## The modular curve $X_{35}$

Curve name $X_{35}$
Index $12$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 4 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 6 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right]$
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 $X_{35}$ minimally covers $X_{11}$
Curves that minimally cover $X_{35}$ $X_{61}$, $X_{63}$, $X_{65}$, $X_{82}$, $X_{147}$, $X_{148}$
Curves that minimally cover $X_{35}$ and have infinitely many rational points. $X_{61}$, $X_{63}$, $X_{65}$, $X_{82}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{35}) = \mathbb{Q}(f_{35}), f_{11} = f_{35}^{2} + 8$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 333x + 2088$, with conductor $300$
Generic density of odd order reductions $2659/10752$