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	Level Index of image Corresponding 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 point Image 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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