Output of /home/aschiem/bin/hn --invar -t --shells_all --herm_lll3 0.8 &K=Q(sqrt(-39)) &Hdim=3 V=K^3 &HNeighbourhood at <2,-1+w> contains 42 classes: mass of the neighbourhood is 11/3 Steinitz class <2,w>: &Hlattice (#1 <- #37) <1> <2,-1+w> <3,1+w> 4 1 1 -4/3-1/3w -1/6-1/6w 1 |Aut| = 2^2*3 #short vectors: 0 0 0 18 6 26 30 24 42 54 &Hlattice (#2 <- #14) <2,w> <2,w> <2,-1+w> 1 1/2 1 -1/2-1/4w 0 3/2 |Aut| = 2^4*3 #short vectors: 0 0 0 12 24 14 24 36 24 84 &Hlattice (#3 <- #12) <1> <3,1+w> <2,-1+w> 3 0 1/3 1+1/2w 0 3/2 |Aut| = 2^3 #short vectors: 0 0 6 10 4 18 32 40 48 58 &Hlattice (#4 <- #40) <1> <1> <2,w> 3 w 5 1 1 3/2 |Aut| = 2^2*3 #short vectors: 0 0 6 6 12 20 36 24 18 96 &Hlattice (#5 <- #41) <1> <1> <2,w> 3 -w 4 -1 1/2-1/2w 1 |Aut| = 2^2*3 #short vectors: 0 0 6 12 0 14 42 42 36 66 &Hlattice (#6 <- #10) <2,w> <1> <1> 1 0 5 -1/2w 2-w 5 |Aut| = 2^3 #short vectors: 0 0 4 8 16 18 32 28 20 92 &Hlattice (#7 <- #17) <2,w> <1> <1> 1 -1/2w 4 1 -2+w 4 |Aut| = 2^2 #short vectors: 0 0 2 12 12 26 30 18 28 78 &Hlattice (#8 <- #38) <1> <2,-1+w> <3,1+w> 2 0 1/2 1 0 2/3 |Aut| = 2^3*3 #short vectors: 0 8 0 12 2 8 24 46 24 84 &Hlattice (#9 <- #13) <2,w> <2,w> <2,-1+w> 1/2 0 1/2 0 0 1/2 |Aut| = 2^4 #short vectors: 0 6 0 12 6 14 24 36 24 84 &Hlattice (#10 <- #16) <1> <2,w> <1> 2 -1 1 -1 1/2w 6 |Aut| = 2^4*3 #short vectors: 0 6 0 12 0 32 24 6 24 84 &Hlattice (#11 <- #35) <2,-1+w> <2,-1+w> <2,-1+w> 1/2 0 1/2 0 0 1/2 |Aut| = 2^4*3 #short vectors: 0 6 0 12 6 14 24 36 24 84 &Hlattice (#12 <- #18) <1> <1> <2,w> 2 1 3 -1/2-1/2w -1 2 |Aut| = 2^4 #short vectors: 0 4 8 4 0 10 40 40 16 100 &Hlattice (#13 <- #7) <2,w> <3,1+w> <3,1+w> 1/2 0 1/3 0 0 1/3 |Aut| = 2^4 #short vectors: 0 2 4 8 10 22 16 36 56 56 &Hlattice (#14 <- #8) <2,w> <2,-1+w> <2,w> 1/2 0 1 0 -1/4-1/4w 1 |Aut| = 2^3*3 #short vectors: 0 2 0 12 14 26 24 16 24 84 &Hlattice (#15 <- #34) <1> <3,1+w> <2,-1+w> 2 0 1/3 -1/2w 0 3/2 |Aut| = 2^3 #short vectors: 0 2 6 6 4 28 28 18 36 78 &Hlattice (#16 <- #36) <1> <2,-1+w> <3,1+w> 2 1-1/2w 2 -1/3-1/3w 2/3-1/3w 1 |Aut| = 2^4 #short vectors: 0 2 0 20 0 12 32 50 48 44 &Hlattice (#17 <- #39) <2,w> <1> <1> 1 1-1/2w 4 -1/2w 2-w 6 |Aut| = 2^3 #short vectors: 0 2 0 16 4 28 28 18 36 64 &Hlattice (#18 <- #1) <1> <1> <2,w> 1 0 1 0 0 1/2 |Aut| = 2^4 #short vectors: 4 6 8 12 10 18 32 16 28 64 &Hlattice (#19 <- #9) <1> <1> <2,w> 1 0 3 0 3/2-1/2w 3/2 |Aut| = 2^3 #short vectors: 2 0 4 16 16 18 32 28 38 54 &Hlattice (#20 <- #11) <1> <2,-1+w> <3,1+w> 1 0 1/2 0 0 1/3 |Aut| = 2^3 #short vectors: 2 2 6 10 14 26 24 16 42 60 &Hlattice (#21 <- #15) <1> <1> <2,w> 1 0 3 0 -1/2+1/2w 1 |Aut| = 2^3 #short vectors: 2 2 8 12 4 16 40 30 34 62 Steinitz class <2,-1+w>: &Hlattice (#22 <- #25) <1> <1> <2,-1+w> 4 2-w 6 -1/2w -1/2w 3/2 |Aut| = 2^2*3 #short vectors: 0 0 0 18 6 26 30 24 42 54 &Hlattice (#23 <- #22) <2,-1+w> <1> <1> 1 1/2-1/2w 5 1/2-1/2w 1+w 7 |Aut| = 2^4*3 #short vectors: 0 0 0 12 24 14 24 36 24 84 &Hlattice (#24 <- #20) <1> <3,1+w> <2,w> 3 0 1/3 3/2-1/2w 0 3/2 |Aut| = 2^3 #short vectors: 0 0 6 10 4 18 32 40 48 58 &Hlattice (#25 <- #30) <1> <1> <2,-1+w> 3 1-w 4 -1 -1/2w 1 |Aut| = 2^2*3 #short vectors: 0 0 6 12 0 14 42 42 36 66 &Hlattice (#26 <- #33) <1> <1> <2,-1+w> 3 -1+w 5 -1 1 3/2 |Aut| = 2^2*3 #short vectors: 0 0 6 6 12 20 36 24 18 96 &Hlattice (#27 <- #24) <1> <1> <2,-1+w> 3 -w 5 1+1/2w -1 5/2 |Aut| = 2^3 #short vectors: 0 0 4 8 16 18 32 28 20 92 &Hlattice (#28 <- #26) <1> <1> <2,-1+w> 3 -w 4 -1-1/2w 2-1/2w 2 |Aut| = 2^2 #short vectors: 0 0 2 12 12 26 30 18 28 78 &Hlattice (#29 <- #31) <2,w> <1> <3,1+w> 1/2 0 2 0 1 2/3 |Aut| = 2^3*3 #short vectors: 0 8 0 12 2 8 24 46 24 84 &Hlattice (#30 <- #3) <2,w> <2,w> <2,w> 1/2 0 1/2 0 0 1/2 |Aut| = 2^4*3 #short vectors: 0 6 0 12 6 14 24 36 24 84 &Hlattice (#31 <- #21) <2,-1+w> <2,w> <2,-1+w> 1/2 0 1/2 0 0 1/2 |Aut| = 2^4 #short vectors: 0 6 0 12 6 14 24 36 24 84 &Hlattice (#32 <- #42) <1> <2,-1+w> <1> 2 1 1 -w 1/2-1/2w 6 |Aut| = 2^4*3 #short vectors: 0 6 0 12 0 32 24 6 24 84 &Hlattice (#33 <- #6) <1> <1> <2,-1+w> 2 -1 6 1-1/2w -2+1/2w 2 |Aut| = 2^4 #short vectors: 0 4 8 4 0 10 40 40 16 100 &Hlattice (#34 <- #19) <2,-1+w> <3,1+w> <3,1+w> 1/2 0 1/3 0 0 1/3 |Aut| = 2^4 #short vectors: 0 2 4 8 10 22 16 36 56 56 &Hlattice (#35 <- #23) <2,-1+w> <2,-1+w> <2,w> 1/2 0 1 0 -1/4-1/4w 1 |Aut| = 2^3*3 #short vectors: 0 2 0 12 14 26 24 16 24 84 &Hlattice (#36 <- #28) <1> <1> <2,-1+w> 4 1 4 1-1/2w 1 1 |Aut| = 2^3 #short vectors: 0 2 0 16 4 28 28 18 36 64 &Hlattice (#37 <- #29) <1> <3,1+w> <2,w> 2 0 1/3 1/2-1/2w 0 3/2 |Aut| = 2^3 #short vectors: 0 2 6 6 4 28 28 18 36 78 &Hlattice (#38 <- #32) <1> <2,w> <3,1+w> 2 -1/2-1/2w 2 2/3-1/3w 2/3+1/6w 1 |Aut| = 2^4 #short vectors: 0 2 0 20 0 12 32 50 48 44 &Hlattice (#39 <- #4) <1> <1> <2,-1+w> 1 0 1 0 0 1/2 |Aut| = 2^4 #short vectors: 4 6 8 12 10 18 32 16 28 64 &Hlattice (#40 <- #2) <1> <2,w> <3,1+w> 1 0 1/2 0 0 1/3 |Aut| = 2^3 #short vectors: 2 2 6 10 14 26 24 16 42 60 &Hlattice (#41 <- #5) <1> <1> <2,-1+w> 1 0 3 0 -1-1/2w 3/2 |Aut| = 2^3 #short vectors: 2 0 4 16 16 18 32 28 38 54 &Hlattice (#42 <- #27) <1> <1> <2,-1+w> 1 0 2 0 -1/2w 3/2 |Aut| = 2^3 #short vectors: 2 2 8 12 4 16 40 30 34 62 &Adjacence matrix (M_ij=#{neighbours of class i isometric to class j}): 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 1 0 0 0 0 0 0 1 0 0 2 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 2 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 3 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 2 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 2 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 2 2 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 2 0 0 0 0 0 0 0 0 0 2 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 3 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 1 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 2 1 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 3 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 1 0 2 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 2 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 1 0 1 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 classes of Z-lattices with respect to the trace form one representative of each class &Dim=6 V=Q^6 &Genus of the trace-forms: det= 59319 = 3^3 *13^3 2-adic symbol: 1^6_II 13-adic symbol: 1^3 13^3 3-adic symbol: 1^3 3^3 -1-adic symbol: +^6 -^0 level=39, weight=3 a_0,..,a_28 determine modular form 20 classes of trace forms &begin_block &Gram (#1 <- H1,H22) 8 4 8 4 0 8 2 0 1 8 1 -1 2 -2 8 3 4 -1 -2 -3 10 |Aut| = 2^2*3 #short vectors: 0 0 0 0 0 0 0 18 0 6 0 26 0 30 0 24 0 42 0 54 0 48 0 66 0 108 0 72 &Gram (#2 <- H2,H23) 8 4 8 4 0 8 3 4 -1 10 3 -1 4 2 10 4 0 3 3 0 10 |Aut| = 2^4*3 #short vectors: 0 0 0 0 0 0 0 12 0 24 0 14 0 24 0 36 0 24 0 84 0 0 0 84 0 120 0 72 &Gram (#3 <- H3,H24) 6 2 6 0 0 6 0 0 3 8 1 2 0 0 8 2 1 0 0 -2 8 |Aut| = 2^4 #short vectors: 0 0 0 0 0 6 0 10 0 4 0 18 0 32 0 40 0 48 0 58 0 36 0 84 0 78 0 72 &Gram (#4 <- H4,H26) 6 2 6 2 2 6 1 -2 2 10 2 -2 -1 3 10 2 -1 -2 2 0 10 |Aut| = 2^2*3 #short vectors: 0 0 0 0 0 6 0 6 0 12 0 20 0 36 0 24 0 18 0 96 0 36 0 96 0 96 0 54 &Gram (#5 <- H5,H25) 6 2 6 2 2 6 1 0 0 8 0 0 1 -2 8 0 -1 0 2 2 8 |Aut| = 2^2*3 #short vectors: 0 0 0 0 0 6 0 12 0 0 0 14 0 42 0 42 0 36 0 66 0 42 0 78 0 84 0 54 &Gram (#6 <- H6,H27) 6 2 6 2 -2 8 0 0 -3 8 1 2 -2 4 10 2 1 2 -4 -4 10 |Aut| = 2^3 #short vectors: 0 0 0 0 0 4 0 8 0 16 0 18 0 32 0 28 0 20 0 92 0 24 0 92 0 104 0 60 &Gram (#7 <- H7,H28) 6 2 8 2 4 8 0 2 3 8 0 -3 -2 2 8 1 2 2 2 -2 8 |Aut| = 2^2 #short vectors: 0 0 0 0 0 2 0 12 0 12 0 26 0 30 0 18 0 28 0 78 0 42 0 82 0 108 0 66 &Gram (#8 <- H8,H29) 4 0 4 0 2 4 1 0 0 10 0 1 2 0 14 0 -1 1 0 -6 14 |Aut| = 2^4*3 #short vectors: 0 0 0 8 0 0 0 12 0 2 0 8 0 24 0 46 0 24 0 84 0 26 0 84 0 120 0 72 &Gram (#9 <- H9,H11,H30,H31) 4 0 4 0 0 4 -1 0 0 10 0 -1 0 0 10 0 0 -1 0 0 10 |Aut| = 2^4*3 #short vectors: 0 0 0 6 0 0 0 12 0 6 0 14 0 24 0 36 0 24 0 84 0 30 0 84 0 120 0 72 &Gram (#10 <- H10,H32) 4 0 4 0 0 4 1 2 -2 12 2 -1 -2 1 12 2 -2 1 -1 1 12 |Aut| = 2^4*3 #short vectors: 0 0 0 6 0 0 0 12 0 0 0 32 0 24 0 6 0 24 0 84 0 72 0 84 0 120 0 72 &Gram (#11 <- H12,H33) 4 0 4 2 2 6 2 -2 -1 12 1 2 3 4 14 2 1 0 -4 -2 14 |Aut| = 2^4 #short vectors: 0 0 0 4 0 8 0 4 0 0 0 10 0 40 0 40 0 16 0 100 0 40 0 100 0 88 0 48 &Gram (#12 <- H13,H34) 4 0 6 0 0 6 0 3 0 8 0 0 3 0 8 1 0 0 0 0 10 |Aut| = 2^6 #short vectors: 0 0 0 2 0 4 0 8 0 10 0 22 0 16 0 36 0 56 0 56 0 26 0 92 0 84 0 96 &Gram (#13 <- H14,H35) 4 0 8 0 -4 8 0 -3 0 8 0 0 -3 4 8 1 0 0 0 0 10 |Aut| = 2^4*3 #short vectors: 0 0 0 2 0 0 0 12 0 14 0 26 0 24 0 16 0 24 0 84 0 38 0 84 0 120 0 72 &Gram (#14 <- H15,H37) 4 2 6 0 0 6 0 0 3 8 0 -1 0 0 8 1 0 0 0 4 12 |Aut| = 2^4 #short vectors: 0 0 0 2 0 6 0 6 0 4 0 28 0 28 0 18 0 36 0 78 0 56 0 96 0 86 0 72 &Gram (#15 <- H16,H38) 4 2 8 2 -2 8 2 2 1 8 2 1 0 -2 8 1 -2 -1 -2 -1 14 |Aut| = 2^4 #short vectors: 0 0 0 2 0 0 0 20 0 0 0 12 0 32 0 50 0 48 0 44 0 32 0 60 0 104 0 72 &Gram (#16 <- H17,H36) 4 2 8 2 -2 8 0 -2 -1 8 0 1 2 -4 8 1 0 1 2 2 12 |Aut| = 2^3 #short vectors: 0 0 0 2 0 0 0 16 0 4 0 28 0 28 0 18 0 36 0 64 0 56 0 72 0 112 0 72 &Gram (#17 <- H18,H39) 2 0 2 0 0 4 0 0 -1 10 1 0 0 0 20 0 -1 0 0 0 20 |Aut| = 2^6 #short vectors: 0 4 0 6 0 8 0 12 0 10 0 18 0 32 0 16 0 28 0 64 0 42 0 64 0 104 0 48 &Gram (#18 <- H19,H41) 2 0 6 0 -2 6 0 1 -2 8 0 -2 1 2 8 1 0 0 0 0 20 |Aut| = 2^4 #short vectors: 0 2 0 0 0 4 0 16 0 16 0 18 0 32 0 28 0 38 0 54 0 24 0 62 0 104 0 60 &Gram (#19 <- H20,H40) 2 0 4 0 0 6 0 0 3 8 0 -1 0 0 10 1 0 0 0 0 20 |Aut| = 2^5 #short vectors: 0 2 0 2 0 6 0 10 0 14 0 26 0 24 0 16 0 42 0 60 0 38 0 78 0 94 0 72 &Gram (#20 <- H21,H42) 2 0 4 0 2 6 0 0 -1 8 0 -1 -1 4 12 1 0 0 0 0 20 |Aut| = 2^4 #short vectors: 0 2 0 2 0 8 0 12 0 4 0 16 0 40 0 30 0 34 0 62 0 44 0 70 0 88 0 48 &end_block