Output of /home/aschiem/Pgm/Hn/hn --invar -t --herm_lll2 0.0001 --shells-6 &K=Q(sqrt(-23)) &Hdim=3 V=K^3 &HNeighbourhood at <2,-1+w> contains 30 classes: mass of the neighbourhood is 3/2 Steinitz class <1,w>: &Hlattice (#1 <-- #16) 2 -1 5 -1+w 1-2w 6 |Aut| = 2^3*3 #short vectors: 0 8 2 20 24 44 &Hlattice (#2 <-- #18) 4 1 5 -2w -2 7 |Aut| = 2^3*3 #short vectors: 0 8 2 20 24 44 &Hlattice (#3 <-- #13) 5 2 6 1-2w 3-2w 7 |Aut| = 2^4*3 #short vectors: 0 6 6 18 24 50 &Hlattice (#4 <-- #17) 5 -2 6 -1+2w -1-2w 7 |Aut| = 2^4*3 #short vectors: 0 6 6 18 24 50 &Hlattice (#5 <-- #14) 2 1 3 1-w 1 4 |Aut| = 2^4 #short vectors: 0 4 8 22 24 46 &Hlattice (#6 <-- #21) 7 -1 8 -2-3w -1+2w 11 |Aut| = 2^4 #short vectors: 0 4 8 22 24 46 &Hlattice (#7 <-- #19) 2 -1 6 -1 2-3w 10 |Aut| = 2^4 #short vectors: 0 2 16 8 24 72 &Hlattice (#8 <-- #1) 1 0 1 0 0 1 |Aut| = 2^4*3 #short vectors: 6 12 8 6 24 36 &Hlattice (#9 <-- #15) 1 0 5 0 2w 5 |Aut| = 2^3 #short vectors: 2 4 12 18 24 44 &Hlattice (#10 <-- #20) 1 0 3 0 -1-w 3 |Aut| = 2^3*3 #short vectors: 2 0 12 38 24 16 Steinitz class <2,w>: &Hlattice (#11 <-- #11) <2,w> <1> <1> 1 1/2w 4 -1/2w -1-w 4 |Aut| = 2^4*3 #short vectors: 0 0 8 42 24 18 &Hlattice (#12 <-- #12) <1> <1> <2,w> 2 -1 2 1/2+1/2w -1/2-1/2w 3/2 |Aut| = 2^4*3 #short vectors: 0 12 0 6 24 62 &Hlattice (#13 <-- #29) <1> <1> <2,w> 5 1-w 5 -1-w 3/2+1/2w 3 |Aut| = 2^3*3 #short vectors: 0 8 2 20 24 44 &Hlattice (#14 <-- #6) <2,w> <1> <1> 1/2 0 5 0 2w 5 |Aut| = 2^4 #short vectors: 0 6 6 18 24 50 &Hlattice (#15 <-- #9) <1> <1> <2,w> 2 -1 4 -1 1/2+1/2w 1 |Aut| = 2^4*3 #short vectors: 0 6 0 36 24 20 &Hlattice (#16 <-- #10) <2,w> <1> <1> 1/2 0 3 0 -1-w 3 |Aut| = 2^3*3 #short vectors: 0 2 14 14 24 62 &Hlattice (#17 <-- #30) <1> <1> <2,w> 3 w 3 -1/2-1/2w -1 1 |Aut| = 2^3 #short vectors: 0 2 12 20 24 52 &Hlattice (#18 <-- #7) <1> <1> <2,w> 1 0 1 0 0 1/2 |Aut| = 2^4 #short vectors: 4 6 10 22 24 26 &Hlattice (#19 <-- #5) <1> <1> <2,w> 3 -w 6 1 -3/2-1/2w 2 |Aut| = 2^4 #short vectors: 2 4 12 18 24 44 &Hlattice (#20 <-- #8) <1> <1> <2,w> 1 0 2 0 -1/2+1/2w 1 |Aut| = 2^3*3 #short vectors: 2 6 12 8 24 58 Steinitz class <2,-1+w>: &Hlattice (#21 <-- #28) <1> <1> <2,-1+w> 3 -1 4 1/2w -2+1/2w 2 |Aut| = 2^4*3 #short vectors: 0 0 8 42 24 18 &Hlattice (#22 <-- #4) <1> <1> <2,-1+w> 2 -1 2 -1+1/2w 1-1/2w 3/2 |Aut| = 2^4*3 #short vectors: 0 12 0 6 24 62 &Hlattice (#23 <-- #24) <1> <1> <2,-1+w> 2 -1 4 1/2w -2+1/2w 5/2 |Aut| = 2^3*3 #short vectors: 0 8 2 20 24 44 &Hlattice (#24 <-- #22) <2,-1+w> <1> <1> 1/2 0 5 0 2w 5 |Aut| = 2^4 #short vectors: 0 6 6 18 24 50 &Hlattice (#25 <-- #23) <1> <1> <2,-1+w> 2 -w 4 1 -1+1/2w 1 |Aut| = 2^4*3 #short vectors: 0 6 0 36 24 20 &Hlattice (#26 <-- #25) <1> <1> <2,-1+w> 3 2-w 5 1-1/2w 1-1/2w 3/2 |Aut| = 2^3 #short vectors: 0 2 12 20 24 52 &Hlattice (#27 <-- #27) <2,-1+w> <1> <1> 1/2 0 3 0 -1-w 3 |Aut| = 2^3*3 #short vectors: 0 2 14 14 24 62 &Hlattice (#28 <-- #2) <1> <1> <2,-1+w> 1 0 1 0 0 1/2 |Aut| = 2^4 #short vectors: 4 6 10 22 24 26 &Hlattice (#29 <-- #3) <1> <1> <2,-1+w> 1 0 5 0 -3+w 5/2 |Aut| = 2^4 #short vectors: 2 4 12 18 24 44 &Hlattice (#30 <-- #26) <1> <1> <2,-1+w> 1 0 2 0 1/2w 1 |Aut| = 2^3*3 #short vectors: 2 6 12 8 24 58 &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 3 1 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 0 0 0 3 1 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 3 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 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 2 0 0 4 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 4 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 1 0 4 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 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 1 1 0 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 3 0 0 0 0 0 1 3 0 0 0 0 1 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 6 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 1 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 2 1 1 0 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 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 3 0 0 0 3 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 2 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 1 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 2 0 1 0 1 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 1 0 0 0 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 0 0 0 4 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 4 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 3 0 0 3 0 0 1 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 1 1 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 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 1 0 0 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 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 1 1 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 1 0 0 1 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 1 0 0 0 3 3 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= 12167 = 23^3 2-adic symbol: 1^6_II 23-adic symbol: 1^3 23^3 -1-adic symbol: +^6 -^0 level=23, weight=3 a_0,..,a_12 determine modular form &Gram (#1 <- H1,H2,H13,H23) 4 0 4 0 -2 4 1 0 0 6 0 1 -1 0 8 0 -1 0 0 -4 8 |Aut| = 2^3*3 #short vectors: 0 0 0 8 0 2 0 20 0 24 0 44 &Gram (#2 <- H3,H4,H14,H24) 4 0 4 0 0 4 1 0 0 6 0 0 -1 0 6 0 1 0 0 0 6 |Aut| = 2^4*3 #short vectors: 0 0 0 6 0 6 0 18 0 24 0 50 &Gram (#3 <- H5,H6) 4 0 4 2 -2 6 2 -2 1 8 0 -1 2 3 8 1 0 2 3 2 8 |Aut| = 2^4 #short vectors: 0 0 0 4 0 8 0 22 0 24 0 46 &Gram (#4 <- H7) 4 2 6 2 0 6 2 1 0 6 2 0 1 2 6 1 3 -2 -1 2 10 |Aut| = 2^5 #short vectors: 0 0 0 2 0 16 0 8 0 24 0 72 &Gram (#5 <- H8) 2 0 2 0 0 2 1 0 0 12 0 1 0 0 12 0 0 1 0 0 12 |Aut| = 2^7*3 #short vectors: 0 6 0 12 0 8 0 6 0 24 0 36 &Gram (#6 <- H9,H19,H29) 2 0 4 0 0 4 0 0 -1 6 0 -1 0 0 6 1 0 0 0 0 12 |Aut| = 2^5 #short vectors: 0 2 0 4 0 12 0 18 0 24 0 44 &Gram (#7 <- H10) 2 0 6 0 -3 6 0 0 2 6 0 -2 0 -3 6 1 0 0 0 0 12 |Aut| = 2^5*3 #short vectors: 0 2 0 0 0 12 0 38 0 24 0 16 &Gram (#8 <- H11,H21) 6 2 6 2 -2 6 3 0 3 8 3 0 0 4 8 2 3 1 4 0 8 |Aut| = 2^4*3 #short vectors: 0 0 0 0 0 8 0 42 0 24 0 18 &Gram (#9 <- H12,H22) 4 2 4 2 2 4 1 -1 1 10 1 2 2 -3 10 2 1 0 3 3 10 |Aut| = 2^4*3 #short vectors: 0 0 0 12 0 0 0 6 0 24 0 62 &Gram (#10 <- H15,H25) 4 0 4 0 0 4 1 -2 2 8 2 2 -1 -1 8 2 1 -2 -1 2 8 |Aut| = 2^4*3 #short vectors: 0 0 0 6 0 0 0 36 0 24 0 20 &Gram (#11 <- H16,H27) 4 0 6 0 -3 6 0 0 2 6 0 -2 2 3 6 1 0 0 0 0 6 |Aut| = 2^4*3 #short vectors: 0 0 0 2 0 14 0 14 0 24 0 62 &Gram (#12 <- H17,H26) 4 2 6 2 2 6 0 2 -1 6 0 -1 2 -2 6 1 0 0 -2 -2 8 |Aut| = 2^3 #short vectors: 0 0 0 2 0 12 0 20 0 24 0 52 &Gram (#13 <- H18,H28) 2 0 2 0 0 4 0 0 -1 6 1 0 0 0 12 0 -1 0 0 0 12 |Aut| = 2^6 #short vectors: 0 4 0 6 0 10 0 22 0 24 0 26 &Gram (#14 <- H20,H30) 2 0 4 0 -2 4 0 1 -1 8 0 1 0 4 8 1 0 0 0 0 12 |Aut| = 2^4*3 #short vectors: 0 2 0 6 0 12 0 8 0 24 0 58