## The modular curve $X_{34}$

Curve name $X_{34}$
Index $12$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 5 & 0 \\ 4 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 7 \\ 4 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 4 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$
Meaning/Special name
Chosen covering $X_{13}$
Curves that $X_{34}$ minimally covers $X_{13}$
Curves that minimally cover $X_{34}$ $X_{75}$, $X_{79}$, $X_{94}$, $X_{100}$, $X_{34a}$, $X_{34b}$, $X_{34c}$, $X_{34d}$, $X_{34e}$, $X_{34f}$, $X_{34g}$, $X_{34h}$
Curves that minimally cover $X_{34}$ and have infinitely many rational points. $X_{75}$, $X_{79}$, $X_{94}$, $X_{100}$, $X_{34a}$, $X_{34b}$, $X_{34c}$, $X_{34d}$, $X_{34e}$, $X_{34f}$, $X_{34g}$, $X_{34h}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{34}) = \mathbb{Q}(f_{34}), f_{13} = 8f_{34}^{2} - 8$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 + 58x - 284$, with conductor $350$
Generic density of odd order reductions $513/3584$