Computing all elements of given index in sextic fields with a cubic subfield
Let be a given real quadratic field. We give a fast algorithm for determining all dihedral quartic fields with mixed signature having power integral bases and containing as a subfield. We also determine all generators of power integral bases in . Our algorithm combines a recent result of Kable [9] with the algorithm of Gaál, Pethö and Pohst [6], [7]. To illustrate the method we performed computations for
We exhibit an algorithm to compute a Dirichlet domain for a Fuchsian group with cofinite area. As a consequence, we compute the invariants of , including an explicit finite presentation for .
We extend the "character sum method" for the computation of densities in Artin primitive root problems given by Lenstra and the authors to the situation of radical extensions of arbitrary rank. Our algebraic set-up identifies the key parameters of the situation at hand, and obviates the lengthy analytic multiplicative number theory arguments that used to go into the computation of actual densities. It yields a conceptual interpretation of the formulas obtained, and enables us to extend their range...
We give an algorithm to compute the modular degree of an elliptic curve defined over . Our method is based on the computation of the special value at of the symmetric square of the -function attached to the elliptic curve. This method is quite efficient and easy to implement.
We develop a recursive method for computing the -removed -orderings and -orderings of order the characteristic sequences associated to these and limits associated to these sequences for subsets of a Dedekind domain This method is applied to compute these objects for and .
Let be an elliptic curve having complex multiplication by a given quadratic order of an imaginary quadratic field . The field of definition of is the ring class field of the order. If the prime splits completely in , then we can reduce modulo one the factors of and get a curve defined over . The trace of the Frobenius of is known up to sign and we need a fast way to find this sign, in the context of the Elliptic Curve Primality Proving algorithm (ECPP). For this purpose, we propose...