The modular curve $X_{86n}$

Curve name	$X_{86n}$
Index	$48$
Level	$8$
Genus	$0$
Does the subgroup contain $-I$?	No
Generating matrices	$\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 7 \end{matrix}\right]$
Images in lower levels	Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13f}$
Meaning/Special name
Chosen covering	$X_{86}$
Curves that $X_{86n}$ minimally covers
Curves that minimally cover $X_{86n}$
Curves that minimally cover $X_{86n}$ and have infinitely many rational points.
Model	$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $$y^2 = x^3 + A(t)x + B(t), \text{ where}$$ $$A(t) = -27t^{12} - 432t^{10} - 1080t^{8} + 11232t^{6} + 65232t^{4} + 93312t^{2} - 6912$$ $$B(t) = 54t^{18} + 1296t^{16} + 20088t^{14} + 211680t^{12} + 1319328t^{10} + 4437504t^{8} + 6604416t^{6} - 41472t^{4} - 7464960t^{2} - 221184$$
Info about rational points
Comments on finding rational points	None
Elliptic curve whose $2$-adic image is the subgroup	$y^2 = x^3 - x^2 + 1992x + 6012$, with conductor $600$
Generic density of odd order reductions	$41/336$

Back to the 2-adic image homepage.