## The modular curve $X_{117i}$

Curve name $X_{117i}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13f}$ $8$ $24$ $X_{36f}$
Meaning/Special name
Chosen covering $X_{117}$
Curves that $X_{117i}$ minimally covers
Curves that minimally cover $X_{117i}$
Curves that minimally cover $X_{117i}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -432t^{16} + 1944t^{12} - 1323t^{8} + 324t^{4} - 27$ $B(t) = 3456t^{24} + 23328t^{20} - 39528t^{16} + 23814t^{12} - 6885t^{8} + 972t^{4} - 54$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 3227352x + 40529065788$, with conductor $142296$
Generic density of odd order reductions $307/2688$