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

Curve name $X_{34a}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 4 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 4 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 0 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13h}$
Meaning/Special name
Chosen covering $X_{34}$
Curves that $X_{34a}$ minimally covers
Curves that minimally cover $X_{34a}$
Curves that minimally cover $X_{34a}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -432t^{8} + 2592t^{6} - 5292t^{4} + 3888t^{2} - 432$ $B(t) = 3456t^{12} - 31104t^{10} + 110160t^{8} - 190512t^{6} + 158112t^{4} - 46656t^{2} - 3456$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + 49x - 686$, with conductor $392$
Generic density of odd order reductions $643/5376$