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.
Each entry of these lists ResultmCNh are tuples of the form < K, [<A1, L1>, <A2, L2>, ... ] > where
> 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.
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
> 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.
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
> 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).
Each entry of the list consists of a triple < K, A, L > where
> 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)
]
> 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))
]
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.
> 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.