Jacobian curve: Difference between revisions

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In [[mathematics|mathematical]] field of  [[algebraic geometry]], an [[elliptic curve]] E over a [[field (mathematics)|field]] K  has an associated '''quadratic twist''', that is another elliptic curve which is [[isomorphism|isomorphic]] to E over an [[algebraic closure]] of K. In particular, an isomorphism between elliptic curves is an [[isogeny]] of degree 1, that is an invertible isogeny. Some curves have higher order twists such as '''cubic'''
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and '''quartic twists'''. The curve and its twists have the same [[j-invariant]].
 
==Quadratic twist==
 
First assume K is a field of [[characteristic (algebra)|characteristic]] different from 2.
Let E be an [[elliptic curve]] over K of the form:
 
: <math>y^2 = x^3 + a_2 x^2 +a_4 x + a_6. \, </math>
 
Given <math>d\in K\setminus K^2</math> and <math>d\neq 0</math>, the '''quadratic twist''' of E is the curve E<sup>d</sup>, defined by the equation:
 
: <math>dy^2 = x^3 + a_2 x^2 + a_4 x + a_6. \, </math>
 
or equivalently
 
: <math>y^2 = x^3 + d a_2 x^2 + d^2 a_4 x + d^3 a_6. \, </math>
 
The two elliptic curves E and E<sup>d</sup> are not isomorphic over K, but over the [[field extension]] <math>K(\sqrt{d})</math>.
 
Now assume K is of characteristic 2. Let E be an [[elliptic curve]] over K of the form:
 
: <math>y^2 + a_1 x y +a_3 y = x^3 + a_2 x^2 +a_4 x + a_6. \, </math>
 
Given <math>d\in K</math> such that <math>X^2+X+d</math> is an [[irreducible polynomial]] over K, the '''quadratic twist''' of E is the curve E<sup>d</sup>, defined by the equation:
 
: <math>y^2 + a_1 x y +a_3 y = x^3 + (a_2 + d a_1^2) x^2 +a_4 x + a_6 + d a_3^2. \, </math>
 
The two elliptic curves E and E<sup>d</sup> are not isomorphic over K, but over the [[field extension]] <math>K[X]/(X^2+X+d)</math>.
 
===Quadratic twist over finite fields===
 
If K is a [[finite field]] with q elements, then for all x there exist a y such that the point <math>(x,y)</math> belongs to either E or E<sup>d</sup>.
In fact there is always exactly two such y unless the point belongs to both curves (which can happen if the characteristic is not 2).
 
As a consequence
 
: <math> |E(K)|+|E^d(K)| = 2 q+2 </math> or equivalently <math> t_{E^d} = - t_E </math>
 
where <math>t_E</math> is the trace of the [[Frobenius endomorphism]] of the curve.
 
==Quartic twist==
 
It is possible to "twist" elliptic curves with j-invariant equal to 1728 by quartic characters; twisting a curve E by a '''quartic twist''', one obtains precisely four  curves: one is isomorphic to E, one is its quadratic twist, and only the other two are really new.
Also in this case, twisted curves are isomorphic over the field extension given by the twist degree.
 
==Cubic twist==
 
Analogously to the quartic twist case, an elliptic curve over K with j-invariant equal to zero can be twisted by cubic characters. The curves obtained are isomorphic to the starting curve over the field extension given by the twist degree.
 
==Examples==
 
1.[[Twisted Hessian curves]]
 
2.[[Twisted Edwards curve]]
 
3.[[Tripling-oriented Doche–Icart–Kohel curve#Equivalence with Weierstrass form|Twisted tripling-oriented Doche–Icart–Kohel curve]]
 
==References==
 
* {{cite book
| author = P. Stevenhagen
| year = 2008
| title = Elliptic Curves
| publisher = Universiteit Leiden
| url = http://websites.math.leidenuniv.nl/algebra/ellcurves.pdf
}}
 
* {{cite book
| author = F. Gouvea, [[Barry Mazur|B.Mazur]]
| year = 1991
| title = The square-free sieve and the rank of elliptic curves
| publisher = Journal of American Mathematical Society, Vol 4, Num 1
| url = http://www.ams.org/jams/1991-04-01/S0894-0347-1991-1080648-7/S0894-0347-1991-1080648-7.pdf
}}
 
* {{cite book
| author = C. L. Stewart and J. Top
| year = 1995
| title = On Ranks of Twists of Elliptic Curves and Power-Free Values of Binary Forms
| publisher = Journal of the American Mathematical Society, Vol. 8, No. 4 (Oct., 1995), pp. 943–973
| url = http://www.jstor.org/stable/pdfplus/2152834.pdf
}}
 
[[Category:Elliptic curves]]
[[Category:Elliptic curve cryptography]]

Latest revision as of 14:46, 11 February 2014

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