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 19 classes: mass of the neighbourhood is 1027/1152 Steinitz class <1,w>: &Hlattice (#1 <- #30) 4 -2 4 1-w -1 6 1-w w 2+w 8 |Aut| = 2^5 #short vectors: 0 0 0 24 0 48 0 216 0 288 0 672 &Hlattice (#2 <- #31) 4 1 6 -w -3-w 6 1-w -2 2 6 |Aut| = 2^5*3 #short vectors: 0 0 0 24 0 48 0 216 0 288 0 672 &Hlattice (#3 <- #56) 4 0 4 1 -w 4 -w 1 -2 6 |Aut| = 2^4 #short vectors: 0 0 0 24 0 48 0 216 0 288 0 672 &Hlattice (#4 <- #57) 4 1 4 0 2+w 4 w -1 -1 4 |Aut| = 2^3*3 #short vectors: 0 0 0 24 0 48 0 216 0 288 0 672 &Hlattice (#5 <- #66) 4 1 4 0 -2+w 4 -2-w -1 1 4 |Aut| = 2^4*3 #short vectors: 0 0 0 24 0 48 0 216 0 288 0 672 &Hlattice (#6 <- #76) 4 -1-w 4 -w 2 4 2 -1+w -1 4 |Aut| = 2^3 #short vectors: 0 0 0 24 0 48 0 216 0 288 0 672 &Hlattice (#7 <- #77) 4 -1-w 4 -1-w 2 4 2 -1+w 0 4 |Aut| = 2^3*3 #short vectors: 0 0 0 24 0 48 0 216 0 288 0 672 &Hlattice (#8 <- #79) <2,w> <1> <2,w> <1> 1 0 4 0 -1/2w 1 -1/2w -1+w -1 6 |Aut| = 2^3 #short vectors: 0 0 0 20 0 68 0 192 0 268 0 704 &Hlattice (#9 <- #78) <2,w> <2,w> <1> <1> 1 0 1 0 1+1/2w 6 1-1/2w -1 -2+w 6 |Aut| = 2^3*3^2 #short vectors: 0 0 0 12 0 108 0 144 0 228 0 768 &Hlattice (#10 <- #4) <1> <1> <2,w> <2,w> 2 1 2 1 0 1 1 1 1/2 1 |Aut| = 2^7*3^2 #short vectors: 0 24 0 24 0 96 0 24 0 168 0 672 &Hlattice (#11 <- #83) 2 0 2 0 1 2 1 1-w 1 8 |Aut| = 2^3*3^2 #short vectors: 0 12 0 36 0 12 0 192 0 288 0 576 &Hlattice (#12 <- #10) 2 0 2 w 1 6 1 0 0 6 |Aut| = 2^7 #short vectors: 0 8 0 24 0 64 0 152 0 248 0 672 &Hlattice (#13 <- #12) 2 0 2 1-w 0 6 0 1-w 0 6 |Aut| = 2^7 #short vectors: 0 8 0 24 0 64 0 152 0 248 0 672 &Hlattice (#14 <- #52) <1> <2,w> <1> <2,w> 2 0 1 1 -1/2w 4 0 1/2 0 1 |Aut| = 2^3*3^2 #short vectors: 0 6 0 24 0 60 0 168 0 258 0 672 &Hlattice (#15 <- #20) <1> <2,w> <1> <2,w> 4 -1/2w 1 w -1 4 1 0 -1/2w 1 |Aut| = 2^5 #short vectors: 0 4 0 24 0 56 0 184 0 268 0 672 &Hlattice (#16 <- #26) <1> <1> <2,w> <2,w> 2 0 2 0 1-1/2w 2 1-1/2w 0 0 2 |Aut| = 2^5 #short vectors: 0 4 0 24 0 56 0 184 0 268 0 672 &Hlattice (#17 <- #33) 2 -1 4 0 -1-w 4 -1+w 2 -2 8 |Aut| = 2^4 #short vectors: 0 4 0 24 0 56 0 184 0 268 0 672 &Hlattice (#18 <- #82) 2 0 2 1 0 4 1 1 1-w 4 |Aut| = 2^3 #short vectors: 0 4 0 28 0 36 0 208 0 288 0 640 &Hlattice (#19 <- #53) 4 0 4 -1 -1-w 4 w 1-w 2 6 |Aut| = 2^3 #short vectors: 0 2 0 24 0 52 0 200 0 278 0 672 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 12-classes of trace forms &Gram (#1 <- H1,H2) 4 2 4 2 0 4 2 0 0 4 1 1 1 -1 6 0 1 -1 1 2 6 1 0 2 0 -2 -3 6 1 0 0 0 3 2 0 6 |Aut| = 2^8*3^2 #short vectors: 0 0 0 24 0 48 0 216 0 288 0 672 &Gram (#2 <- H3,H4,H5,H6,H7) 4 2 4 1 0 4 0 -1 2 4 0 0 -1 0 4 0 0 -1 -1 2 4 1 0 0 0 1 1 4 0 -1 0 0 0 1 2 4 |Aut| = 2^5*3 #short vectors: 0 0 0 24 0 48 0 216 0 288 0 672 &Gram (#3 <- H8) 4 2 4 0 1 4 0 1 -1 4 0 0 -1 -1 4 0 0 1 1 -2 4 0 -1 0 0 -1 1 4 0 -1 0 0 -1 1 -1 4 |Aut| = 2^7 #short vectors: 0 0 0 20 0 68 0 192 0 268 0 704 &Gram (#4 <- H9) 4 2 4 0 0 4 0 0 -2 4 2 2 -2 0 6 2 2 -2 0 1 6 2 0 2 -2 0 0 6 2 0 -2 0 2 2 -1 6 |Aut| = 2^7*3^4 #short vectors: 0 0 0 12 0 108 0 144 0 228 0 768 &Gram (#5 <- H10) 2 1 2 1 1 2 1 0 1 2 0 0 0 0 10 0 0 0 0 5 10 0 0 0 0 5 0 10 0 0 0 0 5 0 0 10 |Aut| = 2^14*3^4 #short vectors: 0 24 0 24 0 96 0 24 0 168 0 672 &Gram (#6 <- H11) 2 1 2 0 0 2 0 0 -1 2 1 1 1 -1 8 1 0 1 0 4 8 1 1 1 0 1 1 8 1 0 -1 0 0 0 3 8 |Aut| = 2^7*3^4 #short vectors: 0 12 0 36 0 12 0 192 0 288 0 576 &Gram (#7 <- H12,H13) 2 0 2 0 0 2 0 0 0 2 0 1 1 0 6 1 0 0 1 0 6 1 0 0 -1 0 0 6 0 1 -1 0 0 0 0 6 |Aut| = 2^13 #short vectors: 0 8 0 24 0 64 0 152 0 248 0 672 &Gram (#8 <- H14) 2 1 2 1 1 4 1 1 -1 4 0 0 0 0 4 0 0 0 0 2 4 0 0 0 0 -2 0 8 0 0 0 0 -2 0 -2 8 |Aut| = 2^6*3^4 #short vectors: 0 6 0 24 0 60 0 168 0 258 0 672 &Gram (#9 <- H15,H16) 2 0 2 0 -1 4 0 1 -2 4 1 0 0 0 4 1 0 0 0 -1 4 0 0 1 1 0 0 6 0 0 0 0 -1 -1 0 6 |Aut| = 2^9 #short vectors: 0 4 0 24 0 56 0 184 0 268 0 672 &Gram (#10 <- H17) 2 0 2 1 1 4 1 -1 0 4 0 0 -1 1 4 0 0 -1 -1 0 4 1 1 0 0 2 2 8 1 1 2 0 -2 -2 -1 8 |Aut| = 2^9 #short vectors: 0 4 0 24 0 56 0 184 0 268 0 672 &Gram (#11 <- H18) 2 0 2 1 1 4 1 -1 0 4 1 0 0 0 4 0 -1 -1 1 0 4 1 -1 -1 1 -1 -1 8 1 -1 0 0 -1 2 1 8 |Aut| = 2^7 #short vectors: 0 4 0 28 0 36 0 208 0 288 0 640 &Gram (#12 <- H19) 2 1 4 1 -1 4 1 1 1 4 1 0 0 1 4 0 0 0 -1 -1 4 0 0 0 1 1 -2 4 1 0 0 -1 2 -1 -1 8 |Aut| = 2^6 #short vectors: 0 2 0 24 0 52 0 200 0 278 0 672