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Matthias Koeppe
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src/sage/schemes/elliptic_curves/hom_frobenius.py: Mark sage.doctest: optional - sage.rings.finite_rings
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src/sage/schemes/elliptic_curves/hom_frobenius.py

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# sage.doctest: optional - sage.rings.finite_rings
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r"""
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Frobenius isogenies of elliptic curves
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sage: E = EllipticCurve([z5,1])
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sage: pi = EllipticCurveHom_frobenius(E); pi
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Frobenius isogeny of degree 17:
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From: Elliptic Curve defined by y^2 = x^3 + z5*x + 1 over Finite Field in z5 of size 17^5
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To: Elliptic Curve defined by y^2 = x^3 + (9*z5^4+7*z5^3+10*z5^2+z5+14)*x + 1 over Finite Field in z5 of size 17^5
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From: Elliptic Curve defined by y^2 = x^3 + z5*x + 1
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over Finite Field in z5 of size 17^5
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To: Elliptic Curve defined by y^2 = x^3 + (9*z5^4+7*z5^3+10*z5^2+z5+14)*x + 1
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over Finite Field in z5 of size 17^5
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By passing `n`, we can also construct higher-power Frobenius maps,
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such as the Frobenius *endo*\morphism::
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sage: z5, = GF(7^5).gens()
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sage: E = EllipticCurve([z5,1])
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sage: pi = EllipticCurveHom_frobenius(E,5); pi
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sage: pi = EllipticCurveHom_frobenius(E, 5); pi
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Frobenius endomorphism of degree 16807 = 7^5:
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From: Elliptic Curve defined by y^2 = x^3 + z5*x + 1 over Finite Field in z5 of size 7^5
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To: Elliptic Curve defined by y^2 = x^3 + z5*x + 1 over Finite Field in z5 of size 7^5
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From: Elliptic Curve defined by y^2 = x^3 + z5*x + 1
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over Finite Field in z5 of size 7^5
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To: Elliptic Curve defined by y^2 = x^3 + z5*x + 1
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over Finite Field in z5 of size 7^5
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The usual :class:`EllipticCurveHom` methods are supported::
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