## The modular curve $X_{11}$

Curve name $X_{11}$
Index $6$
Level $4$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 1 \\ 2 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$
Meaning/Special name Elliptic curves with discriminant $\Delta$ whose $2$-isogenous curve has discriminant in the square class of $-\Delta$
Chosen covering $X_{6}$
Curves that $X_{11}$ minimally covers $X_{6}$
Curves that minimally cover $X_{11}$ $X_{23}$, $X_{24}$, $X_{26}$, $X_{27}$, $X_{28}$, $X_{29}$, $X_{35}$, $X_{39}$, $X_{41}$, $X_{43}$, $X_{45}$, $X_{47}$, $X_{49}$, $X_{50}$, $X_{53}$, $X_{54}$
Curves that minimally cover $X_{11}$ and have infinitely many rational points. $X_{23}$, $X_{24}$, $X_{26}$, $X_{27}$, $X_{28}$, $X_{29}$, $X_{35}$, $X_{39}$, $X_{41}$, $X_{43}$, $X_{45}$, $X_{47}$, $X_{49}$, $X_{50}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{11}) = \mathbb{Q}(f_{11}), f_{6} = f_{11}^{2} - 16$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + 72x + 485$, with conductor $2772$
Generic density of odd order reductions $83/336$