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Displaying similar documents to “Generators and integral points on twists of the Fermat cubic”

On the fundamental units of some cubic orders generated by units

Jun Ho Lee, Stéphane R. Louboutin (2014)

Acta Arithmetica

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Let ϵ be a totally real cubic algebraic unit. Assume that the cubic number field ℚ(ϵ) is Galois. Let ϵ, ϵ' and ϵ'' be the three real conjugates of ϵ. We tackle the problem of whether {ϵ,ϵ'} is a system of fundamental units of the cubic order ℤ[ϵ,ϵ',ϵ'']. Given two units of a totally real cubic order, we explain how one can prove that they form a system of fundamental units of this order. Several explicit families of totally real cubic orders defined by parametrized families of cubic...

Note on autopolar plane cubics

Hari das Bagchi, Manindra Chandra Chaki (1952)

Rendiconti del Seminario Matematico della Università di Padova

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An explicit algebraic family of genus-one curves violating the Hasse principle

Bjorn Poonen (2001)

Journal de théorie des nombres de Bordeaux

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We prove that for any t ∈ 𝐐 , the curve 5 x 3 + 9 y 3 + 10 z 3 + 12 t 2 + 82 t 2 + 22 3 ( x + y + z ) 3 = 0 in 𝐏 2 is a genus 1 curve violating the Hasse principle. An explicit Weierstrass model for its jacobian E t is given. The Shafarevich-Tate group of each E t contains a subgroup isomorphic to 𝐙 / 3 × 𝐙 / 3 .