The modular curve $X_{318}$

Curve name $X_{318}$
Index $48$
Level $16$
Genus $1$
Does the subgroup contain $-I$? Yes
Generating matrices $ \left[ \begin{matrix} 1 & 1 \\ 10 & 15 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 10 & 15 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $12$ $X_{23}$
$8$ $24$ $X_{65}$
Meaning/Special name
Chosen covering $X_{65}$
Curves that $X_{318}$ minimally covers $X_{65}$, $X_{114}$, $X_{155}$
Curves that minimally cover $X_{318}$
Curves that minimally cover $X_{318}$ and have infinitely many rational points.
Model \[y^2 = x^3 + 8x\]
Info about rational points $X_{318}(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}$
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 + x^2 - 189530533x - 998555156245$, with conductor $76614$
Generic density of odd order reductions $12833/57344$

Back to the 2-adic image homepage.