + 5184t^{14} - 84672t^{10} + 497664t^{6} - 442368t^{2}$ $B(t) = 432t^{27} - 31104t^{23} + 881280t^{19} - 12192768t^{15} + 80953344t^{11} - 191102976t^{7} - 113246208t^{3}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 301539042x + 2195354163991$, with conductor $38025$
Generic density of odd order reductions $307/2688$