Determine the rational points on the two remaining Klein-quartic twists

Determine the rational points of the quartic twists Z(β) and Z(β^3) of the Klein quartic, where β = ε₁ε₂⁴ in the notation of the paper, in order to complete the rational-point analysis of the three quartics arising from the Kummer descent.

Background

The Kummer descent associates to the relevant superelliptic equations a family of quartic curves Z(η), indexed by classes in a quotient of K×, where K = Q(ζ₇)⁺. The paper proves that exactly three isomorphism classes arise: the Klein quartic Z(1), together with two non-isomorphic twists Z(β) and Z(β³), with β = ε₁ε₂⁴.

The rational points of the untwisted quartic Z(1), which is isomorphic over Q to the Klein quartic, are determined using Hurwitz’s reduction to Fermat’s Last Theorem for exponent 7. The two other twists are shown to be pairwise non-isomorphic to Z(1) and to one another, partly by comparing their reductions modulo 29, but their rational points are not computed. Determining these points would clarify the arithmetic of the remaining twists, although the result of the paper does not require them.