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Some Quartic Curves with no Points in any Cubic Field
Author(s) -
Bremner Andrew
Publication year - 1986
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-52.2.193
Subject(s) - mathematics , quartic function , elliptic curve , rank (graph theory) , rational number , jacobian matrix and determinant , extension (predicate logic) , field (mathematics) , degree (music) , pure mathematics , twists of curves , rational point , mathematical analysis , schoof's algorithm , combinatorics , algebraic number , quarter period , physics , computer science , acoustics , programming language
A sufficient condition is given such that the curves Γ D : x 4 $ y 4 = D z 4 have no points in any odd‐degree (greater than 1) extension field of the rationals. The condition is in terms of the rational rank of an elliptic curve in the Jacobian of Γ D . Various examples are given.