The modular curve $X_{369}$

Curve name $X_{369}$
Index $48$
Level $16$
Genus $2$
Does the subgroup contain $-I$? Yes
Generating matrices $ \left[ \begin{matrix} 9 & 9 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 11 & 1 \\ 12 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 15 & 15 \\ 4 & 1 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $6$ $X_{13}$
$8$ $24$ $X_{79}$
Meaning/Special name
Chosen covering $X_{79}$
Curves that $X_{369}$ minimally covers $X_{79}$
Curves that minimally cover $X_{369}$
Curves that minimally cover $X_{369}$ and have infinitely many rational points.
Model \[y^2 = x^6 - 5x^4 - 5x^2 + 1\]
Info about rational points
Rational pointImage on the $j$-line
$(1 : -1 : 0)$ \[287496 \,\,(\text{CM by }-16)\]
$(1 : 1 : 0)$ \[287496 \,\,(\text{CM by }-16)\]
$(0 : -1 : 1)$ \[287496 \,\,(\text{CM by }-16)\]
$(0 : 1 : 1)$ \[287496 \,\,(\text{CM by }-16)\]
Comments on finding rational points The rank of the Jacobian is 2. This curve admits a family of etale double covers that map to rank zero elliptic curves.
Elliptic curve whose $2$-adic image is the subgroup None
Generic density of odd order reductions N/A

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