## The modular curve $X_{43}$

Curve name $X_{43}$
Index $12$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 5 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{11}$
Meaning/Special name
Chosen covering $X_{11}$
Curves that $X_{43}$ minimally covers $X_{11}$
Curves that minimally cover $X_{43}$ $X_{64}$, $X_{67}$, $X_{71}$, $X_{74}$, $X_{77}$, $X_{82}$, $X_{91}$, $X_{92}$, $X_{127}$, $X_{137}$, $X_{139}$, $X_{144}$
Curves that minimally cover $X_{43}$ and have infinitely many rational points. $X_{64}$, $X_{67}$, $X_{71}$, $X_{74}$, $X_{77}$, $X_{82}$, $X_{91}$, $X_{92}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{43}) = \mathbb{Q}(f_{43}), f_{11} = -2f_{43}^{2} + 8$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 + 12x + 36$, with conductor $1176$
Generic density of odd order reductions $25/112$