Curve name 
$X_{694}$ 
Index 
$96$ 
Level 
$32$ 
Genus 
$5$ 
Does the subgroup contain $I$? 
Yes 
Generating matrices 
$
\left[ \begin{matrix} 23 & 19 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 13 & 0 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 15 & 15 \\ 2 & 1 \end{matrix}\right],
\left[ \begin{matrix} 25 & 4 \\ 0 & 1 \end{matrix}\right]$ 
Images in lower levels 

Meaning/Special name 

Chosen covering 
$X_{324}$ 
Curves that $X_{694}$ minimally covers 
$X_{324}$ 
Curves that minimally cover $X_{694}$ 

Curves that minimally cover $X_{694}$ and have infinitely many rational
points. 

Model 
\[y^2 = x^3  2x\]\[w^2 = 8x^2y + 8xy + 2y^3\] 
Info about rational points 
Rational point  Image on the $j$line 
$(1/2 : 1/2 : 1/4 : 1)$ 
\[3375 \,\,(\text{CM by }7)\]

$(1/2 : 1/2 : 1/4 : 1)$ 
\[3375 \,\,(\text{CM by }7)\]

$(0 : 0 : 1 : 0)$ 
Singular

$(0 : 0 : 0 : 1)$ 
Singular


Comments on finding rational points 
This curve admits a family of twists of etale double covers that are also
modular curves. Each of these modular curves maps to one we have already
computed that has finitely many rational points. 
Elliptic curve whose $2$adic image is the subgroup 
None 
Generic density of odd order reductions 
N/A 