## The modular curve $X_{115a}$

Curve name $X_{115a}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 9 \\ 4 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 4 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13h}$ $8$ $24$ $X_{32e}$
Meaning/Special name
Chosen covering $X_{115}$
Curves that $X_{115a}$ minimally covers
Curves that minimally cover $X_{115a}$
Curves that minimally cover $X_{115a}$ 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 $y^2 = x^3 - 50x + 125$, with conductor $200$
Generic density of odd order reductions $103/672$