## The modular curve $X_{189}$

Curve name $X_{189}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 6 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 4 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 4 \\ 4 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $24$ $X_{58}$
Meaning/Special name
Chosen covering $X_{58}$
Curves that $X_{189}$ minimally covers $X_{58}$, $X_{98}$, $X_{101}$
Curves that minimally cover $X_{189}$ $X_{449}$, $X_{450}$, $X_{458}$, $X_{465}$, $X_{500}$, $X_{501}$, $X_{502}$, $X_{503}$, $X_{189a}$, $X_{189b}$, $X_{189c}$, $X_{189d}$, $X_{189e}$, $X_{189f}$, $X_{189g}$, $X_{189h}$, $X_{189i}$, $X_{189j}$, $X_{189k}$, $X_{189l}$, $X_{189m}$, $X_{189n}$
Curves that minimally cover $X_{189}$ and have infinitely many rational points. $X_{189a}$, $X_{189b}$, $X_{189c}$, $X_{189d}$, $X_{189e}$, $X_{189f}$, $X_{189g}$, $X_{189h}$, $X_{189i}$, $X_{189j}$, $X_{189k}$, $X_{189l}$, $X_{189m}$, $X_{189n}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{189}) = \mathbb{Q}(f_{189}), f_{58} = \frac{f_{189}^{2} - 8}{f_{189}^{2} + 8f_{189} + 8}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - 913553x + 328508948$, with conductor $25410$
Generic density of odd order reductions $17/168$