Special Units in Real Cyclic Sextic Fields

Abstract

Abstract. We study the real cyclic sextic fields generated by a root w of (X -l)6 -(r2 + 108)(X2 + A')2, t e Z -{0, ±6, ±26}. We show that, when t1 + 108 is square-free (except for powers of 2 and 3), and t =t 0, ± 10. + 54, then w is a generator of the module of relative units. The details of the proofs are given in [3].

Connection between Gaussian periods and cyclic units

“…A similar relation holds between the roots of Marie Gras's quartic [2] for p = I + 16b2 and the cyclotomic quartic, except that the coefficients in the linear combination of the roots are excessively large, as we shall see in Section 4. Section 6 will be devoted to similar results for the sextic with 4p = L2 + 27, given by Marie Gras [3], [4]. We have not studied the case of 4p = 1 + 27M2, but we expect that there exists a similar relation.…”

“…The sextic period polynomials were given in [5]. Recently, Marie Gras [4] has given a sextic whose roots are units in a real sextic field. In the simplest case, in which Similarly, for L < 0 we find that F3((-L + l)/6) = 1.…”

Lattice Sieving in Three Dimensions for Discrete Log in Medium Characteristic

“…We implemented the Joux-Lercier-Smart-Vercauteren (JLSV 1 ) algorithm and ranking polynomials by their 3d Murphy E-score, after about 100 core hours found the following polynomial pair from the cyclic family of degree six described in [9]:…”

A New Angle on Lattice Sieving for the Number Field Sieve

“…We implemented the Joux-Lercier-Smart-Vercauteren (JLSV 1 ) algorithm and ranking polynomials by their 3d Murphy E-score, after about 100 core hours found the following polynomial pair from the cyclic family of degree six described in [7]:…”