## The modular curve $X_{45}$

Curve name $X_{45}$
Index $12$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 5 \\ 2 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 5 \\ 0 & 1 \end{matrix}\right]$
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 $X_{45}$ minimally covers $X_{11}$
Curves that minimally cover $X_{45}$ $X_{61}$, $X_{69}$, $X_{70}$, $X_{73}$, $X_{77}$, $X_{81}$, $X_{94}$, $X_{97}$, $X_{109}$, $X_{110}$, $X_{111}$, $X_{112}$, $X_{130}$, $X_{136}$, $X_{140}$, $X_{143}$, $X_{149}$, $X_{150}$, $X_{151}$, $X_{153}$
Curves that minimally cover $X_{45}$ and have infinitely many rational points. $X_{61}$, $X_{69}$, $X_{70}$, $X_{73}$, $X_{77}$, $X_{81}$, $X_{94}$, $X_{97}$, $X_{109}$, $X_{110}$, $X_{111}$, $X_{112}$, $X_{150}$, $X_{153}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{45}) = \mathbb{Q}(f_{45}), f_{11} = -f_{45}^{2} + 8$
Info about rational points None
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 + 258x + 1791$, with conductor $1575$
Generic density of odd order reductions $2659/10752$