## The Lattice E6*

An entry from the Catalogue of Lattices, which is a joint project of
Gabriele Nebe, RWTH Aachen University (nebe@math.rwth-aachen.de) and
Neil J. A. Sloane (njasloane@gmail.com)

## Contents of this file

NAME
DIMENSION
MINIMAL_NORM
KISSING_NUMBER
GRAM
PROPERTIES
NOTES
REFERENCES
LAST_LINE

• NAME
E6*

• DIMENSION
6

• MINIMAL_NORM
4

• KISSING_NUMBER
54

• GRAM
6 6
4 5 6 4 2 3
5 10 12 8 4 6
6 12 18 12 6 9
4 8 12 10 5 6
2 4 6 5 4 3
3 6 9 6 3 6

• PROPERTIES
INTEGRAL =1

• NOTES
Dual to E6

# maple program to construct the min vecs of E6* in 6-D coords
with(linalg): gc(0):
w:=-1/2 + I*sqrt(3)/2;
f1:= proc(x) [ Re(x[ 1]), Re(x[ 2]), Re(x[ 3]),
Im(x[ 1]), Im(x[ 2]), Im(x[ 3]) ]: end:
# the 54 min vecs of E6*
nov:=0:
for i1 from 1 to 2 do
for i from 1 to 3 do
for j from 1 to 3 do
nov:=nov+1:
mvE6s[ nov]:=f1(evalm(expand( (-1)^i1*[ w^i,-w^j,0] )));
od; od; od;
for i1 from 1 to 2 do
for i from 1 to 3 do
for j from 1 to 3 do
nov:=nov+1:
mvE6s[ nov]:=f1(evalm(expand( (-1)^i1*[ 0, w^i,-w^j] )));
od; od; od;
for i1 from 1 to 2 do
for i from 1 to 3 do
for j from 1 to 3 do
nov:=nov+1:
mvE6s[ nov]:=f1(evalm(expand( (-1)^i1*[ -w^j, 0, w^i] )));
od; od; od;
print(`minimal vectors of E6*`);
for i from 1 to 54 do lprint(evalf(mvE6s[ i])); od;

• REFERENCES
SPLAG p. 127.

• LAST_LINE

Haftungsausschluss/Disclaimer

Gabriele Nebe