Machine Learning Class Numbers of Real Quadratic Fields

We describe implementations for solving the discrete logarithm problem in the class group of an imaginary quadratic field and in the infrastructure of a real quadratic field. The algorithms used incorporate improvements over previously-used algorithms, and extensive numerical results are presented demonstrating their efficiency. This data is used as the basis for extrapolations, used to provide recommendations for parameter sizes providing approximately the same level of secu...

In this paper, we examine the general algorithm for class group computations, when we do not have a small defining polynomial for the number field. Based on a result of Biasse and Fieker, we simplify their algorithm, improve the complexity analysis and identify the optimal parameters to reduce the runtime. We make use of the classes $\mathcal D$ defined in [GJ16] for classifying the fields according to the size of the extension degree and prove that they enable to describe al...

We investigate the class number one problem for a parametric family of real quadratic fields of the form $\mathbb{Q}( \sqrt{m^2+4r})$ for certain positive integers $m$ and $r$.

We construct an infinite family of real cyclotomic fields with non-trivial class group. This result generalizes the result in [1] in the sense that our family includes theirs.

We propose a modification of the predictions of the Cohen--Lenstra heuristic for class groups of number fields in the case where roots of unity are present in the base field. As evidence for this modified formula we provide a large set of computational data which show close agreement.

We present improvements to the index-calculus algorithm for the computation of the ideal class group and regulator of a real quadratic field. Our improvements consist of applying the double large prime strategy, an improved structured Gaussian elimination strategy, and the use of Bernstein's batch smoothness algorithm. We achieve a significant speed-up and are able to compute the ideal class group structure and the regulator corresponding to a number field with a 110-decimal ...

Class field theory furnishes an intrinsic description of the abelian extensions of a number field that is in many cases not of an immediate algorithmic nature. We outline the algorithms available for the explicit computation of such extensions.

We solve unconditionally the class number one problem for the $2$-parameter family of real quadratic fields $\mathbb{Q}(\sqrt{d})$ with square-free discriminant $d=(an)^2+4a$ for positive odd integers $a$ and $n$.

We describe an online database of number fields which accompanies this paper The database centers on complete lists of number fields with prescribed invariants. Our description here focuses on summarizing tables and connections to theoretical issues of current interest.

In this paper we provide criteria for the insolvability of the Diophantine equation $x^2+D=y^n$. This result is then used to determine the class number of the quadratic field $\mathbb{Q}(\sqrt{-D})$. We also determine some criteria for the divisibility of the class number of the quadratic field $\mathbb{Q}(\sqrt{-D})$ and this result is then used to discuss the solvability of the Diophantine equation $x^2+D=y^n$.