Output of /home/aschiem/bin/hn --invar -t --shells-12 --herm_lll3 0.8 &K=Q(sqrt(-10)) &Hdim=4 V=K^4 &HNeighbourhood at <2,w> (even classes) contains 14 classes: mass of the neighbourhood is 79/160 Steinitz class <2,w>: &Hlattice (#1 <- #28) <2,w> <1> <2,w> <2,w> 1 -1/2w 4 1/2 1/2w 1 1/2 0 0 1 |Aut| = 2^4*3*5 #short vectors: 0 0 0 30 0 20 0 230 0 420 0 360 &Hlattice (#2 <- #29) <2,w> <2,w> <2,w> <1> 1 1/2 1 0 1/2 1 1-1/2w -1/2w -1 6 |Aut| = 2^4*3 #short vectors: 0 0 0 22 0 60 0 182 0 380 0 424 &Hlattice (#3 <- #85) <1> <1> <1> <2,w> 4 -w 6 -2+w -3-w 6 -1/2w -1/2w 0 2 |Aut| = 2^2*5 #short vectors: 0 0 0 10 0 120 0 110 0 320 0 520 &Hlattice (#4 <- #86) <1> <1> <1> <2,w> 4 2-w 6 -1-w 3 6 0 -2-1/2w -2+1/2w 3 |Aut| = 2^2*5 #short vectors: 0 0 0 10 0 120 0 110 0 320 0 520 &Hlattice (#5 <- #22) <1> <1> <1> <2,w> 2 1 2 1 1 2 -1/2w 0 0 2 |Aut| = 2^4*3*5 #short vectors: 0 20 0 30 0 60 0 70 0 320 0 360 &Hlattice (#6 <- #6) <1> <1> <1> <2,w> 2 0 2 -1 0 2 -1 1/2w 1 2 |Aut| = 2^5*3 #short vectors: 0 14 0 30 0 48 0 118 0 350 0 360 &Hlattice (#7 <- #23) <1> <1> <1> <2,w> 2 0 4 -1 1-w 4 0 2-1/2w 2+1/2w 2 |Aut| = 2^4*3 #short vectors: 0 12 0 22 0 84 0 86 0 320 0 424 &Hlattice (#8 <- #10) <1> <1> <2,w> <1> 2 1 4 1 1 1 -w 0 -1/2w 6 |Aut| = 2^5*3 #short vectors: 0 6 0 30 0 32 0 182 0 390 0 360 &Hlattice (#9 <- #13) <1> <1> <1> <2,w> 2 0 2 1-w 0 6 0 1-1/2w 0 2 |Aut| = 2^5 #short vectors: 0 6 0 22 0 72 0 134 0 350 0 424 &Hlattice (#10 <- #61) <1> <1> <2,w> <1> 2 0 4 0 1/2w 1 -1 -1 1 4 |Aut| = 2^3*3 #short vectors: 0 6 0 18 0 92 0 110 0 330 0 456 &Hlattice (#11 <- #62) <1> <1> <1> <2,w> 2 1 4 0 1+w 4 0 1 -1/2w 1 |Aut| = 2^3*3 #short vectors: 0 6 0 18 0 92 0 110 0 330 0 456 &Hlattice (#12 <- #35) <1> <1> <2,w> <1> 4 1-w 4 1 0 1 -1-w 2 -1-1/2w 6 |Aut| = 2^3 #short vectors: 0 2 0 14 0 104 0 118 0 330 0 488 &Hlattice (#13 <- #67) <2,w> <1> <1> <1> 1 1 4 1-1/2w -w 6 -1/2w 1 2 6 |Aut| = 2^3*3 #short vectors: 0 2 0 18 0 84 0 142 0 350 0 456 &Hlattice (#14 <- #68) <2,w> <1> <1> <1> 1 1 4 0 -1+w 6 1-1/2w -w -2+w 6 |Aut| = 2^3*3 #short vectors: 0 2 0 18 0 84 0 142 0 350 0 456 classes of Z-lattices with respect to the trace form (scaled by 1/2) one representative of each class &Dim=8 V=Q^8 &Genus of the trace-forms: det= 10000 = 2^4 *5^4 2-adic symbol: 1^-4_II 2^-4_II 5-adic symbol: 1^-4 5^-4 -1-adic symbol: +^8 -^0 level=10, weight=4 a_0,..,a_12 determine modular form 11-classes of trace forms &Gram (#1 <- H1) 4 2 4 2 0 4 2 2 0 4 0 0 0 0 4 0 0 0 0 -1 4 0 0 0 0 -1 -1 4 0 0 0 0 1 1 1 4 |Aut| = 2^8*3^2*5^2 #short vectors: 0 0 0 30 0 20 0 230 0 420 0 360 &Gram (#2 <- H2) 4 2 4 2 0 4 2 0 2 4 0 0 0 0 4 2 0 0 1 -1 6 2 0 0 1 -1 1 6 2 0 0 1 1 3 3 6 |Aut| = 2^8*3^2 #short vectors: 0 0 0 22 0 60 0 182 0 380 0 424 &Gram (#3 <- H3,H4) 4 1 4 1 -1 4 1 -1 -1 4 2 2 1 0 6 2 0 2 -1 3 6 2 0 2 -1 -1 0 6 2 1 -1 2 3 0 0 6 |Aut| = 2^4*5^2 #short vectors: 0 0 0 10 0 120 0 110 0 320 0 520 &Gram (#4 <- H5) 2 1 2 1 1 2 1 0 0 2 0 0 0 0 8 0 0 0 0 -2 8 0 0 0 0 2 2 8 0 0 0 0 -2 -2 2 8 |Aut| = 2^8*3^2*5^2 #short vectors: 0 20 0 30 0 60 0 70 0 320 0 360 &Gram (#5 <- H6) 2 1 2 1 0 2 0 0 0 2 1 0 0 0 6 0 0 0 -1 0 8 0 0 0 -1 0 3 8 0 0 0 -1 0 3 -2 8 |Aut| = 2^10*3^2 #short vectors: 0 14 0 30 0 48 0 118 0 350 0 360 &Gram (#6 <- H7) 2 1 2 1 1 2 1 1 1 4 1 0 0 1 4 0 0 0 1 1 8 0 0 0 -1 -1 2 8 0 0 0 1 1 -2 2 8 |Aut| = 2^8*3^2 #short vectors: 0 12 0 22 0 84 0 86 0 320 0 424 &Gram (#7 <- H8) 2 0 2 0 0 2 1 -1 1 4 0 0 0 0 4 0 0 0 0 2 6 0 0 0 0 -2 -1 6 0 0 0 0 2 1 -1 6 |Aut| = 2^10*3^2 #short vectors: 0 6 0 30 0 32 0 182 0 390 0 360 &Gram (#8 <- H9) 2 0 2 0 0 2 0 -1 0 4 0 -1 0 -1 4 1 0 1 0 0 6 0 0 0 1 1 0 6 1 0 1 0 0 1 0 6 |Aut| = 2^10 #short vectors: 0 6 0 22 0 72 0 134 0 350 0 424 &Gram (#9 <- H10,H11) 2 1 2 1 0 4 1 0 2 4 0 0 -1 -1 4 0 0 -1 -1 -1 4 1 0 0 0 1 1 8 1 0 0 0 1 1 -2 8 |Aut| = 2^5*3^2 #short vectors: 0 6 0 18 0 92 0 110 0 330 0 456 &Gram (#10 <- H12) 2 0 4 0 2 4 0 -1 -2 4 0 1 0 1 4 0 -1 0 1 0 4 1 1 2 0 2 2 6 0 1 2 -1 0 0 1 6 |Aut| = 2^7 #short vectors: 0 2 0 14 0 104 0 118 0 330 0 488 &Gram (#11 <- H13,H14) 2 1 4 1 -1 4 1 -1 2 4 1 -1 0 2 6 1 2 1 -1 0 6 1 -1 0 2 1 0 6 1 -1 0 0 2 -1 2 8 |Aut| = 2^5*3^2 #short vectors: 0 2 0 18 0 84 0 142 0 350 0 456