Output of /home/aschiem/Pgm/Hn/hn --invar -t --shells-6 &K=Q(sqrt(-19)) &Hdim=4 V=K^4 &HNeighbourhood at <5,-1+w> contains 12 classes: mass of the neighbourhood is 1991/5760 Steinitz class <1,w>: &Hlattice (#1 <-- #11) 3 -1 3 0 -2+w 3 0 0 1 3 |Aut| = 2^4*3 #short vectors: 0 0 24 48 96 168 &Hlattice (#2 <-- #8) 3 1 3 -1+w w 3 -1 0 1 3 |Aut| = 2^2*5 #short vectors: 0 0 20 60 100 140 &Hlattice (#3 <-- #4) 2 1 2 -1 -1 2 -1+w -1+w 1-w 4 |Aut| = 2^4*3*5 #short vectors: 0 20 0 40 40 200 &Hlattice (#4 <-- #12) 2 -1 2 1 -1 2 -w w -w 4 |Aut| = 2^4*3*5 #short vectors: 0 20 0 40 40 200 &Hlattice (#5 <-- #10) 2 -1 2 0 0 2 -w 0 1 4 |Aut| = 2^3*3^2 #short vectors: 0 12 0 72 72 120 &Hlattice (#6 <-- #5) 2 0 2 -1 -1 3 1-w 1 -2+w 4 |Aut| = 2^3*3 #short vectors: 0 8 12 52 76 164 &Hlattice (#7 <-- #6) 2 0 2 1-w 0 3 0 -w 0 3 |Aut| = 2^5 #short vectors: 0 8 16 40 72 192 &Hlattice (#8 <-- #9) 2 -1 2 1 0 3 0 -w -1 4 |Aut| = 2^3*3 #short vectors: 0 8 12 52 76 164 &Hlattice (#9 <-- #7) 2 0 2 0 1 3 -1 1 1-w 3 |Aut| = 2^4 #short vectors: 0 4 24 32 80 208 &Hlattice (#10 <-- #1) 1 0 1 0 0 1 0 0 0 1 |Aut| = 2^7*3 #short vectors: 8 24 32 24 64 192 &Hlattice (#11 <-- #2) 1 0 1 0 0 2 0 0 -w 3 |Aut| = 2^5 #short vectors: 4 8 24 64 100 112 &Hlattice (#12 <-- #3) 1 0 2 0 0 3 0 1 -1-w 3 |Aut| = 2^3*3 #short vectors: 2 6 26 42 88 174 &Adjacence matrix (M_ij=#{neighbours of class i isometric to class j}): 0 48 0 0 0 12 12 24 12 0 0 48 20 36 0 0 5 20 5 15 25 0 20 10 0 0 0 31 10 50 15 0 30 0 0 20 0 0 26 0 10 0 0 70 45 5 0 0 0 18 3 3 0 57 0 48 0 0 9 18 12 18 7 0 16 6 27 17 27 0 12 14 8 8 0 2 0 32 20 36 26 2 6 16 6 24 0 5 19 16 24 6 33 2 9 12 4 20 3 2 0 22 13 18 42 0 8 24 0 0 8 0 0 32 24 0 0 16 12 64 0 32 0 0 4 12 6 16 16 1 37 32 24 12 0 2 6 12 12 14 36 4 24 10 classes of Z-lattices with respect to the trace form &Dim=8 V=Q^8 &Genus of the trace-forms: det= 130321 = 19^4 2-adic symbol: 1^8_II 19-adic symbol: 1^4 19^4 -1-adic symbol: +^8 -^0 level=19, weight=4 a_0,..,a_12 determine modular form &Gram (#1 <- H1) 6 3 6 2 2 6 0 2 3 6 0 0 0 2 6 0 0 -2 0 3 6 0 -2 0 0 2 0 6 2 0 0 0 2 2 3 6 |Aut| = 2^5*3 #short vectors: 0 0 0 0 0 24 0 48 0 96 0 168 &Gram (#2 <- H2) 6 2 6 1 -2 6 2 -1 -1 6 2 0 0 0 6 0 0 -2 2 2 6 2 0 2 0 1 -1 6 0 -2 0 2 -2 1 2 6 |Aut| = 2^3*5 #short vectors: 0 0 0 0 0 20 0 60 0 100 0 140 &Gram (#3 <- H3,H4) 4 2 4 2 2 4 2 2 2 4 1 1 1 1 8 1 0 0 0 2 8 0 -1 0 0 -2 2 8 0 0 1 0 2 -2 2 8 |Aut| = 2^4*3*5 #short vectors: 0 0 0 20 0 0 0 40 0 40 0 200 &Gram (#4 <- H5) 4 2 4 0 0 4 0 0 2 4 1 0 2 0 8 1 1 -2 0 2 8 2 2 1 0 1 0 8 2 0 1 1 1 0 4 8 |Aut| = 2^4*3^2 #short vectors: 0 0 0 12 0 0 0 72 0 72 0 120 &Gram (#5 <- H6,H8) 4 0 4 0 -2 4 2 2 0 6 2 -2 0 1 8 1 -2 2 1 -1 8 0 -1 0 1 3 0 8 0 -1 1 -2 -2 1 2 8 |Aut| = 2^3*3 #short vectors: 0 0 0 8 0 12 0 52 0 76 0 164 &Gram (#6 <- H7) 4 0 4 0 0 4 0 0 0 4 1 0 -2 0 6 0 -1 0 2 0 6 2 0 1 0 0 0 6 0 2 0 -1 0 -1 0 6 |Aut| = 2^7 #short vectors: 0 0 0 8 0 16 0 40 0 72 0 192 &Gram (#7 <- H9) 4 0 4 2 -2 6 2 2 0 6 0 2 -1 2 6 2 0 2 1 0 6 0 -1 -1 0 2 2 8 1 0 1 -1 2 3 2 8 |Aut| = 2^5 #short vectors: 0 0 0 4 0 24 0 32 0 80 0 208 &Gram (#8 <- H10) 2 0 2 0 0 2 0 0 0 2 1 0 0 0 10 0 1 0 0 0 10 0 0 1 0 0 0 10 0 0 0 1 0 0 0 10 |Aut| = 2^11*3 #short vectors: 0 8 0 24 0 32 0 24 0 64 0 192 &Gram (#9 <- H11) 2 0 2 0 0 4 0 0 0 4 0 0 -1 2 6 0 0 2 -1 -1 6 1 0 0 0 0 0 10 0 1 0 0 0 0 0 10 |Aut| = 2^8 #short vectors: 0 4 0 8 0 24 0 64 0 100 0 112 &Gram (#10 <- H12) 2 0 4 0 2 4 0 2 2 6 0 -2 0 1 6 0 -1 -2 -2 1 8 0 -2 -1 0 0 3 8 1 0 0 0 0 0 0 10 |Aut| = 2^5*3 #short vectors: 0 2 0 6 0 26 0 42 0 88 0 174