Output of /home/aschiem/bin/hn --invar -t --shells-8 --herm_lll3 0.8 &K=Q(sqrt(-31)) &Hdim=3 V=K^3 &HNeighbourhood at <2,-1+w> contains 39 classes: mass of the neighbourhood is 3 Steinitz class <1,w>: &Hlattice (#1 <- #22) <1> <2,w> <2,-1+w> 3 -1 1 -1-1/2w 1/4w 3/2 |Aut| = 2^2*3 #short vectors: 0 0 6 12 30 26 18 54 &Hlattice (#2 <- #23) <1> <2,-1+w> <2,w> 3 -1 1 3/2-1/2w -1/4+1/4w 3/2 |Aut| = 2^2*3 #short vectors: 0 0 6 12 30 26 18 54 &Hlattice (#3 <- #13) <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 18 6 32 30 42 &Hlattice (#4 <- #17) <2,w> <2,w> <2,w> 1/2 0 1/2 0 0 1/2 |Aut| = 2^4*3 #short vectors: 0 6 0 18 6 32 30 42 &Hlattice (#5 <- #24) 2 0 3 1 -1-w 4 |Aut| = 2^2*3 #short vectors: 0 6 2 12 12 30 24 54 &Hlattice (#6 <- #15) <1> <2,-1+w> <2,w> 2 0 1/2 -1/2-1/2w 0 3/2 |Aut| = 2^3 #short vectors: 0 4 4 12 14 28 30 58 &Hlattice (#7 <- #16) 2 1 3 1-w 0 5 |Aut| = 2^4 #short vectors: 0 4 8 6 8 24 48 82 &Hlattice (#8 <- #18) <1> <2,w> <2,-1+w> 2 0 1/2 1-1/2w 0 3/2 |Aut| = 2^3 #short vectors: 0 4 4 12 14 28 30 58 &Hlattice (#9 <- #21) 2 -1 3 w 0 5 |Aut| = 2^4 #short vectors: 0 4 8 6 8 24 48 82 &Hlattice (#10 <- #19) <1> <2,w> <2,-1+w> 2 1/2-1/2w 3/2 -1/2w 1-1/4w 3/2 |Aut| = 2^4 #short vectors: 0 2 0 24 16 32 24 26 &Hlattice (#11 <- #1) 1 0 1 0 0 1 |Aut| = 2^4*3 #short vectors: 6 12 8 6 24 24 0 24 &Hlattice (#12 <- #14) <1> <2,w> <2,-1+w> 1 0 1/2 0 0 1/2 |Aut| = 2^3 #short vectors: 2 4 8 10 20 24 28 52 &Hlattice (#13 <- #20) 1 0 3 0 -1+w 3 |Aut| = 2^3 #short vectors: 2 0 8 22 12 24 52 36 Steinitz class <2,w>: &Hlattice (#14 <- #11) <1> <2,-1+w> <2,-1+w> 3 1/2w 1 -1/2w -1/2 1 |Aut| = 2^4*3 #short vectors: 0 0 8 18 0 24 72 66 &Hlattice (#15 <- #39) <1> <1> <2,w> 3 1+w 4 1 1/2-1/2w 1 |Aut| = 2^2*3 #short vectors: 0 0 6 18 12 26 48 54 &Hlattice (#16 <- #37) <1> <2,-1+w> <2,-1+w> 3 0 1 -1-1/2w 1/2 3/2 |Aut| = 2^2*3 #short vectors: 0 0 2 24 18 30 30 30 &Hlattice (#17 <- #36) <1> <1> <2,w> 2 1 2 1/2-1/2w 0 3/2 |Aut| = 2^4*3 #short vectors: 0 12 0 6 0 32 24 66 &Hlattice (#18 <- #6) <2,w> <2,w> <2,-1+w> 1/2 0 1/2 0 0 1/2 |Aut| = 2^4 #short vectors: 0 6 0 18 6 32 30 42 &Hlattice (#19 <- #9) <1> <2,w> <1> 2 -1 1 -1 1/2w 5 |Aut| = 2^4*3 #short vectors: 0 6 0 12 24 32 0 42 &Hlattice (#20 <- #12) <1> <1> <2,w> 2 1 3 0 -3/2+1/2w 3/2 |Aut| = 2^2*3 #short vectors: 0 6 6 6 6 26 42 78 &Hlattice (#21 <- #35) <1> <2,-1+w> <2,-1+w> 2 0 1/2 1-1/2w 0 3/2 |Aut| = 2^3 #short vectors: 0 4 4 12 14 28 30 58 &Hlattice (#22 <- #10) <2,w> <1> <1> 1/2 0 3 0 -1+w 3 |Aut| = 2^3 #short vectors: 0 2 8 6 22 24 30 74 &Hlattice (#23 <- #38) <1> <2,w> <1> 3 -1/2+1/2w 1 w 1 4 |Aut| = 2^3 #short vectors: 0 2 4 12 28 28 12 50 &Hlattice (#24 <- #7) <1> <1> <2,w> 1 0 1 0 0 1/2 |Aut| = 2^4 #short vectors: 4 6 8 14 18 24 26 30 &Hlattice (#25 <- #5) <1> <2,-1+w> <2,-1+w> 1 0 1/2 0 0 1/2 |Aut| = 2^4 #short vectors: 2 4 8 10 20 24 28 52 &Hlattice (#26 <- #8) <1> <1> <2,w> 1 0 3 0 1/2+1/2w 1 |Aut| = 2^3 #short vectors: 2 2 8 16 16 24 40 44 Steinitz class <2,-1+w>: &Hlattice (#27 <- #33) <1> <2,w> <2,w> 3 -1/2+1/2w 1 1/2-1/2w -1/2 1 |Aut| = 2^4*3 #short vectors: 0 0 8 18 0 24 72 66 &Hlattice (#28 <- #32) <1> <1> <2,-1+w> 3 2-w 4 -1 -1/2w 1 |Aut| = 2^2*3 #short vectors: 0 0 6 18 12 26 48 54 &Hlattice (#29 <- #34) <1> <2,w> <2,w> 3 0 1 -3/2+1/2w 1/2 3/2 |Aut| = 2^2*3 #short vectors: 0 0 2 24 18 30 30 30 &Hlattice (#30 <- #4) <1> <1> <2,-1+w> 2 -1 2 1/2w -1/2w 3/2 |Aut| = 2^4*3 #short vectors: 0 12 0 6 0 32 24 66 &Hlattice (#31 <- #25) <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 18 6 32 30 42 &Hlattice (#32 <- #26) <1> <2,-1+w> <1> 2 1 1 -w 1/2-1/2w 5 |Aut| = 2^4*3 #short vectors: 0 6 0 12 24 32 0 42 &Hlattice (#33 <- #31) <1> <1> <2,-1+w> 2 1 3 0 1+1/2w 3/2 |Aut| = 2^2*3 #short vectors: 0 6 6 6 6 26 42 78 &Hlattice (#34 <- #27) <1> <2,w> <2,w> 2 0 1/2 -1/2-1/2w 0 3/2 |Aut| = 2^3 #short vectors: 0 4 4 12 14 28 30 58 &Hlattice (#35 <- #28) <2,-1+w> <1> <1> 1/2 0 3 0 -1+w 3 |Aut| = 2^3 #short vectors: 0 2 8 6 22 24 30 74 &Hlattice (#36 <- #30) <1> <1> <2,-1+w> 2 1-w 5 1-1/2w 2 2 |Aut| = 2^3 #short vectors: 0 2 4 12 28 28 12 50 &Hlattice (#37 <- #2) <1> <1> <2,-1+w> 1 0 1 0 0 1/2 |Aut| = 2^4 #short vectors: 4 6 8 14 18 24 26 30 &Hlattice (#38 <- #3) <1> <2,w> <2,w> 1 0 1/2 0 0 1/2 |Aut| = 2^4 #short vectors: 2 4 8 10 20 24 28 52 &Hlattice (#39 <- #29) <1> <1> <2,-1+w> 1 0 3 0 1-1/2w 1 |Aut| = 2^3 #short vectors: 2 2 8 16 16 24 40 44 &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 0 0 0 0 1 0 0 0 0 0 3 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 1 0 0 0 0 0 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 0 0 3 1 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 1 0 0 0 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 0 0 0 0 0 0 1 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 1 1 0 2 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 0 0 0 0 0 0 4 0 0 0 0 0 2 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 2 0 0 0 1 0 2 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 4 0 2 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 4 0 1 0 0 0 0 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 1 0 0 0 0 0 0 3 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 1 0 0 1 1 1 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 0 0 0 1 0 2 0 0 0 0 0 1 0 1 0 2 0 0 1 0 0 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 0 0 0 1 0 0 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 0 0 0 1 0 0 0 0 0 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 0 0 0 0 0 6 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 1 1 0 0 0 2 0 1 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 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 3 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 2 2 0 1 1 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 2 0 0 0 0 1 2 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 2 2 0 0 2 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 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 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 0 0 0 0 0 1 0 0 2 1 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 0 0 2 0 0 0 0 1 0 0 0 1 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 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 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 3 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 0 3 0 0 0 1 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 3 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 1 2 0 0 2 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 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 1 0 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 2 0 1 0 0 0 2 1 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 2 2 1 1 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 1 1 4 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 1 1 0 0 0 0 1 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 0 0 0 1 0 0 0 1 0 0 0 2 0 2 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 2 0 0 0 0 0 1 0 1 1 0 2 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= 29791 = 31^3 2-adic symbol: 1^6_II 31-adic symbol: 1^3 31^3 -1-adic symbol: +^6 -^0 level=31, weight=3 a_0,..,a_16 determine modular form 19-classes of trace forms &Gram (#1 <- H1,H2) 6 2 6 2 2 6 1 -2 -2 8 2 -1 2 2 8 2 2 -1 2 -2 8 |Aut| = 2^2*3 #short vectors: 0 0 0 0 0 6 0 12 0 30 0 26 0 18 0 54 &Gram (#2 <- H3,H4,H18,H31) 4 0 4 0 0 4 -1 0 0 8 0 -1 0 0 8 0 0 -1 0 0 8 |Aut| = 2^4*3 #short vectors: 0 0 0 6 0 0 0 18 0 6 0 32 0 30 0 42 &Gram (#3 <- H5) 4 2 4 0 0 6 2 2 -3 8 1 -1 -2 1 12 1 2 2 0 4 12 |Aut| = 2^3*3 #short vectors: 0 0 0 6 0 2 0 12 0 12 0 30 0 24 0 54 &Gram (#4 <- H6,H8,H21,H34) 4 0 4 2 0 6 2 0 -1 8 0 -1 0 0 8 1 0 2 3 0 10 |Aut| = 2^3 #short vectors: 0 0 0 4 0 4 0 12 0 14 0 28 0 30 0 58 &Gram (#5 <- H7,H9) 4 0 4 2 -2 6 0 0 -1 8 1 0 0 4 10 0 1 0 -4 -2 10 |Aut| = 2^4 #short vectors: 0 0 0 4 0 8 0 6 0 8 0 24 0 48 0 82 &Gram (#6 <- H10) 4 2 8 2 -2 8 2 4 1 8 0 3 -4 3 8 1 4 -3 0 4 12 |Aut| = 2^5 #short vectors: 0 0 0 2 0 0 0 24 0 16 0 32 0 24 0 26 &Gram (#7 <- H11) 2 0 2 0 0 2 1 0 0 16 0 1 0 0 16 0 0 1 0 0 16 |Aut| = 2^7*3 #short vectors: 0 6 0 12 0 8 0 6 0 24 0 24 0 0 0 24 &Gram (#8 <- H12,H25,H38) 2 0 4 0 0 4 0 -1 0 8 0 0 -1 0 8 1 0 0 0 0 16 |Aut| = 2^5 #short vectors: 0 2 0 4 0 8 0 10 0 20 0 24 0 28 0 52 &Gram (#9 <- H13) 2 0 6 0 2 6 0 -1 0 6 0 0 -1 -2 6 1 0 0 0 0 16 |Aut| = 2^5 #short vectors: 0 2 0 0 0 8 0 22 0 12 0 24 0 52 0 36 &Gram (#10 <- H14,H27) 6 2 6 2 -2 6 1 1 0 8 1 0 1 4 8 1 0 0 4 4 8 |Aut| = 2^4*3 #short vectors: 0 0 0 0 0 8 0 18 0 0 0 24 0 72 0 66 &Gram (#11 <- H15,H28) 6 2 6 2 2 6 3 2 2 8 2 2 3 0 8 2 3 2 0 0 8 |Aut| = 2^2*3 #short vectors: 0 0 0 0 0 6 0 18 0 12 0 26 0 48 0 54 &Gram (#12 <- H16,H29) 6 2 8 2 -3 8 2 4 -2 8 2 -2 0 -3 8 1 2 1 -2 2 8 |Aut| = 2^2*3 #short vectors: 0 0 0 0 0 2 0 24 0 18 0 30 0 30 0 30 &Gram (#13 <- H17,H30) 4 2 4 2 0 4 1 0 1 12 1 1 0 4 12 0 0 -1 -4 4 12 |Aut| = 2^4*3 #short vectors: 0 0 0 12 0 0 0 6 0 0 0 32 0 24 0 66 &Gram (#14 <- H19,H32) 4 0 4 0 0 4 1 2 -2 10 2 -1 2 -1 10 2 -2 1 -1 2 10 |Aut| = 2^4*3 #short vectors: 0 0 0 6 0 0 0 12 0 24 0 32 0 0 0 42 &Gram (#15 <- H20,H33) 4 2 4 2 2 6 2 0 1 8 1 1 -1 -2 12 1 0 2 3 4 12 |Aut| = 2^2*3 #short vectors: 0 0 0 6 0 6 0 6 0 6 0 26 0 42 0 78 &Gram (#16 <- H22,H35) 4 0 6 0 2 6 0 -1 0 6 0 0 -1 -2 6 1 0 0 0 0 8 |Aut| = 2^4 #short vectors: 0 0 0 2 0 8 0 6 0 22 0 24 0 30 0 74 &Gram (#17 <- H23,H36) 4 0 6 0 -2 6 2 2 1 8 2 1 2 4 8 1 2 2 2 2 10 |Aut| = 2^3 #short vectors: 0 0 0 2 0 4 0 12 0 28 0 28 0 12 0 50 &Gram (#18 <- H24,H37) 2 0 2 0 0 4 0 0 -1 8 1 0 0 0 16 0 -1 0 0 0 16 |Aut| = 2^6 #short vectors: 0 4 0 6 0 8 0 14 0 18 0 24 0 26 0 30 &Gram (#19 <- H26,H39) 2 0 4 0 -2 6 0 2 -3 8 0 -1 -1 -3 10 1 0 0 0 0 16 |Aut| = 2^4 #short vectors: 0 2 0 2 0 8 0 16 0 16 0 24 0 40 0 44