## The modular curve $X_{441}$

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
 Level Index of image Corresponding curve $2$ $1$ $X_{1}$ $4$ $4$ $X_{7}$ $8$ $16$ $X_{55}$
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 $y^2 = x^6 + x^4 - 3x^2 + 1$
 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 four-fold 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 $45875/86016$