Genera of quadratic and hermitian lattices with small class number

Currently, the following lists of definite quadratic and hermitian lattices are available:

binary rational quadratic forms with class number one or two
quadratic lattices with class number one or two
hermitian lattices with class number one or two
quaternionic hermitian lattices with class number one or two
maximal integral latticess of class number one
ternary lattices of class number one and two
unimodular lattices with mass at most 1/2

Some notation and conventions used

The tables given here are simple text files. The following conventions are used in these text files.

A number field K is represented by a list of integers [f[0],f[1],...,f[n]] which corresponds to the extension K:= Q[X]/(f) with f(X) = f[n] * X^n + ... + f[0].
Every element in K is represented by a list [c[0], ..., c[n-1]] of rationals which stands for c[0] + c[1] * alpha + ... + c[n-1] *alpha^(n-1) where alpha denotes the image of X in K.
A diagonal matrix over K is simply written as the list of its diagonal entries (which again are lists of rationals just as explained before).
A Z_K-module L in K^m is written as a list [L[1,1],...,L[1,m],L[2,1],...,L[r,m]] of elements in K. The module L is then generated by the vectors { (L[i,1],...,L[i,m]) | 1 ≤ i ≤ r }.

Rational binary quadratic forms with class number one or two

Any imaginary quadratic order O together with its trace bilinear form yield a definite binary quadratic lattice. Let h^+(O) and h(O) denote the (proper) class number of this quadratic form. Then

h(O)=1 iff h^+(O)=1
h(O)=2 iff h^+(O)=2 or 3

For k=1,2,3 the file contains a list Resultk which gives the discriminants of all imaginary quadratic orders O such that h^+(O)=k. The tables are complete assuming (GRH). Each entry of the list Resultk is a pair < d, [c1,c2,...] > where d denotes the (negative) fundamental discriminant of some maximal order M and the ci are the conductors of the suborders of M with proper class number k.

Quadratic lattices with class number one or two

Definite quadratic lattices of class number at most 2 exists only up to rank 16. For 3 ≤ m ≤ 16 and h=1,2 this file contains (after uncompressing) a list ResultmCNh which gives representatives of the similarity classes of all genera of definite quadratic lattices over number fields of rank m and class number h.

Each entry of these lists ResultmCNh are tuples of the form < K, [<A1, L1>, <A2, L2>, ... ] > where

K is a totally real number field.
Ai is a mxm-diagonal matrix over K.
Li is a Z_K-module of rank m.

See the notation section for how to read these entries. The file also has a function called ReadEntry which extracts the pairs < Ai, Li > for you. For example:


  > load "res_orth.m";
  > X:= ReadEntry( Result5CN1[2]); 
  > BaseRing(X[1,1]); #X;
  Number Field with defining polynomial $.1^2 - $.1 - 1 over the Rational Field
  2

shows that there are 2 similarity classes of one-class genera of definite lattices of rank 5 over Q(sqrt(5)) = Q[X]/(X^2-X-1). Further, X[i] (i=1,2) contains a diagonal matrix and generators for a Z_K-module representing these two genera.

Hermitian lattices with class number one or two

Definite hermitian lattices of class number at most 2 exists only up to rank 9. For 2 ≤ m ≤ 9 and h=1,2 this file contains (after uncompressing) a list ResultmCNh which gives representatives of the similarity classes of all genera of definite hermitian lattices over number fields of rank m and class number h.

Warning: The lists Result2CN1 and Result2CN2 only contain the one and two-class genera over imaginary quadratic number fields!

Each entry of these lists ResultmCNh are tuples of the form < K, f, [<A1, L1>, <A2, L2>, ... ] > where

K is a totally real number field.
f is a list of elements in K. The lattices will be over E = K[X]/(f[0] + f[1]X + X^2). The elements a+bX with a,b in K is given written as a list [a,b] (where a,b are represented by lists of rationals, as explained here.)
Ai is a mxm-diagonal matrix over K.
Li is a Z_E-module of rank m. Given as a sequence of generators. Each generator is a list of m elements over E.

See the notation section for how to read these entries. The file also has a function called ReadEntry which extracts the pairs < Ai, Li > for you. For example:


  > load "res_orth.m";
  > X:= ReadEntry( Result3CN1[8] );
  > BaseRing(X[1,1]); BaseRing(BaseRing(X[1,1])); #X;
  Number Field with defining polynomial $.1^2 + 1 over its ground field
  Number Field with defining polynomial $.1^2 - 2 over the Rational Field
  6

shows that there 6 similarity classes of genera of definite hermitian lattices over E=Q(sqrt(2), sqrt(-1)). Further, X[i] (1 ≤ i ≤6) contains a diagonal matrix and generators for a Z_E-module representing these genera.

Quaternionic hermitian lattices with class number one or two

Definite quaternionic hermitian lattices of class number at most 2 exist only up to rank 5. If the rank is at least 2, these are given in Section 9.3 of my Habilitation in terms of genus symbols.

The unary lattices are listed in this file. It contains two lists Result1CN1 and Result1CN2 representing the definite quaternion algebras E that admit unary lattices with class number one and two respectively. The entries of these lists are of the form [K, d_K, [<dE1, N1>, <dE2, N2>, ...]] where

K is a totally number field with discriminant d_K.
The relative reduced discrimiant of E turned out to be principal in all cases. The dEi give generators (as elements in K) of the reduced discriminants of all definite quaternion algebras over K that admit unary lattices of class number one or two.
For convenience, the norm of dEi is also given in the field Ni.

See the notation section for how to read these entries. Note that the actual lattices of rank 1 over E can be obtained as explained in Theorem 9.2.1 of my Habilitation. Again, the file contains a function ReadEntry to extract these algebras E for a fixed field K. In addition, there is a function Lattices which given a maximal order M in such an algebra, it returns representatives of the similarity classes of genera of M-lattices as well as their class numbers. Here is an example:


> load "res_quat1.m";
> Es:= ReadEntry(Result1CN1[7]);
> #Es; BaseField(Es[1]); Discriminant(Es[1]);
1
Number Field with defining polynomial x^2 - x - 5 over the Rational Field
Principal Ideal
Generator:
  [1, 0]
[ 1st place at infinity, 2nd place at infinity ]
> L:= Lattices(MaximalOrder(Es[1]));
> #L, L[1,2], L[2,2];
2, 1, 2

So over Q(sqrt(21)) = Q(x)/(x^2-x-5) there exists only one definite quaternion order E which admits unary lattices of class number one. It is the algebra ramified at no finite place. Any maximal order M in E yields 2 similarity classes of genera of unary lattices. One has class number one, the other has class number two. L[1,1] and L[2,1] contain representatives of these genera (i.e. left ideals of M).

One-class genera of maximal integral lattices

A list of all one-class genera of maximal integral lattices of degree ≥ 3 is available here.

Each entry of the list consists of a triple < K, A, L > where

K is a totally real number field.
A is a mxm diagonal matrix over K.
L is a Z_K-module of rank m.

See the notation section for how to read these entries.

Here is an example how to read the 3rd entry in the table with Magma.


> load "maxgen.m";
> entry:= MaxGen[3];
> K:= NumberField(Polynomial(entry[1]));
> A:= ChangeUniverse(entry[2], K);
> L:= [ Vector(ChangeUniverse(x, K)) : x in entry[3] ];
> K, A, L;
Rational Field
[ 1, 1, 3 ]
[
  (  1   0   0),
  (  0   1   0),
  (  0 1/2 1/2)
]

Similarly some quaternary lattice over Q(sqrt{3}):


> entry:= MaxGen[508];
> K:= NumberField(Polynomial(entry[1]));
> A:= ChangeUniverse(entry[2], K);
> L:= [ Vector(ChangeUniverse(x, K)) : x in entry[3] ];
> K, A, L;
Number Field with defining polynomial $.1^2 - 3 over the Rational Field
[
    1,
    K.1 + 2,
    -3*K.1 + 7,
    5*K.1 + 25
]
[
    (1 0 0 0),
    (0 1 0 0),
    (  1/2*(-2*K.1 + 23)    1/2*(3*K.1 - 36) 1/22*(-53*K.1 - 87)    1/22*(-K.1 + 38)),
    (   1/2*(-17*K.1 - 394)     1/2*(29*K.1 + 618) 1/22*(1139*K.1 + 1906)   1/22*(-69*K.1 - 667))
]

Unimodular lattices with mass at most 1/2

A list of all unimodular lattices of rank >= 3 over number fields K ≠ Q is available here. (For the classification of unimodular lattices over the integers, see for example Conway&Sloane, Sphere packings, lattices and groups.)

Each entry of the list consists of a triple

< K, A, [L1, ..., Ln] >

where K is the base field, A is some diagonal quadratic form and L1,...,Ln represent the isometry classes in the genus. Details how to read these entries are given above.

Here is an example how to read the 5th entry in this list.


> load "unimod.m";
> entry:= Unimodular[5];
> K:= NumberField(Polynomial(entry[1]));
> A:= ChangeUniverse(entry[2], K);
> LL:= [ Matrix([Vector(ChangeUniverse(x, K)) : x in X]): X in entry[3] ];
> K, A, LL;
Number Field with defining polynomial x^2 - 3 over the Rational Field
[
    1,
    1,
    1
]
[
    [                 1                  0                  0]
    [                 0   1/2*(7*K.1 - 12)    1/2*(4*K.1 + 5)]
    [                 0 1/2*(-32*K.1 + 55)       -5/2*K.1],

    [          5*K.1 + 2          -8*K.1 - 4                   0]
    [  1/3*(20*K.1 + 15)  1/6*(-43*K.1 + 51)     1/6*(-K.1 + 15)]
    [       -65*K.1 - 31 1/2*(170*K.1 - 227)    1/2*(9*K.1 - 54)]
]

So this is a genus over Q(sqrt{3}) of rank 3. Its envelopping quadratic space has Gram matrix <1,1,1> and the genus consists of two isometry classes represented by the row-spans of the two matrices above.