## The modular curve $X_{34b}$

Curve name $X_{34b}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 4 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 0 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$
Meaning/Special name
Chosen covering $X_{34}$
Curves that $X_{34b}$ minimally covers
Curves that minimally cover $X_{34b}$
Curves that minimally cover $X_{34b}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{8} + 648t^{6} - 1323t^{4} + 972t^{2} - 108$ $B(t) = 432t^{12} - 3888t^{10} + 13770t^{8} - 23814t^{6} + 19764t^{4} - 5832t^{2} - 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + 196x - 5488$, with conductor $3136$
Generic density of odd order reductions $289/1792$