The zero vector of the number field lattice L.
Object e is coerced into the number field lattice L. The possibilities for the coerced object e are vectors of number field lattices, vectors in the proper degree ambient, and sequences of the proper length.
The ith pseudobasis vector of a number field lattice L. The given integer i must be nonnegative (0 gives the zero vector, also obtainable by Zero), and not exceed the rank of L. The ith coefficient ideal must also be trivial.
Given a sequence (or vector) S coercible into the coefficient field of number field lattice L whose length is equal to the rank of L, return the lattice vector with these coordinates. A check is made as to whether the vector (S or v) is in L.
Given a vector in an ambient space A of the number field lattice L where A has the same degree as L, determine whether v is in L. If so, then the coordinates of v with respect to the pseudobasis of L will also be returned. The coordinates of v will actually be returned whenever v lies in the K-span of the pseudobasis.
The parent number field lattice to which the given lattice vector v belongs.
Addition, subtraction, negation, and (non)equality of the number field lattice elements v and w.
Given a vector v belonging to the number field lattice L defined over the number field K and an element s of K, scale v by s as indicated. The result is checked for membership of L.
Given a vector v belonging to the number field lattice L defined over the number field K, and a matrix T defined over K, the pseudobasis coordinates of v are transformed by T. The result is checked for membership of L.
Given a vector v belonging to the number field lattice L defined over the field K, and a matrix T acting on the ambient space of L, transform v by T. The result is checked for membership of L.
Given an element v belonging to the number field lattice L and a matrix M acting on the ambient space of L, return the image of v under the transformation M. Here the action is on the coordinates of the vector (so M must be square, of dimensions equal to the rank of L), and the resulting vector must belong to the lattice.
Given an element v belonging to the number field lattice L and a matrix group G acting on L, return the orbit of v under the action of G. This operation is also available if v is replaced by a set or sequence of elements of L. The user is responsible for ensuring that the orbit is finite.
Given an element v belonging to the number field lattice L and a matrix group G acting on the coordinates of the vectors of L, return the stabilizer of v under the action of G. This operation is also available if v is replaced by a set or sequence of elements of L. The user is responsible for ensuring that the group G is finite.
The norm of a given number field lattice element v.
The inner product of two number field lattice elements v and w.
> K<s13> := NumberField(Polynomial([-13,0,1])); // Q(sqrt(13))
> L := NumberFieldLattice(K,3);
> v := Zero(L);
> assert IsZero(v);
> w1 := L.1;
> w2 := L.2-L.3;
> CoordinatesToLattice(L,Vector(5*w1-s13*w2));
( 5 -s13 s13)
> assert w2 in L;
> assert not Vector(w2)/2 in L; // cannot divide w2 by 2 directly
> assert Parent(v) eq L;
> Norm(w2);
2
> InnerProduct(w1,w2);
0
> T := Matrix(3,3,[K|s13,1,0, 3,-1,1+s13, s13,-s13,2+s13]);
> T*w2;
(-s13 + 3 s13 - 1 -1)
> w2*T; // same, as basis is standard
(-s13 + 3 s13 - 1 -1)
> S := sub<L|[w1,w2]>;
> Submatrix(T,1,1,2,2)*(S.1); // random input data, 2x2 mat in T*v
(s13 1 -1)
> G := AutomorphismGroup(L);
> assert #G eq 48;
> w2^G; // Orbit
{@
( 0 1 -1),
(-1 1 0),
( 1 0 -1),
(0 1 1),
( 1 -1 0),
(-1 0 -1),
(1 1 0),
(1 0 1),
( 0 -1 -1),
(-1 0 1),
(-1 -1 0),
( 0 -1 1)
@}
> assert #$1 eq 12;
> #Stabilizer(G,w2); // 4*12 is 48
4
> #Stabilizer(G,w1);
8
> #Orbit(G,{w1,w2});
24
Given an element v of the number field lattice L, return the underlying vector of the ambient space associated with v.
Given an element v of the number field lattice L, return the sequence corresponding to the Vector of the element.
Given an element v of the number field lattice L, return the coordinates of v, with respect to the pseudobasis of the parent lattice.