## The modular curve $X_{123}$

Curve name $X_{123}$
Index $24$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 11 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 13 & 13 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 6 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{9}$ $8$ $12$ $X_{37}$
Meaning/Special name
Chosen covering $X_{37}$
Curves that $X_{123}$ minimally covers $X_{37}$
Curves that minimally cover $X_{123}$ $X_{239}$, $X_{123a}$, $X_{123b}$, $X_{123c}$, $X_{123d}$, $X_{123e}$, $X_{123f}$
Curves that minimally cover $X_{123}$ and have infinitely many rational points. $X_{239}$, $X_{123a}$, $X_{123b}$, $X_{123c}$, $X_{123d}$, $X_{123e}$, $X_{123f}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{123}) = \mathbb{Q}(f_{123}), f_{37} = \frac{f_{123} + 1}{f_{123}^{2} - 2}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 6281x - 186919$, with conductor $10880$
Generic density of odd order reductions $85091/344064$