Sage Reference Manual: Elliptic and Plane Curves - Mirrors

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Sage Reference Manual: Elliptic and Plane Curves, Release 6.1.1sage: EllipticCurve(’37b2’)Elliptic Curve defined by y^2 + y = x^3 + x^2 - 1873*x - 31833 over Rational Fieldsage: EllipticCurve(’5077a’)Elliptic Curve defined by y^2 + y = x^3 - 7*x + 6 over Rational Fieldsage: EllipticCurve(’389a’)Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational FieldOld Cremona labels are allowed:sage: EllipticCurve(’2400FF’)Elliptic Curve defined by y^2 = x^3 + x^2 + 2*x + 8 over Rational FieldUnicode labels are allowed:sage: EllipticCurve(u’389a’)Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational FieldWe create curves over a finite field as follows:sage: EllipticCurve([GF(5)(0),0,1,-1,0])Elliptic Curve defined by y^2 + y = x^3 + 4*x over Finite Field of size 5sage: EllipticCurve(GF(5), [0, 0,1,-1,0])Elliptic Curve defined by y^2 + y = x^3 + 4*x over Finite Field of size 5Elliptic curves over Z/NZ with N prime are of type “elliptic curve over a finite field”:sage: F = Zmod(101)sage: EllipticCurve(F, [2, 3])Elliptic Curve defined by y^2 = x^3 + 2*x + 3 over Ring of integers modulo 101sage: E = EllipticCurve([F(2), F(3)])sage: type(E)sage: E.category()Category of schemes over Ring of integers modulo 101In contrast, elliptic curves over Z/NZ with N composite are of type “generic elliptic curve”:sage: F = Zmod(95)sage: EllipticCurve(F, [2, 3])Elliptic Curve defined by y^2 = x^3 + 2*x + 3 over Ring of integers modulo 95sage: E = EllipticCurve([F(2), F(3)])sage: type(E)sage: E.category()Category of schemes over Ring of integers modulo 95The following is a curve over the complex numbers:sage: E = EllipticCurve(CC, [0,0,1,-1,0])sage: EElliptic Curve defined by y^2 + 1.00000000000000*y = x^3 + (-1.00000000000000)*x over Complex Fisage: E.j_invariant()2988.97297297297We can also create elliptic curves by giving the Weierstrass equation:sage: x, y = var(’x,y’)sage: EllipticCurve(y^2 + y == x^3 + x - 9)Elliptic Curve defined by y^2 + y = x^3 + x - 9 over Rational Field42 Chapter 10. Elliptic curve constructor
Sage Reference Manual: Elliptic and Plane Curves, Release 6.1.1sage: R. = GF(5)[]sage: EllipticCurve(x^3 + x^2 + 2 - y^2 - y*x)Elliptic Curve defined by y^2 + x*y = x^3 + x^2 + 2 over Finite Field of size 5We can explicitly specify the j-invariant:sage: E = EllipticCurve(j=1728); E; E.j_invariant(); E.label()Elliptic Curve defined by y^2 = x^3 - x over Rational Field1728’32a2’sage: E = EllipticCurve(j=GF(5)(2)); E; E.j_invariant()Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 52See trac ticket #6657sage: EllipticCurve(GF(144169),j=1728)Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 144169By default, when a rational value of j is given, the constructed curve is a minimal twist (minimal conductor forcurves with that j-invariant). This can be changed by setting the optional parameter minimal_twist, whichis True by default, to False:sage: EllipticCurve(j=100)Elliptic Curve defined by y^2 = x^3 + x^2 + 3392*x + 307888 over Rational Fieldsage: E =EllipticCurve(j=100); EElliptic Curve defined by y^2 = x^3 + x^2 + 3392*x + 307888 over Rational Fieldsage: E.conductor()33129800sage: E.j_invariant()100sage: E =EllipticCurve(j=100, minimal_twist=False); EElliptic Curve defined by y^2 = x^3 + 488400*x - 530076800 over Rational Fieldsage: E.conductor()298168200sage: E.j_invariant()100Without this option, constructing the curve could take a long time since both j and j − 1728 have to be factoredto compute the minimal twist (see trac ticket #13100):sage: E = EllipticCurve_from_j(2^256+1,minimal_twist=False)sage: E.j_invariant() == 2^256+1TrueTESTS:sage: R = ZZ[’u’, ’v’]sage: EllipticCurve(R, [1,1])Elliptic Curve defined by y^2 = x^3 + x + 1 over Multivariate Polynomial Ring in u, vover Integer RingWe create a curve and a point over QQbar (see #6879):sage: E = EllipticCurve(QQbar,[0,1])sage: E(0)(0 : 1 : 0)sage: E.base_field()Algebraic Field43
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CHAPTERTHIRTYNINEINDICES AND TABLES
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BIBLIOGRAPHY[WpJacobianVariety] htt
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PYTHON MODULE INDEXisage.interfaces
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Sage Reference Manual: Elliptic and Plane Curves - Mirrors

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