\newcommand{\Bold}[1]{\mathbf{#1}}y^2 + a_{1}xy + a_{3}y = x^3 + a_{2}x^2 + a_{4}x + a_{6}
}}}
{{{id=13|
E = EllipticCurve_from_j(1728); E
///
Elliptic Curve defined by y^2 = x^3 - x over Rational Field
}}}
{{{id=12|
E.j_invariant()
///
1728
}}}
{{{id=8|
E = EllipticCurve_from_j(389); E
///
Elliptic Curve defined by y^2 + x*y = x^3 + x^2 + 10852*x + 811751 over Rational Field
}}}
{{{id=16|
E.j_invariant()
///
389
}}}
{{{id=69|
E = EllipticCurve_from_c4c6(8,19); E
///
Elliptic Curve defined by y^2 = x^3 - 1/6*x - 19/864 over Rational Field
}}}
{{{id=71|
E.c_invariants()
///
(8, 19)
}}}
{{{id=72|
E = EllipticCurve_from_cubic('x^3 + y^3 + z^3', [1,-1,0])
///
Traceback (most recent call last):
File "The Group Law
Example over QQ
{{{id=34| E = EllipticCurve([0,0,1,-1,0]); E /// Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field }}} {{{id=99| E.integral_points() /// [(-1 : 0 : 1), (0 : 0 : 1), (1 : 0 : 1), (2 : 2 : 1), (6 : 14 : 1)] }}} {{{id=100| 14*(14+1) == 6*(6-1)*(6+1) /// True }}} {{{id=101| 14*15 == 5*6*7 /// True }}} {{{id=102| P = E([6,14]); P /// (6 : 14 : 1) }}} {{{id=103| -P /// (6 : -15 : 1) }}} {{{id=43| show(plot(E, thickness=1), gridlines=True) ///
}}}
{{{id=30|
P = E([0,0]); P
///
(0 : 0 : 1)
}}}
{{{id=35|
P.curve()
///
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
}}}
{{{id=40|
R = P + P; R
///
(1 : 0 : 1)
}}}
{{{id=39|
R + P
///
(-1 : -1 : 1)
}}}
{{{id=38|
30*P
///
(79799551268268089761/62586636021357187216 : -754388827236735824355996347601/495133617181351428873673516736 : 1)
}}}
{{{id=37|
@interact
def f(n=(1..200)):
print n*P
show(plot(E) + plot(n*P,pointsize=100,color='red') )
///
|
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Example over Complex Field CC
{{{id=104| ComplexField(200) /// Complex Field with 200 bits of precision }}} {{{id=105| CC /// Complex Field with 53 bits of precision }}} {{{id=44| E = EllipticCurve(CC, [pi, e+I]); E /// Elliptic Curve defined by y^2 = x^3 + 3.14159265358979*x + (2.71828182845905+1.00000000000000*I) over Complex Field with 53 bits of precision }}} {{{id=42| P = E.lift_x(2010); P /// (2010.00000000000 : 90114.4123729358 + 5.54850203018320e-6*I : 1.00000000000000) }}} {{{id=41| P+P /// (502.498826251294 - 5.56916330211630e-7*I : 11264.3278918597 + 0.0000256617911193528*I : 1.00000000000000) }}} {{{id=36| 10*P /// (20.0677989357789 - 0.000353445046938822*I : 90.2629379753835 + 0.00316782900257474*I : 1.00000000000000) }}} {{{id=49| /// }}}Example over Finite Field
{{{id=51| E = EllipticCurve(GF(7), [2,3]); E /// Elliptic Curve defined by y^2 = x^3 + 2*x + 3 over Finite Field of size 7 }}} {{{id=58| E.plot(pointsize=100) ///
}}}
{{{id=48|
P = E.lift_x(2); P
///
(2 : 1 : 1)
}}}
{{{id=52|
P + P
///
(3 : 6 : 1)
}}}
{{{id=21|
3*P
///
(6 : 0 : 1)
}}}
{{{id=53|
4*P
///
(3 : 1 : 1)
}}}
{{{id=55|
5*P
///
(2 : 6 : 1)
}}}
{{{id=56|
P.additive_order()
///
6
}}}
{{{id=54|
6*P
///
(0 : 1 : 0)
}}}
{{{id=63|
///
}}}
{{{id=62|
///
}}}
Example over "Generic Base"
(this is too much of a pain in Sage still. :-( )
{{{id=77| /// }}} {{{id=19| /// }}}Visualizing Elliptic Curves
{{{id=107| var('foobar') /// foobar }}} {{{id=106| E = EllipticCurve([1.4, foobar^2]); E /// Elliptic Curve defined by y^2 = x^3 + 1.40000000000000*x + foobar^2 over Symbolic Ring }}} {{{id=60| E = EllipticCurve([1,2,3,4,5]) E.plot(color=hue(.8), thickness=5, xmax=5) ///
}}}
{{{id=59|
E = EllipticCurve(RR, [pi, e])
plot(E)
///
}}}
The above looks crappy on the left -- somebody could fix that by increasing the default plot_points parameter. See below:
{{{id=79| E = EllipticCurve(RR, [pi, e]) plot(E, plot_points=500) ///
}}}
{{{id=81|
E.plot??
///
}}}
We can also plot over prime finite fields:
{{{id=78| EllipticCurve(GF(101), [1,7]).plot() ///
}}}
Nontrivial prime power order finite fields aren't yet supported...
{{{id=22| EllipticCurve(GF(27,'a'), [1,7]).plot() /// Traceback (most recent call last): File "It's fun to start with a curve over the rational numbers, and vary the finite field:
{{{id=84| @interact def _(p=primes(1000)): E = EllipticCurve(GF(p),[0,0,1,-1,0]) show(E.plot()) ///
|
|||||||||||||||
Elliptic Curves define curves over the complex numbers, which we can (try to) plot in various ways, e.g., consider the plot of $\sqrt{x^3+ax+b}$ associated to $y^2=x^3+ax+b$.
{{{id=89| var('x') complex_plot(sqrt(x^3+2*x+3), (-2,1), (-1,1)) ///
}}}
{{{id=91|
E = EllipticCurve([2,3])
E.plot(xmin=-2,xmax=1).show(xmin=-2,xmax=1)
///
}}}
{{{id=92|
def f(x1,x2):
z = CDF(x1,x2)
return abs((z^3+2*z+3).sqrt())
plot3d(f, [-2,1], [-2,2])
///
}}}
{{{id=88|
///
}}}
{{{id=87|
///
}}}
{{{id=24|
///
}}}
{{{id=25|
///
}}}
Some of What is Missing or Slow
- Cubic to curve (see above)
- Other coordinates
- 3d visualization could be improved
- speed of arithmetic (too technical for class?)