This worksheet accompanies these slides.
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Pythagorean triples
@interact
def _(t=(1/4,(1/16,1/8,..,1))):
t0 = t
x,y,t=var('x,y,t')
show([x==(1-t^2)/(1+t^2), y==2*t/(1+t^2)])
t = t0
(x,y) = ((1-t^2)/(1+t^2), 2*t/(1+t^2))
a = 1/3
G = circle((0,0), 1, color='blue', thickness=3)
G += arrow((-1-a,-t*a), (x+a,y+t*a), head=2, color='red')
G += point((0,t), pointsize=150, color='black', zorder=100)
G += point((-1,0), pointsize=150, color='black', zorder=100)
G += point((x,y), pointsize=190, color='lightgreen', zorder=100)
G += text("$(0, %s)$"%t, (-.3, t+.2), fontsize=16, color='black')
G += text(r"$(%s,\,%s)$"%(x,y), (x+.35, y+.25), fontsize=16, color='black')
G.show(aspect_ratio=1, ymin=-1.1, ymax=1.1, xmax=1.3, xmin=-1.3, fontsize=0, figsize=6)
Click to the left again to hide and once more to show the dynamic interactive window |
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Cubic equations
var('x,y')
implicit_plot(x^3 + y^3 == 1, (x,-2,2), (y,-2,2), aspect_ratio=1, figsize=5, gridlines=True)
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var('x,y')
implicit_plot(y^2 + y == x^3 - x, (x,-2,3), (y,-4.5,4), aspect_ratio=1/2, figsize=5, gridlines=True)
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@interact
def _(equation='x^3 + y^3 == 1',
xmin=-2, xmax=2, ymin=-2, ymax=2, aspect_ratio=1, gridlines=True):
x,y = var('x,y')
eqn = sage_eval(equation, {'x':x, 'y':y})
print eqn
G = implicit_plot(eqn, (x,xmin,xmax), (y,ymin,ymax), aspect_ratio=aspect_ratio, gridlines=gridlines)
G.show(figsize=4)
Click to the left again to hide and once more to show the dynamic interactive window |
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The Group Law
How the figure in the slide was made:
E = EllipticCurve([0,0,1,-1,0]); print E
G = E.plot(plot_points=600, thickness=2)
G += arrow((-2,1), (3,-4), head=2, color='red', width=2)
G += points([(-1,0), (0,-1), (2,-3)], color='black', pointsize=70, zorder=50)
G += text("$(2,-3)$", (1.3,-3.1), fontsize=18, color='black')
G += text("$(-1,0)$", (-.9,1), fontsize=18, color='black')
G += text("$(0,-1)$", (-.7,-1.85), fontsize=18, color='black')
G.show(gridlines=True, frame=True, aspect_ratio=1/2, xmax=3.1, xmin=-2, figsize=5)
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field |
E = EllipticCurve([0,0,1,-1,0])
P = E([-1,0]); Q = E([0,-1]); R = E([2,-3])
print P + Q + R
print -(P+Q)
print 7*P
(0 : 1 : 0) (2 : -3 : 1) (1849037896/6941055969 : -260151768440137/578280195945297 : 1) (0 : 1 : 0) (2 : -3 : 1) (1849037896/6941055969 : -260151768440137/578280195945297 : 1) |
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