# Tutorial for plotting complex numbers, roots of nuity, Gauss and Jacobi sums
#
# Defines a graph, axes etc.
a = plot([],figsize=(4,4),title='Here is a Graph',frame=True,axes_labels=['$x$-axis, units','$y$-axis, units'])
A=[(i,i^2 % 10) for i in range(11)] # defines a list of 2D-points
a+=sage.plot.line.line2d(A) # adds the corresponding polygonal line
a+=point((3,0), color='red', size=30) # adds a single point
x=3; y=-1
a+=text("p="+str(x), (x,y), rgbcolor=(1,0,0)) # adds text
show(a)

# From complex numbers to 2D-points
list_plot([1, I, pi + I/2, CC(.25, .25)], color='red', size=30)

# Plotting roots of unity
p=3;
A=[exp(2*pi*I*k/p) for k in range(p)]
a = plot([],xmin=-1, xmax=1, ymin=-1, ymax=1, figsize=(3,3),title='Cubic Roots of Unity',frame=True,axes_labels=['$x$-axis','$y$-axis'], )
a+=list_plot(A, color='red', size=30, xmin=-1,xmax=1)
a+=circle((0.0,0.0),1,color='blue')
show(a)

# Primitive Roots of unity in Fp
p=7; Zp=Integers(p)
for k in factor(p-1):
    print k
    root_list = Zp.zeta(k[0],all=True); root_list

# Plotting the above roots in Zp via exponentiation
a = plot([],figsize=(3,3),title='Roots of Unity of order 7',frame=True,axes_labels=['$x$-axis','$y$-axis'])
# Use exp:Zp -> C to map the Z/pZ cycle onto the unit circle
p=7; n=3 # divisor of p-1
root_list = Integers(p).zeta(n,all=True); root_list
A=[exp(2*pi*I*ZZ(k)/p) for k in root_list] # must convert Z/pZ elements into integers ZZ(k)
a+=list_plot(A, color='red', size=30, xmin=-1,xmax=1)
a+=circle((0.0,0.0),1,color='blue')
for k in range(p): # plotting all Z/pZ via exp
    a+=list_plot([exp(2*pi*I*k/p) for k in range(p)], size=10)
show(a)

# Plottinq Jacobi sums ... a challenge!
G=DirichletGroup(7); c=G[1]; c
J=c.jacobi_sum(c); J  # Jacobi sum J(c,c) belomgs to Q(zeta_(p-1))

# It plots the terms of the Jacobi sums
list_plot(J) # joins two lists of points

# Plotting Jacobi and Gauss sums
c.gauss_sum()
list_plot(c.gauss_sum())
list_plot(zip(J,c.gauss_sum()))

# ... with different colors ...
a = plot([],figsize=(3,3),title='Gauss and Jacobi Sums',frame=True,axes_labels=['$x$-axis','$y$-axis'])
a+=list_plot(J,color='red', size=30)
a+=list_plot(c.gauss_sum(),color='blue', size=10)
show(a)