Conway lattice in rank 24

Parameters: D = 24, A = 1, B = 24, X = 70.

This page records the compact basis and Gram matrix for the lattice obtained from the primitive isotropic vector w = (0,1,2,...,24;70) in II_{25,1}.

Reference: A Lorentzian Construction in Dimension 88 and Infinitely Many Further Ranks.

Ambient basis

We use the ambient basis [g, delta_1, ..., delta_24, h] of II_{25,1}, where g=(1/2,...,1/2,-1/2), delta_j=e_{j+1}-e_{j+2} for j=1,...,D, and h=e_{D+1}+e_{D+2}.

Explicit basis of the quotient lattice

Download: conway-lattice-D24-basis.txt

b_1 = 1:1, 2:185
b_2 = 3:-1, 4:1
b_3 = 4:-1, 5:1
b_4 = 5:-1, 6:1
b_5 = 6:-1, 7:1
b_6 = 7:-1, 8:1
b_7 = 8:-1, 9:1
b_8 = 9:-1, 10:1
b_9 = 10:-1, 11:1
b_10 = 11:-1, 12:1
b_11 = 12:-1, 13:1
b_12 = 13:-1, 14:1
b_13 = 14:-1, 15:1
b_14 = 15:-1, 16:1
b_15 = 16:-1, 17:1
b_16 = 17:-1, 18:1
b_17 = 18:-1, 19:1
b_18 = 19:-1, 20:1
b_19 = 20:-1, 21:1
b_20 = 21:-1, 22:1
b_21 = 22:-1, 23:1
b_22 = 23:-1, 24:1
b_23 = 24:94, 25:1
b_24 = 24:-46, 26:1

Gram matrix

Download: conway-lattice-D24-gram.txt

68456 185 0 0 0 0 0 0
185 6 -4 1 0 0 0 0
0 -4 6 -4 1 0 0 0
0 1 -4 6 -4 1 0 0
0 0 1 -4 6 -4 1 0
0 0 0 1 -4 6 -4 1
0 0 0 0 1 -4 6 -4
0 0 0 0 0 1 -4 6

Shown above: the upper-left 8×8 block. The full Gram matrix is in the text file.

SageMath script

Download: conway-lattice-D24.sage