## The modular curve $X_{227a}$

Curve name $X_{227a}$
Index $96$
Level $32$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 5 & 0 \\ 16 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 16 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 7 \\ 16 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 16 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{84}$ $16$ $48$ $X_{227}$
Meaning/Special name
Chosen covering $X_{227}$
Curves that $X_{227a}$ minimally covers
Curves that minimally cover $X_{227a}$
Curves that minimally cover $X_{227a}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -452984832t^{24} + 13589544960t^{22} - 15231614976t^{20} + 4246732800t^{18} - 26542080t^{16} - 318504960t^{14} + 122535936t^{12} - 19906560t^{10} - 103680t^{8} + 1036800t^{6} - 232416t^{4} + 12960t^{2} - 27$ $B(t) = 3710851743744t^{36} + 233783659855872t^{34} - 965053381607424t^{32} + 847465766977536t^{30} - 301796614471680t^{28} + 12785043898368t^{26} + 26457938067456t^{24} - 11757674299392t^{22} + 2582353281024t^{20} - 161397080064t^{16} + 45928415232t^{14} - 6459457536t^{12} - 195084288t^{10} + 287815680t^{8} - 50512896t^{6} + 3595104t^{4} - 54432t^{2} - 54$
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 + 583749x + 55853398$, with conductor $6150$
Generic density of odd order reductions $73/672$