About: Imaginary hyperelliptic curve

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A hyperelliptic curve is a particular kind of algebraic curve. There exist hyperelliptic curves of every genus . If the genus of a hyperelliptic curve equals 1, we simply call the curve an elliptic curve. Hence we can see hyperelliptic curves as generalizations of elliptic curves. There is a well-known group structure on the set of points lying on an elliptic curve over some field , which we can describe geometrically with chords and tangents. Generalizing this group structure to the hyperelliptic case is not straightforward. We cannot define the same group law on the set of points lying on a hyperelliptic curve, instead a group structure can be defined on the so-called Jacobian of a hyperelliptic curve. The computations differ depending on the number of points at infinity. This article is

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rdfs:label	Imaginary hyperelliptic curve (en)
rdfs:comment	A hyperelliptic curve is a particular kind of algebraic curve. There exist hyperelliptic curves of every genus . If the genus of a hyperelliptic curve equals 1, we simply call the curve an elliptic curve. Hence we can see hyperelliptic curves as generalizations of elliptic curves. There is a well-known group structure on the set of points lying on an elliptic curve over some field , which we can describe geometrically with chords and tangents. Generalizing this group structure to the hyperelliptic case is not straightforward. We cannot define the same group law on the set of points lying on a hyperelliptic curve, instead a group structure can be defined on the so-called Jacobian of a hyperelliptic curve. The computations differ depending on the number of points at infinity. This article is (en)
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dcterms:subject	Algebraic curves
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Link from a Wikipage to another Wikipage	Projective plane Monic polynomial Multiplicity (mathematics) Algebraic closure Algorithm David G. Cantor Integral domain Jacobian variety Genus (mathematics) Quotient group Elliptic curve Georg Cantor Singular point of a curve Closure (mathematics) Function field of an algebraic variety Kernel (algebra) Picard group Subgroup Bézout's theorem Irreducible polynomial Algebraic curve Extended Euclidean algorithm Field (mathematics) Abel-Jacobi map Uniqueness quantification Group (mathematics) Group homomorphism Hyperelliptic curve Hyperelliptic curve cryptography Abelian group Algebraic curves Characteristic (algebra) Polynomial Field of fractions Group isomorphism Neal Koblitz Real hyperelliptic curve Variable (mathematics) Formal sum Leading coefficient Weierstrass point Point at infinity Polynomial function
sameAs	Imaginary hyperelliptic curve Imaginary hyperelliptic curve Imaginary hyperelliptic curve Imaginary hyperelliptic curve
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has abstract	A hyperelliptic curve is a particular kind of algebraic curve. There exist hyperelliptic curves of every genus . If the genus of a hyperelliptic curve equals 1, we simply call the curve an elliptic curve. Hence we can see hyperelliptic curves as generalizations of elliptic curves. There is a well-known group structure on the set of points lying on an elliptic curve over some field , which we can describe geometrically with chords and tangents. Generalizing this group structure to the hyperelliptic case is not straightforward. We cannot define the same group law on the set of points lying on a hyperelliptic curve, instead a group structure can be defined on the so-called Jacobian of a hyperelliptic curve. The computations differ depending on the number of points at infinity. This article is about imaginary hyperelliptic curves, these are hyperelliptic curves with exactly 1 point at infinity. Real hyperelliptic curves have two points at infinity. (en)
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foaf:isPrimaryTopicOf	wikipedia-en:Imaginary_hyperelliptic_curve
is Link from a Wikipage to another Wikipage of	David G. Cantor Division polynomials Hyperelliptic curve Hyperelliptic curve cryptography Real hyperelliptic curve Schoof's algorithm
is foaf:primaryTopic of	wikipedia-en:Imaginary_hyperelliptic_curve

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