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Package sage :: Package ellcurve :: Module ellcurve :: Class EllipticCurve |
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object
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EllipticCurve
EllipticCurve_generic
Elliptic curve. There are several forms of this function. -- EllipticCurve([a1,a2,a3,a4,a6]): Elliptic curve with given a-invariants. The invariants are coerced into a the parent of the first element. If all are integers, they are coerced into the rational numbers. -- EllipticCurve([a4,a6]): Same as above, but a1=a2=a3=0. -- EllipticCurve(label): Returns the elliptic curve over Q from the Cremoa database with the given label. The label is a string, such as"11A" or "37B2". -- 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. EXAMPLES: We illustrate creating elliptic curves. >>> EllipticCurve([0,0,1,-1,0]) Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field >>> EllipticCurve([GF(5)(0),0,1,-1,0]) Elliptic Curve defined by y^2 + y = x^3 - x over Finite field of size 5 Alternatively, one can create the curve over the finite field as follows: >>> EllipticCurve(GF(5), [0, 0,1,-1,0]) Elliptic Curve defined by y^2 + y = x^3 - x over Finite field of size 5 >>> E = EllipticCurve([C(0),0,1,-1,0]) >>> E Elliptic Curve defined by y^2 + y = x^3 - x over Complex Field >>> E.j_invariant() 2988.97297297297297297 >>> E = EllipticCurve(ZZ, [0, 0,1,-1,0]) >>> E Elliptic Curve defined by y^2 + y = x^3 - x over Integer Ring Of course, arithmetic on elliptic curves over Z need not be defined: >>> P = E([0,0]) >>> P + P + P + P (2, -3) >>> P + P + P + P + P Traceback (most recent call last): ... ArithmeticError: Point (1/4, -5/8) is not on curve.
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__init__(self,
*args,
**kwds)
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__new__(cls,
*args,
**kwds)
(Static method) | |
Inherited from object | |
x.__delattr__('name') <==> del x.name | |
x.__getattribute__('name') <==> x.name | |
x.__hash__() <==> hash(x) | |
helper for pickle | |
helper for pickle | |
x.__repr__() <==> repr(x) | |
x.__setattr__('name', value) <==> x.name = value | |
x.__str__() <==> str(x) |
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