The modular curve $X_{227g}$

Curve name $X_{227g}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 1 & 1 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $12$ $X_{13h}$
$8$ $48$ $X_{84p}$
Meaning/Special name
Chosen covering $X_{227}$
Curves that $X_{227g}$ minimally covers
Curves that minimally cover $X_{227g}$
Curves that minimally cover $X_{227g}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -1811939328t^{24} + 54358179840t^{22} - 61379444736t^{20} + 30576476160t^{18} - 15394406400t^{16} + 4671406080t^{14} - 1445658624t^{12} + 291962880t^{10} - 60134400t^{8} + 7464960t^{6} - 936576t^{4} + 51840t^{2} - 108\] \[B(t) = 29686813949952t^{36} + 1870269278846976t^{34} - 7709294497628160t^{32} + 7481077115387904t^{30} - 5307445706489856t^{28} + 2776180960788480t^{26} - 1236293451251712t^{24} + 438344362229760t^{22} - 141323113857024t^{20} + 36699925118976t^{18} - 8832694616064t^{16} + 1712282664960t^{14} - 301829455872t^{12} + 42361159680t^{10} - 5061574656t^{8} + 445906944t^{6} - 28719360t^{4} + 435456t^{2} + 432\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 + 121148415x - 2427314507775$, with conductor $277440$
Generic density of odd order reductions $5/42$

Back to the 2-adic image homepage.