New Extremal Lattice
Extremal Lattices in
Patience and perseverance have a magical effect before which difficulties disappear and obstacles vanish.
After walking through a thick forest of lattices for a few weeks, Clara Oswald, the impossible girl, found an extremal lattice
To traverse through this genus, Clara was entrusted with a modified version of TN, implemented in MAGMA. This new version features a heuristic search strategy, which runs through the genus of a lattice without caring about isometry at all; it simply creates a suitable number of neighbours in each iteration and then chooses one of those neighbours with the highest minimum
and
1. Standard Lattice of rank 20 and degree 20Determinant: 9765625Factored Determinant: 5^10Minimum: 6Inner Product Matrix:[ 6 1 -2 0 -2 -3 1 1 1 -3 -2 -1 -1 2 1 -3 2 -3 -1 -1][ 1 6 -1 -2 -3 2 0 -1 1 -1 -1 -3 -3 3 -1 0 -1 -1 -2 -3][-2 -1 6 2 0 3 -2 -2 -3 3 0 -1 0 -3 -2 -1 -3 1 2 0][ 0 -2 2 6 1 2 0 1 1 3 -2 2 -1 0 2 1 1 -1 0 1][-2 -3 0 1 6 1 -1 -1 1 3 1 2 2 0 -1 3 2 3 -1 1][-3 2 3 2 1 8 0 -3 -1 5 0 -2 -2 -1 -2 2 -1 1 -2 0][ 1 0 -2 0 -1 0 6 1 1 0 2 2 -1 -2 2 0 2 -3 -2 -1][ 1 -1 -2 1 -1 -3 1 6 1 -1 -1 1 -1 2 1 0 0 -1 -1 0][ 1 1 -3 1 1 -1 1 1 6 -1 -2 2 -2 4 1 1 2 -2 -3 0][-3 -1 3 3 3 5 0 -1 -1 8 2 1 1 -2 -1 2 -1 1 -2 1][-2 -1 0 -2 1 0 2 -1 -2 2 6 1 2 -3 0 0 0 1 0 -1][-1 -3 -1 2 2 -2 2 1 2 1 1 6 1 -1 2 2 1 0 0 1][-1 -3 0 -1 2 -2 -1 -1 -2 1 2 1 6 -2 1 0 0 1 2 2][ 2 3 -3 0 0 -1 -2 2 4 -2 -3 -1 -2 8 0 1 1 0 -3 -1][ 1 -1 -2 2 -1 -2 2 1 1 -1 0 2 1 0 6 1 3 -2 1 0][-3 0 -1 1 3 2 0 0 1 2 0 2 0 1 1 6 1 3 -1 0][ 2 -1 -3 1 2 -1 2 0 2 -1 0 1 0 1 3 1 6 -1 -1 -1][-3 -1 1 -1 3 1 -3 -1 -2 1 1 0 1 0 -2 3 -1 6 1 0][-1 -2 2 0 -1 -2 -2 -1 -3 -2 0 0 2 -3 1 -1 -1 1 6 1][-1 -3 0 1 1 0 -1 0 0 1 -1 1 2 -1 0 0 -1 0 1 6]
|Aut| = [ <2, 1> ]Modular = falseLatDB = -
2. Standard Lattice of rank 20 and degree 20Determinant: 9765625Factored Determinant: 5^10Minimum: 6Inner Product Matrix:[ 6 -2 3 -2 1 0 -2 0 0 -2 -1 0 -1 1 1 -1 0 -1 1 0][-2 6 -1 3 -1 -1 2 -1 2 3 -2 1 2 1 0 -3 -1 1 1 -1][ 3 -1 6 0 -2 1 -1 2 -1 1 -2 -1 -3 -1 -1 0 1 1 0 1][-2 3 0 6 0 -3 1 1 1 3 -2 0 1 2 -3 -2 1 3 1 -1][ 1 -1 -2 0 6 -1 -1 -1 -1 -2 0 2 0 3 -1 -2 -2 -2 1 0][ 0 -1 1 -3 -1 6 -1 1 -2 -1 0 1 0 -3 2 3 -1 -2 -3 3][-2 2 -1 1 -1 -1 6 1 0 2 -1 1 -1 -2 2 -1 -1 3 -1 2][ 0 -1 2 1 -1 1 1 6 1 2 -2 0 0 -2 -1 1 2 3 -2 4][ 0 2 -1 1 -1 -2 0 1 6 1 -2 -2 2 1 -1 -1 0 1 2 1][-2 3 1 3 -2 -1 2 2 1 6 -2 1 1 -1 -2 0 0 2 1 0][-1 -2 -2 -2 0 0 -1 -2 -2 -2 6 1 1 -1 0 2 1 -2 -2 -3][ 0 1 -1 0 2 1 1 0 -2 1 1 6 2 0 1 0 0 -1 -2 1][-1 2 -3 1 0 0 -1 0 2 1 1 2 6 0 0 0 1 -1 -1 -1][ 1 1 -1 2 3 -3 -2 -2 1 -1 -1 0 0 6 -2 -3 -1 0 3 -2][ 1 0 -1 -3 -1 2 2 -1 -1 -2 0 1 0 -2 6 0 0 0 -1 1][-1 -3 0 -2 -2 3 -1 1 -1 0 2 0 0 -3 0 6 1 -1 -2 2][ 0 -1 1 1 -2 -1 -1 2 0 0 1 0 1 -1 0 1 6 1 -1 0][-1 1 1 3 -2 -2 3 3 1 2 -2 -1 -1 0 0 -1 1 6 0 1][ 1 1 0 1 1 -3 -1 -2 2 1 -2 -2 -1 3 -1 -2 -1 0 6 -2][ 0 -1 1 -1 0 3 2 4 1 0 -3 1 -1 -2 1 2 0 1 -2 8]
|Aut| = [ <2, 1> ]Modular = falseLatDB = -