Number Field Lattice Elements#

Creation#

Zero(L): LatNF -> LatNFElt#: The zero vector of the number field lattice \(L\).

L ! e: LatNF, Any -> LatNFElt#: 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.

L . i: LatNF, RngIntElt -> LatNFElt#: The \(i\)th 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 \(i\)th coefficient ideal must also be trivial.

CoordinatesToLattice(L, S): LatNF, SeqEnum -> LatNFElt#
CoordinatesToLattice(L, v): LatNF, ModTupFldElt -> LatNFElt#: 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 \(L\).

Parent and Element Relations#

v in L: ModTupRngElt, LatNF -> BoolElt, ModTupFldElt#
v in L: LatNFElt, LatNF -> BoolElt, ModTupFldElt#: 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.

Parent(v): LatNFElt -> LatNF#: The parent number field lattice to which the given lattice vector \(v\) belongs.

Arithmetic#

v + w: LatNFElt, LatNFElt -> LatNFElt#
v - w: LatNFElt, LatNFElt -> LatNFElt#
- v: LatNFElt -> LatNFElt#
v eq w: LatNFElt, LatNFElt -> BoolElt#
v ne w: LatNFElt, LatNFElt -> BoolElt#
IsZero(v): LatNFElt -> BoolElt#: Addition, subtraction, negation, and (non)equality of the number field lattice elements \(v\) and \(w\).

s * v: RngElt, LatNFElt -> LatNFElt#
v * s: LatNFElt, RngElt -> LatNFElt#
v / s: LatNFElt, RngElt -> LatNFElt#: 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 \(L\).

T * v: Mtrx, LatNFElt -> LatNFElt#: 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 \(L\).

v * T: LatNFElt, Mtrx -> LatNFElt#: 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 \(L\).

v ^ M: LatNFElt, Mtrx -> LatNFElt#: 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.

v ^ G: LatNFElt, GrpMat -> Setq[LatNFElt]#
Orbit(G, v): GrpMat, LatNFElt -> Setq[LatNFElt]#: 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.

Stabilizer(G, v): GrpMat, LatNFElt -> GrpMat#: 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.

Norm(v): LatNFElt -> FldNumElt#: The norm of a given number field lattice element \(v\).

InnerProduct(v, w): LatNFElt, LatNFElt -> FldNumElt#: The inner product of two number field lattice elements \(v\) and \(w\).

Example: Nflatelt Ex (ex-9da2f4)#

Here are some examples with number field lattice elements.

> 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

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Access Functions#

Vector(v): LatNFElt -> ModTupFldElt#: Given an element \(v\) of the number field lattice \(L\), return the underlying vector of the ambient space associated with \(v\).

Eltseq(v): LatNFElt -> SeqEnum#: Given an element \(v\) of the number field lattice \(L\), return the sequence corresponding to the Vector of the element.

Coordinates(v): LatNFElt -> ModTupFldElt#: Given an element \(v\) of the number field lattice \(L\), return the coordinates of \(v\), with respect to the pseudobasis of the parent lattice.