## The modular curve $X_{46d}$

Curve name $X_{46d}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 0 \\ 2 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 2 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 6 \\ 6 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{8d}$
Meaning/Special name
Chosen covering $X_{46}$
Curves that $X_{46d}$ minimally covers
Curves that minimally cover $X_{46d}$
Curves that minimally cover $X_{46d}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{4} - 54t^{2} - 108$ $B(t) = 54t^{6} + 162t^{4} - 324t^{2} - 432$
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 + x^2 - 22x - 49$, with conductor $66$
Generic density of odd order reductions $643/5376$