Sage Reference Manual: Elliptic and Plane Curves - Mirrors

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Sage Reference Manual: Elliptic and Plane Curves, Release 6.1.140 Chapter 9. Projective plane conics over prime finite fields
CHAPTERTENELLIPTIC CURVE CONSTRUCTORAUTHORS:• William Stein (2005): Initial version• John Cremona (2008-01): EllipticCurve(j) fixed for all casessage.schemes.elliptic_curves.constructor.EllipticCurve(x=None, y=None, j=None,minimal_twist=True)Construct an elliptic curve.In Sage, an elliptic curve is always specified by its a-invariantsINPUT:There are several ways to construct an elliptic curve:y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 .•EllipticCurve([a1,a2,a3,a4,a6]): Elliptic curve with given a-invariants. The invariants arecoerced into the parent of the first element. If all are integers, they are coerced into the rational numbers.•EllipticCurve([a4,a6]): Same as above, but a 1 = a 2 = a 3 = 0.•EllipticCurve(label): Returns the elliptic curve over Q from the Cremona database with thegiven label. The label is a string, such as "11a" or "37b2". The letters in the label must be lower case(Cremona’s new labeling).•EllipticCurve(R, [a1,a2,a3,a4,a6]): Create the elliptic curve over R with given a-invariants. Here R can be an arbitrary ring. Note that addition need not be defined.•EllipticCurve(j=j0) or EllipticCurve_from_j(j0): Return an elliptic curve with j-invariant j0.•EllipticCurve(polynomial): Read off the a-invariants from the polynomial coefficients, seeEllipticCurve_from_Weierstrass_polynomial().In each case above where the input is a list of length 2 or 5, one can instead give a 2 or 5-tuple instead.EXAMPLES:We illustrate creating elliptic curves:sage: EllipticCurve([0,0,1,-1,0])Elliptic Curve defined by y^2 + y = x^3 - x over Rational FieldWe create a curve from a Cremona label:41
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CHAPTERTHIRTYNINEINDICES AND TABLES
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BIBLIOGRAPHY[WpJacobianVariety] htt
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Sage Reference Manual: Elliptic and Plane Curves - Mirrors

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