# Class Groups In Magma, the class group of binary quadratic forms of discriminant $D$ is defined by the equivalence relation `IsEquivalent(f1, f2 : Narrow := false)` for forms `f1, f2`. It is the same as the ideal class group (`PicardGroup`) of the quadratic order of discriminant $D$. The class group of binary forms for $D > 0$ is usually defined using standard (“narrow”) equivalence of forms, which agrees with the narrow ideal class group of the order. For $D < 0$, there is no difference. For $D > 0$, the two class groups differ if and only if all units in the quadratic order have norm $1$. When they are different, each class is the union of two narrow classes: forms $\langle a,b,c \rangle$ and $\langle -a,b,-c \rangle$ are equivalent but not narrowly equivalent. ## `ClassNumber(Q: parameters): QuadBin -> RngIntElt` ## `ClassNumber(D: parameters): RngIntElt -> RngIntElt` ```magma Al : MonStgElt Default: "Automatic" FactorBasisBound: FldReElt Default: 0.1 ProofBound : FldReElt Default: 6 ExtraRelations : RngIntElt Default: 1 ``` This returns the class number of binary quadratic forms $Q$ of discriminant $D$. The parameter `Al` may be supplied to select the method used to calculate the class number. The possible values are `"ReducedForms"` (enumerating all reduced forms), `"Shanks"` (using a Shanks-based algorithm in the class group), `"Sieve"` or `"NoSieve"`. The default is to use reduced form enumeration for small discriminants, the Shanks algorithm for the middle range, a class group index-calculus for large discriminants and the sieve method described in [[Jacobson, Jr., 1999](../../references.md#cite-jacobson)] for very large discriminants. The parameters `FactorBasisBound`, `ProofBound` and `ExtraRelations` are relevant only for the index calculus algorithm. They are described (below) under [`ClassGroup`](#function-quadbin-classgroup). ## `ClassGroup(Q: parameters): QuadBin -> GrpAb, Map` ```magma Al : MonStgElt Default: "Automatic" FactorBasisBound: FldReElt Default: 0.1 ProofBound : FldReElt Default: 6 ExtraRelations : RngIntElt Default: 1 ``` This computes the class group of binary quadratic forms of discriminant $D$. The function returns the class group as an abelian group, together with a map from the group to quadratic forms. **Important:** the class group is defined using non-strict equivalence (see the introduction to this subsection). Depending on the size of $D$, either a Shanks-based method is used ([[Buchmann *et al.*, 1997](../../references.md#cite-teske-bsgs), [Teske, 1998](../../references.md#cite-teske-prho)]), an index calculus variant ([[Cohen *et al.*, 1993](../../references.md#cite-cohdiol), [Cohen, 1993](../../references.md#cite-cohen1), [Hafner and McCurley, 1989](../../references.md#cite-hafnermccurley)]) or a sieving method ([[Jacobson, Jr., 1999](../../references.md#cite-jacobson)]). If the parameter `Al` is set to `"Sieve"` or $D$ is larger than $10^{20},$ then the sieving method is used. The results are only proven under GRH. If the parameter `Al` is set to `"NoSieve"` then the sieving method will not be used. The parameters `FactorBasisBound`, `ProofBound` and `ExtraRelations` apply to the index calculus method only. This method performs two steps. In the first step a factor basis containing prime forms of norm $< B_1$ is built. Next, one looks for generators for the full lattice of relations between the forms in the factor basis. The determinant of this lattice will be the class number and the Smith-form of the relation matrix gives the structure of the class group. Once the matrix is of full rank, the algorithm will look for `ExtraRelations` more relations to potentially decrease the discriminant. In the second step, for all prime forms of norm $< B_2$ it is verified that they are in the class group generated by the forms of the first step. The bounds are chosen to be $B_1 :=$ `FactorBasisBound`$\cdot \log^2 |D|$ and $B_2 :=$ `ProofBound`$\cdot \log^2 |D|$, so the result is correct under the assumption of GRH. The final result is then checked against the Euler product over the first $30000$ primes. If the quotient becomes too large, a warning is issued. In this case one should increase the `ExtraRelations` parameter. ## `ClassGroupStructure(Q: parameters): QuadBin -> [ RngIntElt ]` The structure of the class group of the binary quadratic forms $Q$ of discriminant $D$ returned as a sequence of integers giving the abelian invariants. The parameters are the same as for `ClassGroup`. ## `AmbiguousForms(Q): QuadBin -> SeqEnum` Enumerates the ambiguous forms of negative discriminant $D$, where $D$ is the discriminant of the magma of binary quadratic forms $Q$. ## `TwoTorsionSubgroup(Q): QuadBin -> GrpAb, Map` The subgroup of 2–torsion elements of in the class group of $Q$. ## `Example: Forms (ex-d1a38f)` We give an example of some computations in the class group of a magma of quadratic forms. Elements in the class group are not really represented by single reduced forms but by cycles of equivalent reduced forms. ```magma > Q := QuadraticField(7537543); // arbitrary choice > Q := QuadraticForms(Discriminant(Q)); > C, m := ClassGroup(Q); > C; Abelian Group isomorphic to Z/2 + Z/76 Defined on 2 generators Relations: 76*C.1 = 0 2*C.2 = 0 > // get the generators as quadratic forms: > f := m(C.1); > g := m(C.2); > h := g^2; > g, h; <-1038,4894,1493> <-887,4340,3189> > c := []; // create the cycle of forms equivalent to g^2: > repeat > h := ReductionStep(h); > Append(~c, h); > until h eq g^2; > P := Parent(g); > Identity(P) in c; true // this proves that the second class has order dividing 2 > for d in Divisors(76) do > c := []; > h := f^d; > repeat h := ReductionStep(h); Append(~c, h); > until h eq f^d; > d, Identity(P) in c, #c; > end for; 1 false 16 2 false 14 4 false 12 // the cycle lengths vary 19 false 18 38 false 16 76 true 14 // so the true order of this class is 76 ```