Curve name 
$X_{513}$ 
Index 
$96$ 
Level 
$16$ 
Genus 
$2$ 
Does the subgroup contain $I$? 
Yes 
Generating matrices 
$
\left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 15 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 15 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 0 & 7 \end{matrix}\right]$ 
Images in lower levels 

Meaning/Special name 

Chosen covering 
$X_{211}$ 
Curves that $X_{513}$ minimally covers 
$X_{211}$ 
Curves that minimally cover $X_{513}$ 

Curves that minimally cover $X_{513}$ and have infinitely many rational
points. 

Model 
\[y^2 = 2x^5 + 4x^4 + 4x^2  2x\] 
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 