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| KASH3 Reference Manual

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### Keyword

Class Groups

### Description

<shortdoc> The task of computing the ideal class group is solved by invoking the 'ClassGroup' function. As a result an abelian group 'G' that the class group is isomorphic to, extended the by the map from 'G' into the set of Ideals in 'O'. Using the 'ClassGroupCyclicFactorGenerators' routine one can obtain a list of ideals representing generators of the cyclic factors of ideal class group. Notice that you must first compute the class group before you can use the 'ClassGroupCyclicFactorGenerators' routine.
The Minkowski bound is used to compute a class group in KASH3. This bound always guarantees correct results. However, when the field discriminant is 'large', the Minkowski bound causes very time consuming computations requiring a large amount of memory. You can pass a smaller bound to the 'OrderClassGroup' function calling it with optional arguments. </shortdoc>

### Examples

f := X^5 + 4*X^4 - 56*X^2 -16*X + 192; o := EquationOrder(f); O := MaximalOrder(o); Cl := ClassGroup(O);

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