An Elliptic curve is the union of its affine part Eaff(Z/nZ)={ [x,y,1] in P2(Z/nZ) such that y2 =x3+ax+b } and the point at infinity O = [0,1,0]
P2(Z/nZ) is the projective plane:
T = { (x,y,z) in (Z/nZ)^3 such that gcd(x,y,z,n) = 1}
P2(Z/nZ) = (T / ~) where ~ is the equivalence relation defined by (x,y,z) ~ (x0,y0,z0) iff there is an INVERSIBLE element u in Z/nZ ⇤ such that (x,y,z) = u(x0,y0,z0).
I found those doc pages
http://fe.math.kobe-u.ac.jp/icms2010-dvd/SAGE/www.sagemath.org/doc/reference/sage/schemes/elliptic_curves/ell_generic.html
http://doc.sagemath.org/html/en/reference/curves/sage/schemes/elliptic_curves/constructor.html
I think to define an Elliptic Curve here with
q=10
E1 = EllipticCurve(Zmod(q),[0,1])
E1
Elliptic Curve defined by y^2 = x^3 + 1 over Ring of integers modulo 10
but
E1.points()
is not working. is there a method to get all the points of this Elliptic Curve ? its cardinality etc ????
