Output of /home/aschiem/Pgm/Hn/hn --invar -t --shells-12 &K=Q(sqrt(-5)) &Hdim=4 V=K^4 &HNeighbourhood at <3,-1+w> contains 18 classes: mass of the neighbourhood is 65/128 Steinitz class <1,w>: &Hlattice (#1 <-- #12) <1> <1> <2,-1+w> <2,-1+w> 3 -1-w 3 1/2-1/2w 1/2+1/2w 1 -1/2+1/2w -1/2-1/2w -1/2 1 |Aut| = 2^4*3 #short vectors: 0 0 16 56 96 112 208 344 480 672 800 1248 &Hlattice (#2 <-- #17) 3 0 3 1-w 1 3 -w -1+w 1 4 |Aut| = 2^4 #short vectors: 0 0 16 56 96 112 208 344 480 672 800 1248 &Hlattice (#3 <-- #18) 3 1 3 w 0 4 -w -1-w -2-w 5 |Aut| = 2^5 #short vectors: 0 0 16 56 96 112 208 344 480 672 800 1248 &Hlattice (#4 <-- #4) 2 -1 2 -1 1 3 -1 1 1-w 3 |Aut| = 2^4*3 #short vectors: 0 12 16 8 96 136 208 440 480 612 800 1152 &Hlattice (#5 <-- #5) 2 0 2 0 -w 3 -w 0 0 3 |Aut| = 2^7 #short vectors: 0 8 16 24 96 128 208 408 480 632 800 1184 &Hlattice (#6 <-- #11) <2,-1+w> <2,-1+w> <2,-1+w> <2,-1+w> 1/2 0 1/2 0 0 1/2 0 0 0 1/2 |Aut| = 2^7*3 #short vectors: 0 8 16 24 96 128 208 408 480 632 800 1184 &Hlattice (#7 <-- #13) <1> <1> <2,-1+w> <2,-1+w> 2 1 2 0 0 1/2 -1/2-1/2w 1/2-1/2w 0 3/2 |Aut| = 2^3*3 #short vectors: 0 8 16 24 96 128 208 408 480 632 800 1184 &Hlattice (#8 <-- #16) <1> <2,-1+w> <2,-1+w> <1> 2 0 1/2 0 0 1/2 -w 0 0 3 |Aut| = 2^6 #short vectors: 0 8 16 24 96 128 208 408 480 632 800 1184 &Hlattice (#9 <-- #7) 2 1 3 -1 w 3 0 -1 0 3 |Aut| = 2^4 #short vectors: 0 4 16 40 96 120 208 376 480 652 800 1216 &Hlattice (#10 <-- #10) 2 0 3 -1 w 3 -1-w -1 1+w 4 |Aut| = 2^5 #short vectors: 0 4 16 40 96 120 208 376 480 652 800 1216 &Hlattice (#11 <-- #14) 2 0 2 0 -1 3 -1 0 -1+w 3 |Aut| = 2^4 #short vectors: 0 4 16 40 96 120 208 376 480 652 800 1216 &Hlattice (#12 <-- #15) 2 -1 3 0 -w 3 -1 1+w -1 4 |Aut| = 2^5 #short vectors: 0 4 16 40 96 120 208 376 480 652 800 1216 &Hlattice (#13 <-- #1) 1 0 1 0 0 1 0 0 0 1 |Aut| = 2^7*3 #short vectors: 8 24 32 24 56 160 256 280 296 552 1056 1184 &Hlattice (#14 <-- #2) 1 0 1 0 0 2 0 0 -w 3 |Aut| = 2^6 #short vectors: 4 8 24 56 76 128 232 280 388 632 928 1248 &Hlattice (#15 <-- #8) <1> <1> <2,-1+w> <2,-1+w> 1 0 1 0 0 1/2 0 0 0 1/2 |Aut| = 2^6 #short vectors: 4 8 24 56 76 128 232 280 388 632 928 1248 &Hlattice (#16 <-- #3) <1> <1> <2,-1+w> <2,-1+w> 1 0 2 0 0 1/2 0 -1/2-1/2w 0 1 |Aut| = 2^4*3 #short vectors: 2 8 20 40 86 128 220 344 434 632 864 1216 &Hlattice (#17 <-- #6) <1> <2,-1+w> <1> <2,-1+w> 1 0 1/2 0 0 2 0 0 1/2-1/2w 1 |Aut| = 2^4*3 #short vectors: 2 8 20 40 86 128 220 344 434 632 864 1216 &Hlattice (#18 <-- #9) 1 0 3 0 1 3 0 1+w w 3 |Aut| = 2^3*3 #short vectors: 2 0 20 72 86 112 220 280 434 672 864 1280 &Adjacence matrix (M_ij=#{neighbours of class i isometric to class j}): 4 6 0 0 3 1 0 0 0 12 6 0 0 0 0 0 4 4 2 4 6 0 0 0 0 2 8 1 4 5 0 0 0 4 0 4 0 12 0 0 0 0 8 0 12 0 0 0 0 0 0 0 0 8 0 0 0 4 3 0 4 0 6 0 6 12 1 0 0 4 0 0 8 0 0 8 0 0 0 4 0 0 16 0 0 4 0 0 0 0 0 0 0 0 0 16 0 16 0 0 0 0 0 0 0 0 8 0 0 0 6 2 0 0 4 3 6 6 6 0 0 0 3 1 3 0 0 8 0 0 2 2 8 2 0 0 0 8 0 0 2 8 0 0 0 8 6 2 0 0 4 0 4 5 4 1 0 2 0 0 4 0 0 10 0 8 0 0 0 4 2 0 8 0 0 0 0 0 0 8 0 6 0 0 2 0 8 0 0 2 10 1 0 0 4 0 0 7 8 2 0 0 0 0 8 0 10 0 8 0 0 4 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 4 12 0 16 0 0 0 0 0 0 2 0 0 0 8 8 0 0 2 2 2 0 8 8 0 0 0 0 0 0 8 2 0 0 8 0 0 2 12 0 0 8 6 0 0 0 0 2 0 0 18 0 0 0 0 9 0 2 1 2 0 12 0 4 0 0 2 6 0 0 0 0 2 0 0 4 4 6 2 6 6 0 0 0 0 0 0 0 6 6 0 3 3 3 1 4 classes of Z-lattices with respect to the trace form (scaled by 1/2) &Dim=8 V=Q^8 &Genus of the trace-forms: det= 625 = 5^4 2-adic symbol: 1^8_0 5-adic symbol: 1^4 5^4 -1-adic symbol: +^8 -^0 level(of 2-scaled form)=20, weight=4 a_0,..,a_12 determine modular form &Gram (#1 <- H1,H2,H3) 3 1 3 0 1 3 1 0 1 3 1 0 -1 0 3 0 -1 0 1 1 3 0 1 0 -1 1 0 3 1 0 -1 0 0 -1 -1 3 |Aut| = 2^7*3 #short vectors: 0 0 16 56 96 112 208 344 480 672 800 1248 &Gram (#2 <- H4) 2 1 2 1 0 2 1 0 1 3 1 0 1 1 3 1 0 0 0 1 5 1 0 0 0 1 0 5 1 0 0 0 1 0 0 5 |Aut| = 2^8*3^2 #short vectors: 0 12 16 8 96 136 208 440 480 612 800 1152 &Gram (#3 <- H5,H6,H8) 2 0 2 0 0 2 0 0 0 2 0 0 0 1 3 0 0 1 0 0 3 0 1 0 0 0 0 3 1 0 0 0 0 0 0 3 |Aut| = 2^11*3 #short vectors: 0 8 16 24 96 128 208 408 480 632 800 1184 &Gram (#4 <- H7) 2 1 2 0 0 2 1 1 0 3 1 0 0 1 3 0 0 -1 0 0 3 0 0 0 -1 -1 0 4 0 0 0 1 1 0 1 4 |Aut| = 2^6*3^2 #short vectors: 0 8 16 24 96 128 208 408 480 632 800 1184 &Gram (#5 <- H9,H10,H11,H12) 2 0 2 0 -1 3 1 0 1 3 1 0 0 1 3 0 1 -1 0 1 3 0 -1 0 -1 1 0 4 1 0 -1 0 0 -1 0 4 |Aut| = 2^7 #short vectors: 0 4 16 40 96 120 208 376 480 652 800 1216 &Gram (#6 <- H13) 1 0 1 0 0 1 0 0 0 1 0 0 0 0 5 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 0 5 |Aut| = 2^14*3^2 #short vectors: 8 24 32 24 56 160 256 280 296 552 1056 1184 &Gram (#7 <- H14,H15) 1 0 1 0 0 2 0 0 0 2 0 0 0 1 3 0 0 1 0 0 3 0 0 0 0 0 0 5 0 0 0 0 0 0 0 5 |Aut| = 2^11 #short vectors: 4 8 24 56 76 128 232 280 388 632 928 1248 &Gram (#8 <- H16,H17) 1 0 2 0 0 2 0 0 1 2 0 -1 0 0 3 0 0 1 1 0 4 0 0 1 1 0 -1 4 0 0 0 0 0 0 0 5 |Aut| = 2^7*3^2 #short vectors: 2 8 20 40 86 128 220 344 434 632 864 1216 &Gram (#9 <- H18) 1 0 3 0 1 3 0 1 -1 3 0 1 0 0 3 0 1 0 1 -1 3 0 0 0 -1 -1 1 3 0 0 0 0 0 0 0 5 |Aut| = 2^6*3*5 #short vectors: 2 0 20 72 86 112 220 280 434 672 864 1280