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lines changed Original file line number Diff line number Diff line change 81
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sage: zero(E.random_point())
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(0 : 1 : 0)
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- Due to a bug (:trac:`6413`), retrieving multiplication-by-`m` maps
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- when `m` is divisible by the characteristic currently fails ::
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+ Retrieving multiplication-by-`m` maps when `m` is divisible by the
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+ characteristic also works (since :issue:`37096`) ::
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sage: E = EllipticCurve(GF(7), [1,0])
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sage: phi = E.scalar_multiplication(7); phi
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Scalar-multiplication endomorphism [7] of Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 7
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- sage: phi.rational_maps() # known bug -- #6413
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- (x^49, y^49)
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+ sage: phi.rational_maps()
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+ (x^49, - y^49)
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sage: phi.x_rational_map()
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x^49
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+ sage: psi = E.scalar_multiplication(-2); psi
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+ Scalar-multiplication endomorphism [-2] of Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 7
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+ sage: chi = E.scalar_multiplication(-14); chi
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+ Scalar-multiplication endomorphism [-14] of Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 7
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+ sage: chi == psi * phi
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+ True
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+ sage: chi.rational_maps()
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+ ((x^196 - 2*x^98 + 1)/(-3*x^147 - 3*x^49),
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+ (-x^294*y^49 + 2*x^196*y^49 - 2*x^98*y^49 + y^49)/(-x^294 - 2*x^196 - x^98))
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+ sage: chi.x_rational_map()
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+ (2*x^196 + 3*x^98 + 2)/(x^147 + x^49)
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+ sage: chi.rational_maps() == tuple(f(*phi.rational_maps()) for f in psi.rational_maps())
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+ True
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+ sage: chi.x_rational_map() == psi.x_rational_map()(phi.x_rational_map())
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+ True
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::
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