Mathematical curves that are isomorphic over algebraic closures
In the mathematical field of algebraic geometry, an elliptic curve E over a field K has an associated quadratic twist, that is another elliptic curve which is 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 and quartic twists. The curve and its twists have the same j-invariant.
Applications of twists include cryptography,[1] the solution of Diophantine equations,[2][3] and when generalized to hyperelliptic curves, the study of the Sato–Tate conjecture.[4]
^Bos, Joppe W.; Halderman, J. Alex; Heninger, Nadia; Moore, Jonathan; Naehrig, Michael; Wustrow, Eric (2014). "Elliptic Curve Cryptography in Practice". In Christin, Nicolas; Safavi-Naini, Reihaneh (eds.). Financial Cryptography and Data Security. Lecture Notes in Computer Science. Vol. 8437. Berlin, Heidelberg: Springer. pp. 157–175. doi:10.1007/978-3-662-45472-5_11. ISBN 978-3-662-45471-8. Retrieved 2022-04-10.
^Mazur, B.; Rubin, K. (September 2010). "Ranks of twists of elliptic curves and Hilbert's tenth problem". Inventiones Mathematicae. 181 (3): 541–575. arXiv:0904.3709. Bibcode:2010InMat.181..541M. doi:10.1007/s00222-010-0252-0. ISSN 0020-9910. S2CID 3394387.
^Poonen, Bjorn; Schaefer, Edward F.; Stoll, Michael (2007-03-15). "Twists of X(7) and primitive solutions to x2+y3=z7". Duke Mathematical Journal. 137 (1). arXiv:math/0508174. doi:10.1215/S0012-7094-07-13714-1. ISSN 0012-7094. S2CID 2326034.
^Lombardo, Davide; Lorenzo García, Elisa (February 2019). "Computing twists of hyperelliptic curves". Journal of Algebra. 519: 474–490. arXiv:1611.04856. Bibcode:2016arXiv161104856L. doi:10.1016/j.jalgebra.2018.08.035. S2CID 119143097.
and 23 Related for: Twists of elliptic curves information
enough to include all non-singular cubic curves; see § Ellipticcurves over a general field below.) An ellipticcurve is an abelian variety – that is, it has...
In algebraic geometry, the twisted Edwards curves are plane models ofellipticcurves, a generalisation of Edwards curves introduced by Bernstein, Birkner...
mathematics, the Edwards curves are a family ofellipticcurves studied by Harold Edwards in 2007. The concept ofellipticcurves over finite fields is widely...
In mathematics, the Twisted Hessian curve represents a generalization of Hessian curves; it was introduced in ellipticcurve cryptography to speed up the...
This curve was suggested for application in ellipticcurve cryptography, because arithmetic in this curve representation is faster and needs less memory...
along with Jaap Top, the existence of infinitely many quadratic, cubic, and sextic twistsofellipticcurvesof large rank. In 1991 and 2001 respectively...
psychology, ecology, etc. Rational curves are subdivided according to the degree of the polynomial. Line Plane curvesof degree 2 are known as conics or...
Dual_EC_DRBG (Dual EllipticCurve Deterministic Random Bit Generator) is an algorithm that was presented as a cryptographically secure pseudorandom number...
an ellipticcurve used in elliptic-curve cryptography (ECC) offering 128 bits of security (256-bit key size) and designed for use with the Elliptic-curve...
Taniyama-Shimura-Weil conjecture or modularity conjecture for ellipticcurves) states that ellipticcurves over the field of rational numbers are related to modular forms...
In mathematics, elliptic cohomology is a cohomology theory in the sense of algebraic topology. It is related to ellipticcurves and modular forms. Historically...
Edwards-curve Digital Signature Algorithm (EdDSA) is a digital signature scheme using a variant of Schnorr signature based on twisted Edwards curves. It is...
covered by a continuous family of images ofellipticcurves. (These curves are singular in X, unless X happens to be an elliptic K3 surface.) A stronger question...
converge. The Tate curve was introduced by John Tate (1995) in a 1959 manuscript originally titled "Rational Points on EllipticCurves Over Complete Fields";...
points of order dividing n of an ellipticcurve E, taking values in nth roots of unity. More generally there is a similar Weil pairing between points of order...
Ellipticcurve cryptography is a popular form of public key encryption that is based on the mathematical theory ofellipticcurves. Points on an elliptic...
an ellipticcurve is an abelian variety. Given an integer g ≥ 0 {\displaystyle g\geq 0} , the set of isomorphism classes of smooth complete curvesof genus...
open-source software portal mwrank is one in a suite of programs for computing ellipticcurves over rational numbers. Other programs in the suite compute...
that most algebraic curves are of general type: in the moduli space ofcurves, two connected components correspond to curves not of general type, while...
Global Information