Sage Reference Manual: Elliptic and Plane Curves - Mirrors

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Sage Reference Manual: Elliptic and Plane Curves, Release 6.1.1sage: E = EllipticCurve(RR,[1,2]); E; E.base_field()Elliptic Curve defined by y^2 = x^3 + 1.00000000000000*x + 2.00000000000000 over Real Field withReal Field with 53 bits of precisionsage: EllipticCurve(CC,[3,4]); E; E.base_field()Elliptic Curve defined by y^2 = x^3 + 3.00000000000000*x + 4.00000000000000 over Complex Field wElliptic Curve defined by y^2 = x^3 + 1.00000000000000*x + 2.00000000000000 over Real Field withReal Field with 53 bits of precisionsage: E = EllipticCurve(QQbar,[5,6]); E; E.base_field()Elliptic Curve defined by y^2 = x^3 + 5*x + 6 over Algebraic FieldAlgebraic FieldSee trac ticket #6657sage: EllipticCurve(3,j=1728)Traceback (most recent call last):...ValueError: First parameter (if present) must be a ring when j is specifiedsage: EllipticCurve(GF(5),j=3/5)Traceback (most recent call last):...ValueError: First parameter must be a ring containing 3/5If the universe of the coefficients is a general field, the object constructed has type EllipticCurve_field. Otherwiseit is EllipticCurve_generic. See trac ticket #9816sage: E = EllipticCurve([QQbar(1),3]); EElliptic Curve defined by y^2 = x^3 + x + 3 over Algebraic Fieldsage: type(E)sage: E = EllipticCurve([RR(1),3]); EElliptic Curve defined by y^2 = x^3 + 1.00000000000000*x + 3.00000000000000 over Real Field withsage: type(E)sage: E = EllipticCurve([i,i]); EElliptic Curve defined by y^2 = x^3 + I*x + I over Symbolic Ringsage: type(E)sage: E.category()Category of schemes over Symbolic Ringsage: SR in Fields()Truesage: F = FractionField(PolynomialRing(QQ,’t’))sage: t = F.gen()sage: E = EllipticCurve([t,0]); EElliptic Curve defined by y^2 = x^3 + t*x over Fraction Field of Univariate Polynomial Ring in tsage: type(E)sage: E.category()Category of schemes over Fraction Field of Univariate Polynomial Ring in t over Rational FieldSee trac ticket #12517:sage: E = EllipticCurve([1..5])sage: EllipticCurve(E.a_invariants())Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field44 Chapter 10. Elliptic curve constructor
Sage Reference Manual: Elliptic and Plane Curves, Release 6.1.1See trac ticket #11773:sage: E = EllipticCurve()Traceback (most recent call last):...TypeError: invalid input to EllipticCurve constructorsage.schemes.elliptic_curves.constructor.EllipticCurve_from_Weierstrass_polynomial(f )Return the elliptic curve defined by a cubic in (long) Weierstrass form.INPUT:•f – a inhomogeneous cubic polynomial in long Weierstrass form.OUTPUT:The elliptic curve defined by it.EXAMPLES:sage: R. = QQ[]sage: f = y^2 + 1*x*y + 3*y - (x^3 + 2*x^2 + 4*x + 6)sage: EllipticCurve(f)Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 6 over Rational Fieldsage: EllipticCurve(f).a_invariants()(1, 2, 3, 4, 6)The polynomial ring may have extra variables as long as they do not occur in the polynomial itself:sage: R. = QQ[]sage: EllipticCurve(-y^2 + x^3 + 1)Elliptic Curve defined by y^2 = x^3 + 1 over Rational Fieldsage: EllipticCurve(-x^2 + y^3 + 1)Elliptic Curve defined by y^2 = x^3 + 1 over Rational Fieldsage: EllipticCurve(-w^2 + z^3 + 1)Elliptic Curve defined by y^2 = x^3 + 1 over Rational FieldTESTS:sage: from sage.schemes.elliptic_curves.constructor import EllipticCurve_from_Weierstrass_polynosage: EllipticCurve_from_Weierstrass_polynomial(-w^2 + z^3 + 1)Elliptic Curve defined by y^2 = x^3 + 1 over Rational Fieldsage.schemes.elliptic_curves.constructor.EllipticCurve_from_c4c6(c4, c6)Return an elliptic curve with given c 4 and c 6 invariants.EXAMPLES:sage: E = EllipticCurve_from_c4c6(17, -2005)sage: EElliptic Curve defined by y^2 = x^3 - 17/48*x + 2005/864 over Rational Fieldsage: E.c_invariants()(17, -2005)sage.schemes.elliptic_curves.constructor.EllipticCurve_from_cubic(F, P, morphism=True)Construct an elliptic curve from a ternary cubic with a rational point.If you just want the Weierstrass form and are not interested in the morphism then it is easier to use Jacobian()instead. This will construct the same elliptic curve but you don’t have to supply the point P.INPUT:45
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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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