The modular curve $X_{12}$

Curve name $X_{12}$
Index $6$
Level $4$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $ \left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 2 & 3 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding 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 $X_{12}$ minimally covers $X_{6}$
Curves that minimally cover $X_{12}$ $X_{25}$, $X_{30}$, $X_{31}$, $X_{40}$, $X_{51}$, $X_{52}$
Curves that minimally cover $X_{12}$ and have infinitely many rational points. $X_{25}$, $X_{30}$, $X_{31}$, $X_{40}$
Model \[\mathbb{P}^{1}, \mathbb{Q}(X_{12}) = \mathbb{Q}(f_{12}), f_{6} = -f_{12}^{2} - 16\]
Info about rational points None
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 203x - 1048$, with conductor $2200$
Generic density of odd order reductions $13/42$

Back to the 2-adic image homepage.