Curve name 
$X_{441}$ 
Index 
$64$ 
Level 
$16$ 
Genus 
$2$ 
Does the subgroup contain $I$? 
Yes 
Generating matrices 
$
\left[ \begin{matrix} 4 & 7 \\ 15 & 12 \end{matrix}\right],
\left[ \begin{matrix} 7 & 14 \\ 7 & 9 \end{matrix}\right],
\left[ \begin{matrix} 2 & 1 \\ 11 & 9 \end{matrix}\right]$ 
Images in lower levels 

Meaning/Special name 
$X_{ns}^{+}(16)$ 
Chosen covering 
$X_{55}$ 
Curves that $X_{441}$ minimally covers 
$X_{55}$ 
Curves that minimally cover $X_{441}$ 

Curves that minimally cover $X_{441}$ and have infinitely many rational
points. 

Model 
\[y^2 = x^6 + x^4  3x^2 + 1\] 
Info about rational points 
Rational point  Image on the $j$line 
$(1 : 1 : 0)$ 
\[147197952000 \,\,(\text{CM by }67)\]

$(1 : 1 : 0)$ 
\[32768 \,\,(\text{CM by }11)\]

$(1 : 0 : 1)$ 
\[0 \,\,(\text{CM by }3)\]

$(0 : 1 : 1)$ 
\[12288000 \,\,(\text{CM by }27)\]

$(0 : 1 : 1)$ 
\[884736 \,\,(\text{CM by }19)\]

$(1 : 0 : 1)$ 
\[884736000 \,\,(\text{CM by }43)\]

$(3 : 28 : 1)$ 
\[\frac{18234932071051198464000}{48661191875666868481}\]

$(3 : 28 : 1)$ 
\[262537412640768000 \,\,(\text{CM by }163)\]

$(3 : 28 : 1)$ 
\[0 \,\,(\text{CM by }3)\]

$(3 : 28 : 1)$ 
\[\frac{35817550197738955933474532061609984000}
{2301619141096101839813550846721}
\]


Comments on finding rational points 
The rational points on this curve were first resolved by Burcu Baran in a 2010 Journal
of Number Theory paper. The rank of the Jacobian is 2. We construct a family of
etale double covers, but one of these maps to an elliptic curve of rank 1. We
construct an etale fourfold cover over $\mathbb{Q}(\sqrt{2})$ that maps to an
elliptic curve over $\mathbb{Q}(\sqrt{2})$ and use elliptic curve Chabauty. 
Elliptic curve whose $2$adic image is the subgroup 
$y^2 + y = x^3  3014658660x + 150916472601529$, with conductor
$356257899$ 
Generic density of odd order reductions 
$45875/86016$ 