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| 1 | +# sage.doctest: optional - sage.rings.finite_rings |
1 | 2 | r"""
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2 | 3 | Frobenius isogenies of elliptic curves
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3 | 4 |
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17 | 18 | sage: E = EllipticCurve([z5,1])
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18 | 19 | sage: pi = EllipticCurveHom_frobenius(E); pi
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19 | 20 | Frobenius isogeny of degree 17:
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20 |
| - From: Elliptic Curve defined by y^2 = x^3 + z5*x + 1 over Finite Field in z5 of size 17^5 |
21 |
| - 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 |
| 21 | + From: Elliptic Curve defined by y^2 = x^3 + z5*x + 1 |
| 22 | + over Finite Field in z5 of size 17^5 |
| 23 | + To: Elliptic Curve defined by y^2 = x^3 + (9*z5^4+7*z5^3+10*z5^2+z5+14)*x + 1 |
| 24 | + over Finite Field in z5 of size 17^5 |
22 | 25 |
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23 | 26 | By passing `n`, we can also construct higher-power Frobenius maps,
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24 | 27 | such as the Frobenius *endo*\morphism::
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25 | 28 |
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26 | 29 | sage: z5, = GF(7^5).gens()
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27 | 30 | sage: E = EllipticCurve([z5,1])
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28 |
| - sage: pi = EllipticCurveHom_frobenius(E,5); pi |
| 31 | + sage: pi = EllipticCurveHom_frobenius(E, 5); pi |
29 | 32 | Frobenius endomorphism of degree 16807 = 7^5:
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30 |
| - From: Elliptic Curve defined by y^2 = x^3 + z5*x + 1 over Finite Field in z5 of size 7^5 |
31 |
| - To: Elliptic Curve defined by y^2 = x^3 + z5*x + 1 over Finite Field in z5 of size 7^5 |
| 33 | + From: Elliptic Curve defined by y^2 = x^3 + z5*x + 1 |
| 34 | + over Finite Field in z5 of size 7^5 |
| 35 | + To: Elliptic Curve defined by y^2 = x^3 + z5*x + 1 |
| 36 | + over Finite Field in z5 of size 7^5 |
32 | 37 |
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33 | 38 | The usual :class:`EllipticCurveHom` methods are supported::
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34 | 39 |
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