/* 1. Standard Lattice of rank 10 and degree 10 Determinant: 16807 Factored Determinant: 7^5 Minimum: 4 Inner Product Matrix: [ 4 2 2 2 -2 0 1 1 -1 2] [ 2 4 2 2 0 0 1 2 -2 2] [ 2 2 4 2 -2 1 1 2 -2 2] [ 2 2 2 4 -2 0 1 2 -1 2] [-2 0 -2 -2 4 0 -1 -1 1 -1] [ 0 0 1 0 0 4 0 2 -2 2] [ 1 1 1 1 -1 0 4 -1 1 -1] [ 1 2 2 2 -1 2 -1 6 -4 4] [-1 -2 -2 -1 1 -2 1 -4 6 -2] [ 2 2 2 2 -1 2 -1 4 -2 6] |Aut| = [ <2, 8>, <3, 1>, <5, 1> ] Modular = true LatDB = - 2. Standard Lattice of rank 10 and degree 10 Determinant: 16807 Factored Determinant: 7^5 Minimum: 4 Inner Product Matrix: [ 4 1 -1 0 -1 0 1 1 1 0] [ 1 4 -2 -1 1 -1 0 0 -1 -2] [-1 -2 4 -1 0 2 1 0 0 1] [ 0 -1 -1 4 0 -2 0 -1 1 -1] [-1 1 0 0 4 0 -1 1 1 -1] [ 0 -1 2 -2 0 4 0 0 0 2] [ 1 0 1 0 -1 0 4 1 1 0] [ 1 0 0 -1 1 0 1 4 1 0] [ 1 -1 0 1 1 0 1 1 4 0] [ 0 -2 1 -1 -1 2 0 0 0 4] |Aut| = [ <2, 5>, <3, 2>, <5, 1> ] Modular = true LatDB = - 3. Standard Lattice of rank 10 and degree 10 Determinant: 16807 Factored Determinant: 7^5 Minimum: 4 Inner Product Matrix: [ 4 0 2 1 2 1 -1 1 -2 1] [ 0 4 0 2 0 1 0 -1 0 -2] [ 2 0 4 0 2 0 1 0 -1 2] [ 1 2 0 4 -1 2 -1 1 -1 -1] [ 2 0 2 -1 4 -1 1 -1 0 2] [ 1 1 0 2 -1 4 -1 0 -1 0] [-1 0 1 -1 1 -1 4 -1 0 0] [ 1 -1 0 1 -1 0 -1 4 -1 1] [-2 0 -1 -1 0 -1 0 -1 4 -1] [ 1 -2 2 -1 2 0 0 1 -1 6] |Aut| = [ <2, 10> ] Modular = true LatDB = - 4. Standard Lattice of rank 10 and degree 10 Determinant: 16807 Factored Determinant: 7^5 Minimum: 4 Inner Product Matrix: [ 4 -2 -1 1 -1 0 -1 -1 -1 -1] [-2 4 1 -1 -1 1 1 0 1 -1] [-1 1 4 -2 1 1 0 1 1 1] [ 1 -1 -2 4 1 -1 -1 -1 0 1] [-1 -1 1 1 4 -1 -1 0 0 3] [ 0 1 1 -1 -1 4 1 0 1 1] [-1 1 0 -1 -1 1 4 1 0 1] [-1 0 1 -1 0 0 1 4 1 2] [-1 1 1 0 0 1 0 1 4 2] [-1 -1 1 1 3 1 1 2 2 6] |Aut| = [ <2, 6>, <5, 1> ] Modular = true LatDB = - */ // Data Liste:=[ PowerStructure(Lat) | LatticeWithGram(MatrixRing(IntegerRing(), 10) ! \[ 4, 2, 2, 2, -2, 0, 1, 1, -1, 2, 2, 4, 2, 2, 0, 0, 1, 2, -2, 2, 2, 2, 4, 2, -2, 1, 1, 2, -2, 2, 2, 2, 2, 4, -2, 0, 1, 2, -1, 2, -2, 0, -2, -2, 4, 0, -1, -1, 1, -1, 0, 0, 1, 0, 0, 4, 0, 2, -2, 2, 1, 1, 1, 1, -1, 0, 4, -1, 1, -1, 1, 2, 2, 2, -1, 2, -1, 6, -4, 4, -1, -2, -2, -1, 1, -2, 1, -4, 6, -2, 2, 2, 2, 2, -1, 2, -1, 4, -2, 6 ]:CheckPositive:=false), LatticeWithGram(MatrixRing(IntegerRing(), 10) ! \[ 4, 1, -1, 0, -1, 0, 1, 1, 1, 0, 1, 4, -2, -1, 1, -1, 0, 0, -1, -2, -1, -2, 4, -1, 0, 2, 1, 0, 0, 1, 0, -1, -1, 4, 0, -2, 0, -1, 1, -1, -1, 1, 0, 0, 4, 0, -1, 1, 1, -1, 0, -1, 2, -2, 0, 4, 0, 0, 0, 2, 1, 0, 1, 0, -1, 0, 4, 1, 1, 0, 1, 0, 0, -1, 1, 0, 1, 4, 1, 0, 1, -1, 0, 1, 1, 0, 1, 1, 4, 0, 0, -2, 1, -1, -1, 2, 0, 0, 0, 4 ]:CheckPositive:=false), LatticeWithGram(MatrixRing(IntegerRing(), 10) ! \[ 4, 0, 2, 1, 2, 1, -1, 1, -2, 1, 0, 4, 0, 2, 0, 1, 0, -1, 0, -2, 2, 0, 4, 0, 2, 0, 1, 0, -1, 2, 1, 2, 0, 4, -1, 2, -1, 1, -1, -1, 2, 0, 2, -1, 4, -1, 1, -1, 0, 2, 1, 1, 0, 2, -1, 4, -1, 0, -1, 0, -1, 0, 1, -1, 1, -1, 4, -1, 0, 0, 1, -1, 0, 1, -1, 0, -1, 4, -1, 1, -2, 0, -1, -1, 0, -1, 0, -1, 4, -1, 1, -2, 2, -1, 2, 0, 0, 1, -1, 6 ]:CheckPositive:=false), LatticeWithGram(MatrixRing(IntegerRing(), 10) ! \[ 4, -2, -1, 1, -1, 0, -1, -1, -1, -1, -2, 4, 1, -1, -1, 1, 1, 0, 1, -1, -1, 1, 4, -2, 1, 1, 0, 1, 1, 1, 1, -1, -2, 4, 1, -1, -1, -1, 0, 1, -1, -1, 1, 1, 4, -1, -1, 0, 0, 3, 0, 1, 1, -1, -1, 4, 1, 0, 1, 1, -1, 1, 0, -1, -1, 1, 4, 1, 0, 1, -1, 0, 1, -1, 0, 0, 1, 4, 1, 2, -1, 1, 1, 0, 0, 1, 0, 1, 4, 2, -1, -1, 1, 1, 3, 1, 1, 2, 2, 6 ]:CheckPositive:=false) ];