## The modular curve $X_{10}$

Curve name $X_{10}$
Index $6$
Level $4$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 2 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$
Meaning/Special name Elliptic curves that acquire full $2$-torsion over $\mathbb{Q}(i)$
Chosen covering $X_{6}$
Curves that $X_{10}$ minimally covers $X_{3}$, $X_{6}$
Curves that minimally cover $X_{10}$ $X_{27}$, $X_{42}$, $X_{10a}$, $X_{10b}$, $X_{10c}$, $X_{10d}$
Curves that minimally cover $X_{10}$ and have infinitely many rational points. $X_{27}$, $X_{42}$, $X_{10a}$, $X_{10b}$, $X_{10c}$, $X_{10d}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{10}) = \mathbb{Q}(f_{10}), f_{6} = \frac{48f_{10}^{2} - 16}{f_{10}^{2} + 1}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + 33x - 74$, with conductor $180$
Generic density of odd order reductions $83/336$