The modular curve $X_{224}$

Curve name $X_{224}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $ \left[ \begin{matrix} 15 & 0 \\ 2 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 2 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $12$ $X_{23}$
$8$ $24$ $X_{80}$
Meaning/Special name
Chosen covering $X_{80}$
Curves that $X_{224}$ minimally covers $X_{80}$, $X_{105}$, $X_{107}$
Curves that minimally cover $X_{224}$ $X_{525}$, $X_{526}$
Curves that minimally cover $X_{224}$ and have infinitely many rational points.
Model \[\mathbb{P}^{1}, \mathbb{Q}(X_{224}) = \mathbb{Q}(f_{224}), f_{80} = \frac{f_{224}^{2} - \frac{1}{32}}{f_{224}}\]
Info about rational points None
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 17303x + 881475$, with conductor $13056$
Generic density of odd order reductions $12833/57344$

Back to the 2-adic image homepage.