The modular curve $X_{170}$

Curve name $X_{170}$
Index $24$
Level $16$
Genus $2$
Does the subgroup contain $-I$? Yes
Generating matrices $ \left[ \begin{matrix} 1 & 3 \\ 12 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 12 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $6$ $X_{11}$
$8$ $12$ $X_{50}$
Meaning/Special name
Chosen covering $X_{50}$
Curves that $X_{170}$ minimally covers $X_{50}$
Curves that minimally cover $X_{170}$
Curves that minimally cover $X_{170}$ and have infinitely many rational points.
Model \[y^2 = x^5 - 4x\]
Info about rational points
Rational pointImage on the $j$-line
$(1 : 0 : 0)$ \[ \infty \]
$(0 : 0 : 1)$ \[ \infty \]
Comments on finding rational points The rank of the Jacobian is 0. We use the method of Chabauty.
Elliptic curve whose $2$-adic image is the subgroup None
Generic density of odd order reductions N/A

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