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

Curve name $X_{227e}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13f}$ $8$ $48$ $X_{84o}$
Meaning/Special name
Chosen covering $X_{227}$
Curves that $X_{227e}$ minimally covers
Curves that minimally cover $X_{227e}$
Curves that minimally cover $X_{227e}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -1811939328t^{24} + 54358179840t^{22} - 61379444736t^{20} + 30576476160t^{18} - 15394406400t^{16} + 4671406080t^{14} - 1445658624t^{12} + 291962880t^{10} - 60134400t^{8} + 7464960t^{6} - 936576t^{4} + 51840t^{2} - 108$ $B(t) = -29686813949952t^{36} - 1870269278846976t^{34} + 7709294497628160t^{32} - 7481077115387904t^{30} + 5307445706489856t^{28} - 2776180960788480t^{26} + 1236293451251712t^{24} - 438344362229760t^{22} + 141323113857024t^{20} - 36699925118976t^{18} + 8832694616064t^{16} - 1712282664960t^{14} + 301829455872t^{12} - 42361159680t^{10} + 5061574656t^{8} - 445906944t^{6} + 28719360t^{4} - 435456t^{2} - 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 + 39251315x + 30772225697$, with conductor $50430$
Generic density of odd order reductions $11/112$