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Algebraic Number Fields
New Features:
- A new implementation of the standard (NoSieve) algorithm for
ClassGroup of general fields is now used. Fields of larger
degree are handled much better by the new implementation.
For example, (conditional) class groups and units can be obtained
for the degree 24
fields arising in 5
-descent on elliptic curves,
for a reasonable range of conductors.
In addition, running times are much more stable than before. There
will be further improvements, leading to significant gains in speed.
(The default choice of algorithm, Sieve or NoSieve,
has not changed: Sieve is used by default for fields of
degree at most 5
with large discriminant. The algorithm may be
selected by the user, in all cases except that Sieve is not
allowed for very small discriminants.)
- ClassGroupExpectedNumberOfTrials estimates the difficulty of
finding relations in the standard class group algorithm for a given
factor base bound. Based on this, ClassGroupSuggestedBound
gives a suitable value for the factor base bound.
- Automorphisms can now be applied to a number field that is
represented as an extension of another number field.
Changes and Removals:
- The computation of coercions between orders and fields has been made
more efficient.
Previously when an order or field was constructed all possible coercion paths were computed and
stored. In some cases this caused the graph which stored this information to be
excessively large and cause Magma to use an excessive amount of memory. In all cases there was the time cost
of computing coercion paths which may not have been used. Both of these problems have been addressed by computing and storing coercion paths only when they are required. Examples
which ran into problems with memory usage no longer have these problems, other examples have seen significant speed-ups.
- The choice of algorithm used to compute Subfields of simple extensions
of
Q
has been improved (V2.18-3, V2.18-8).
- Improvements have been made to the implementation of the Klüners-van Hoeij-Novocin algorithm for the computation of Subfields of simple extensions of
Q
. These improvements
were made to the LLL computation, the prime selection and the computation of the subfield polynomial. Some of these improvements were patched in V2.18-8.
- When applied to an element of a number field or a field of fractions of an order
of a number field IsIntegral now returns (as a second return value) a denominator
such that the denominator times the input is integral.
- CoveringStructure is now provided for rings of straight line polynomials
which are used in the computation of Galois groups of polynomials over global
arithmetic fields.
- A straight line polynomial (as used in the computation of Galois groups of
polynomials over global arithmetic fields) can now be evaluated using a coefficient ring map
which maps the coefficients of the straight line polynomial into the universe of
the point at which the polynomial is being evaluated.
Bug Fixes:
- A number of bugs have been fixed in the Sieve algorithm for ClassGroup.
In addition, problems with the computation of S
-units in cases which
used the Sieve algorithm have been corrected.
- A bug with incompatible sequence elements in SplittingField
has been fixed. Thanks to A. Elkin for the correction.
- Compatibility of modules over maximal orders has been fixed.
- The PrimitiveElement of an order is now integral.
Next: Characters and Artin Representations
Up: Arithmetic Fields (Global)
Previous: Arithmetic Fields (Global)