# Examples of Elliptic Curves, with Graphical Representations
#
# plot([],figsize=(4,4),title='Here is a Graph',frame=True,axes_labels=['$x$-axis, units','$y$-axis, units'])
#
# The examples are from Ash and Gros book Fearless symmetry
#
E=EllipticCurve([-1,0]); E # y2=x^3-x (implicitely) over the rational field
E.plot(figsize=(4,4), thickness=4, rgbcolor=(0.1,0.7,0.1),title="Elliptic Curve",frame=True,axes_labels=['$x$-axis, units','$y$-axis, units'] )

# Addition on Elliptic Curves
#
# The Group Law
R=E([-1,0]); P=E([0,0]); Q=E([1,0]) # defining the obvious points on the EC <-> roots of f(x)
print P, Q, R                       # notice the use of homogeneous coordinates

# Addition Law
print P+P, Q+Q, R+R                 # The point at infinity is (0:1:0)

O=E([0,1,0]); print O               # ... Let's label it O

2*P == O                            # simple way of comparing ... ("if ... then ..." not needed)

(P+Q)==R