--- title: "Magma V2.21-6 Patch Notes" canonical: "https://docs.magma-maths.org/patchnotes/2.21-6.html" source: "https://docs.magma-maths.org/patchnotes/2.21-6.md" --- # Magma V2.21-6 Patch Notes ## Aggregates - `MultisetToSequence` has been added. Requested by M. Zieve. ## Algebraic Function Fields - Computations of `Subfields` have had some efficiency improved by not checking known irreducible factors for irreducibility when they are used in the calculation of subfields to define a function field. - The computation of the `GaloisGroup` of a polynomial has been improved by choosing a prime which has been checked to be fine to use with all subfields involved in the calculation. - `WeakApproximation` has been implemented for ideals of orders represented in a relative representation. Requested by M. Kirschmer. - A crash in `WronskianOrders` has been fixed. ## Algebraic Number Fields - Computations of `Subfields` and `GaloisGroups` of number fields defined by a polynomial over an order rather than a field have benefited from improved denominator handling. This includes computations of `GaloisGroups` of polynomials over orders. - Computations of `Subfields` have had some efficiency improved by not checking known irreducible factors for irreducibility when they are used in the calculation of subfields to define a number field. - The computation of the `GaloisGroup` of a polynomial (especially a reducible polynomial) has been improved by choosing a prime which has been checked to be fine to use with all subfields involved in the calculation. - `WeakApproximation` has been implemented for ideals of orders represented in a relative representation. Requested by M. Kirschmer. - A fix has been made to `CyclotomicUnits`. Reported by M. Kirschmer. - A fix has been made to [`GaloisProof`](../GlobalFields/GaloisGroupsAndAutomorphisms/galois.md#function-rngord-galoisproof). Reported by J. Klüners. ## Dirichlet and Hecke Characters - A problem with a `Grossencharacter` of negative weight has been fixed. - Dirichlet and Hecke characters should now retain their ambient subgroups when multiplied by elements in the same subgroup. - The `TateTwist` intrinsic can now be used to get a Hecke character corresponding to twisting by the norm, which internally is of the type `GrossenChar`. This can be done over any field (not just CM). - The printing with Grossencharacters has been changed. - Intrinsics for the local `Components` of characters have now been added, and allow a place or prime ideal to be specified. - Intrinsics for the local [`RootNumbers`](../GlobalFields/DirichletAndHeckeCharacters/introduction.md#function-rootnumbers-grossenchar) of Hecke and Grossencharacters have now been added, and allow a place or prime ideal to be specified. The global root number can also be obtained in this manner. - A new intrinsic [`QuadraticCharacter`](../GlobalFields/DirichletAndHeckeCharacters/introduction.md#function-quadraticcharacter-fldnumelt) has been added, to be able to obtain the Hecke character corresponding to a quadratic extension. ## Graphs - `Spectrum` of a graph no longer returns the wrong type of complex numbers. ## Groups - The code for finding the `PCGenerators` of a `GrpMatUnip` has been improved to remove considerable redundancy. Reported by E. O’Brien. - Fixed a rare crash in `UpperCentralSeries`. Reported by S. Elsenhans. - Characters or sequences of characters may now be coerced into appropriate matrix spaces over the complex numbers, and this is considerably faster than coercing each element of the character table. - A problem with `CohomologicalDimension` involving symplectic groups has been fixed. Reported by A. Brumer. ## Hodge Structures - The convention for `RootNumber` of a `HodgeStructure` has been changed, taking the reciprocal of that given by Deligne. The code for Artin representations has been changed to reflect this. - The output of [`CriticalPoints`](../ArithmeticGeometry/LFunctions/arithmetic.md#function-criticalpoints-hodgestruc) has been modified to be symmetric, and the intrinsic is now documented. Reported by D. Loeffler. ## Hypergeometric Motives - A complementary package for Jacobi sum motives and their Kummer twists has now been added. The most notable feature of this is the ability to identify these motives with a `Grossencharacter`, following Weil. - A incorrect answer for `EulerFactor` at a tame prime when 1 was in the cyclotomic data for the alpha’s has been corrected. ## Local Fields - [`Gamma`](../BasicRings/RealAndComplexFields/gamma-bessel.md#function-gamma-rngserelt) of a 2-adic integer now has its output precision calculated more precisely. Determined in discussions with B. Allombert. - More error checking has been added to the computation of `Roots` of a polynomial over a local ring or field. - Various speed improvements have been made to p-adic [`Gamma`](../BasicRings/RealAndComplexFields/gamma-bessel.md#function-gamma-rngserelt). - The formula for the `Discriminant` of a relative extension has been corrected. - An error when using `AutomorphismGroup` has been remedied (the Generic Group code was called with a nontrivial automorphism being specifed as the group identity). - An incorrect increase in default precision of a `Completion` of a number field, seen when computing preimages using the completion mapping has been fixed. ## Polynomial Rings - `div:=` has been fixed to make it consistent with `div`. Reported by M. Zieve. - The previous functionality of `Denominator` for multivariate polynomials is now implemented by [`CoefficientDenominator`](../BasicRings/MultivariatePolynomialRings/operation-element.md#function-coefficientdenominator-rngmpolelt). The intrinsic [`CoefficientNumerator`](../BasicRings/MultivariatePolynomialRings/operation-element.md#function-coefficientnumerator-rngmpolelt) has been added to match. The intrinsics `Denominator` and `Numerator` for multivariate polynomials now work with the polynomial interpreted as a rational function with denominator 1. Reported by M. Zieve and M. Kirschmer. - A crash in `Roots` for polynomials over number fields has been fixed. Reported by M. Kirschmer. - The `XGCD` algorithm has been improved for univariate polynomials over rational function fields to avoid bad coefficient blowup. Issue reported by M. Massierer. ## System - Fixed a bug that could erroneously cause out of memory errors when using [`printf`](../MagmaLanguage/InputOutput/print.md#literal-literal-printf-printf-format-expression-expression) or similar to produce strings of over a gigabyte in size. - Improved the speed of concatenating a sequence of many strings.