Output of /home/aschiem/Pgm/Hn/hn --invar -t --shells-8 &K=Q(sqrt(-19)) &Hdim=5 V=K^5 &HNeighbourhood at <2,2w> contains 32 classes: mass of the neighbourhood is 2959/2304 Steinitz class <1,w>: &Hlattice (#1 <- #15) 3 -1 3 -1 1 3 -w w -1 4 1 -1+w -2+w 1 4 |Aut| = 2^3 #short vectors: 0 0 20 70 162 340 670 1020 &Hlattice (#2 <- #28) 3 w 3 0 -1 3 -1 0 1 4 1-w -w -1+w 1+w 5 |Aut| = 2^2*5 #short vectors: 0 0 20 70 162 340 670 1020 &Hlattice (#3 <- #29) 3 -1 3 -1+w -1 3 1 -w -1 3 0 -1 1 -1 3 |Aut| = 2^2*5 #short vectors: 0 0 20 70 162 340 670 1020 &Hlattice (#4 <- #30) 3 -1 3 w -1 3 -1 1 -1 3 -w -1+w -1 w 4 |Aut| = 2^2*5 #short vectors: 0 0 20 70 162 340 670 1020 &Hlattice (#5 <- #11) 2 -1 2 1 0 2 1 -1 0 3 0 0 -1 1-w 3 |Aut| = 2^4*3*5 #short vectors: 0 20 20 30 202 220 510 1100 &Hlattice (#6 <- #17) 2 -1 3 -1 1 3 0 w -1 3 0 1 w -w 3 |Aut| = 2^4*3 #short vectors: 0 12 24 42 158 288 654 1056 &Hlattice (#7 <- #20) 2 -1 2 -1 0 2 0 -1 1 3 1-w w -1 -w 5 |Aut| = 2^4*3 #short vectors: 0 12 24 42 158 288 654 1056 &Hlattice (#8 <- #25) 2 0 2 0 -1 3 0 -1 -w 3 1-w 0 0 0 3 |Aut| = 2^4*3 #short vectors: 0 10 22 48 168 290 630 1054 &Hlattice (#9 <- #24) 2 -1 2 0 -1 3 1 -1 0 3 0 1 0 -1+w 3 |Aut| = 2^3*3 #short vectors: 0 8 24 50 150 312 686 1040 &Hlattice (#10 <- #27) 2 0 2 0 1 3 0 -1 -w 3 -w -1 -1 1 4 |Aut| = 2^3*3 #short vectors: 0 8 24 50 150 312 686 1040 &Hlattice (#11 <- #23) 2 0 3 1 1 3 0 -1-w -1 3 1 0 w 1 4 |Aut| = 2^3*3 #short vectors: 0 6 22 56 160 314 662 1038 &Hlattice (#12 <- #32) 2 0 3 0 1 3 -1 0 w 3 0 -1-w 0 1 3 |Aut| = 2^3*3 #short vectors: 0 6 22 56 160 314 662 1038 &Hlattice (#13 <- #12) 2 -1 3 0 -w 3 0 -1 -1 3 -1+w 1 -1 -1 4 |Aut| = 2^4 #short vectors: 0 4 16 66 198 296 558 1048 &Hlattice (#14 <- #13) 2 0 3 0 -1+w 3 -w 1 0 4 1 -1+w 1 -2+w 4 |Aut| = 2^4 #short vectors: 0 4 16 66 198 296 558 1048 &Hlattice (#15 <- #14) 2 0 3 0 -1+w 3 1 1-w -1 3 -1+w 1 -1 -1+w 4 |Aut| = 2^4 #short vectors: 0 4 20 62 170 316 638 1036 &Hlattice (#16 <- #16) 3 -1 3 1-w -1 4 1 -1 1+w 4 1 -1 2 0 5 |Aut| = 2^4 #short vectors: 0 4 20 62 170 316 638 1036 &Hlattice (#17 <- #18) 2 0 3 0 1 3 0 1 -1+w 3 -1 -w -1 1 3 |Aut| = 2^4 #short vectors: 0 4 24 58 142 336 718 1024 &Hlattice (#18 <- #19) 2 0 2 1 1 3 0 0 0 3 -w -1 -1 -1 4 |Aut| = 2^4 #short vectors: 0 4 24 58 142 336 718 1024 &Hlattice (#19 <- #26) 2 0 3 0 -1+w 3 -w 1 -1 4 -1 -1-w -2+w 1-w 4 |Aut| = 2^3 #short vectors: 0 4 20 62 170 316 638 1036 &Hlattice (#20 <- #31) 3 -2+w 3 -1 1 3 1 0 w 4 1 0 -2+w 1+w 4 |Aut| = 2^3 #short vectors: 0 2 18 68 180 318 614 1034 &Hlattice (#21 <- #1) 1 0 1 0 0 1 0 0 0 1 0 0 0 0 1 |Aut| = 2^8*3*5 #short vectors: 10 40 80 90 132 400 820 1000 &Hlattice (#22 <- #2) 1 0 1 0 0 1 0 0 0 2 0 0 0 1-w 3 |Aut| = 2^6*3 #short vectors: 6 16 40 114 240 344 504 1024 &Hlattice (#23 <- #10) 1 0 1 0 0 2 0 0 -1 3 0 0 -1 -w 3 |Aut| = 2^5*3 #short vectors: 4 10 38 96 180 370 658 1006 &Hlattice (#24 <- #3) 1 0 2 0 0 2 0 -1 -1 2 0 w 0 -w 4 |Aut| = 2^5*3*5 #short vectors: 2 20 40 42 124 320 764 1040 &Hlattice (#25 <- #4) 1 0 2 0 0 2 0 0 -w 3 0 w 0 0 3 |Aut| = 2^6 #short vectors: 2 8 32 74 156 352 700 1016 &Hlattice (#26 <- #5) 1 0 2 0 -1 3 0 -1 1-w 3 0 -1 w -1 3 |Aut| = 2^5 #short vectors: 2 4 32 82 148 376 732 1000 &Hlattice (#27 <- #6) 1 0 2 0 0 2 0 1 1 2 0 0 1-w 0 4 |Aut| = 2^5*3*5 #short vectors: 2 20 40 42 124 320 764 1040 &Hlattice (#28 <- #7) 1 0 2 0 0 2 0 0 -1 2 0 1 -1+w 0 4 |Aut| = 2^4*3^2 #short vectors: 2 12 24 74 220 288 508 1056 &Hlattice (#29 <- #8) 1 0 2 0 0 2 0 -1 -1 3 0 -1 -1+w 2-w 4 |Aut| = 2^4*3 #short vectors: 2 8 28 78 184 332 620 1028 &Hlattice (#30 <- #9) 1 0 2 0 0 2 0 1 1 3 0 -1+w 0 -2+w 4 |Aut| = 2^4*3 #short vectors: 2 8 28 78 184 332 620 1028 &Hlattice (#31 <- #21) 1 0 3 0 w 3 0 1 1 3 0 1 2-w -w 4 |Aut| = 2^3*5 #short vectors: 2 0 20 102 224 340 524 1020 &Hlattice (#32 <- #22) 1 0 3 0 1 3 0 0 1+w 3 0 -2+w -1 -1 3 |Aut| = 2^5*3 #short vectors: 2 0 24 98 196 360 604 1008 classes of Z-lattices with respect to the trace form &Dim=10 V=Q^10 &Genus of the trace-forms: det= 2476099 = 19^5 2-adic symbol: 1^-10_II 19-adic symbol: 1^-5 19^-5 -1-adic symbol: +^10 -^0 level=19, weight=5 a_0,..,a_16 determine modular form &Gram (#1 <- H1) 6 2 6 2 -2 6 2 2 2 6 2 0 0 -2 6 0 -1 1 0 -2 6 3 2 1 1 0 -2 8 2 0 -1 0 0 -2 2 8 0 0 -2 1 -1 0 2 -1 8 3 2 0 2 0 0 4 -1 2 8 |Aut| = 2^4 #short vectors: 0 0 0 0 0 20 0 70 0 162 0 340 0 670 0 1020 &Gram (#2 <- H2,H3) 6 2 6 1 -2 6 2 2 -2 6 0 2 -2 2 6 2 1 2 0 -2 6 0 2 0 0 -1 2 6 0 -2 2 -2 -2 0 -2 6 0 0 0 2 2 0 -2 1 6 2 0 2 1 0 0 0 2 2 6 |Aut| = 2^2*5 #short vectors: 0 0 0 0 0 20 0 70 0 162 0 340 0 670 0 1020 &Gram (#3 <- H4) 6 2 6 2 2 6 1 2 2 6 2 2 0 2 6 2 0 2 2 2 6 2 0 1 0 2 2 6 0 1 2 2 0 2 2 6 2 2 2 0 1 0 2 2 6 0 -2 0 -2 -2 -1 -2 -2 -2 6 |Aut| = 2^3*5 #short vectors: 0 0 0 0 0 20 0 70 0 162 0 340 0 670 0 1020 &Gram (#4 <- H5) 4 2 4 2 0 4 2 0 2 4 2 0 2 2 6 2 0 2 2 1 6 1 2 -1 1 1 -1 10 1 -1 2 0 2 0 -3 10 2 1 0 2 0 2 3 1 10 2 0 0 1 0 2 2 2 0 10 |Aut| = 2^5*3*5 #short vectors: 0 0 0 20 0 20 0 30 0 202 0 220 0 510 0 1100 &Gram (#5 <- H6,H7) 4 2 4 2 0 4 2 0 0 6 2 0 2 0 6 0 0 0 -2 -1 6 0 0 0 -1 2 1 6 1 1 0 0 2 -2 -2 10 1 0 1 1 -1 2 2 0 10 0 -1 0 0 2 -2 -2 0 0 10 |Aut| = 2^4*3 #short vectors: 0 0 0 12 0 24 0 42 0 158 0 288 0 654 0 1056 &Gram (#6 <- H8) 4 0 4 0 -2 4 0 0 0 4 0 -2 0 0 6 0 -2 2 0 -1 6 1 0 0 -2 0 0 6 2 0 0 -1 0 0 1 6 0 -1 -1 0 0 -1 0 0 8 0 -2 1 0 0 0 0 0 -2 8 |Aut| = 2^6*3 #short vectors: 0 0 0 10 0 22 0 48 0 168 0 290 0 630 0 1054 &Gram (#7 <- H9,H10) 4 0 4 0 2 4 2 2 2 6 0 2 0 2 6 0 -2 0 0 -1 6 0 -2 -2 -3 -2 2 8 1 -2 0 -1 -2 2 4 8 2 1 1 3 2 2 1 1 10 2 1 0 0 -1 -3 -2 -1 2 10 |Aut| = 2^3*3 #short vectors: 0 0 0 8 0 24 0 50 0 150 0 312 0 686 0 1040 &Gram (#8 <- H11,H12) 4 2 4 2 2 6 2 0 2 6 0 0 -2 -2 6 0 0 -2 -2 3 6 0 0 -1 -2 0 2 6 0 0 2 1 0 -2 0 6 1 0 0 1 0 0 -2 -2 8 0 1 0 0 0 0 -2 -2 -2 8 |Aut| = 2^3*3 #short vectors: 0 0 0 6 0 22 0 56 0 160 0 314 0 662 0 1038 &Gram (#9 <- H13,H14) 4 0 4 2 0 6 0 0 2 6 0 -2 0 -2 6 0 0 1 -2 0 6 0 0 -2 -1 2 1 6 0 0 0 2 1 -2 -1 6 1 2 2 2 -2 2 0 0 8 2 1 0 -2 1 0 0 2 0 8 |Aut| = 2^4 #short vectors: 0 0 0 4 0 16 0 66 0 198 0 296 0 558 0 1048 &Gram (#10 <- H15,H16) 4 0 4 2 -2 6 2 0 2 6 0 0 1 0 6 0 2 0 -2 0 6 2 -2 2 2 2 -3 8 2 -2 2 -1 2 0 4 8 1 0 0 0 2 -2 1 -1 8 0 -1 0 -2 2 -1 -1 1 2 8 |Aut| = 2^4 #short vectors: 0 0 0 4 0 20 0 62 0 170 0 316 0 638 0 1036 &Gram (#11 <- H17,H18) 4 0 4 2 -2 6 2 0 0 6 0 -2 0 -2 6 0 0 0 -1 2 6 0 0 0 2 -1 2 6 0 0 1 0 0 2 2 6 2 -1 2 0 1 0 -2 2 8 1 -2 2 1 0 -2 0 2 2 8 |Aut| = 2^4 #short vectors: 0 0 0 4 0 24 0 58 0 142 0 336 0 718 0 1024 &Gram (#12 <- H19) 4 0 4 2 2 6 0 2 0 6 2 0 2 1 6 2 2 0 2 2 8 2 2 0 0 0 1 8 0 -2 -1 0 2 1 0 8 1 0 1 2 1 2 -2 -2 8 0 -1 0 -1 -2 -3 1 2 -2 8 |Aut| = 2^4 #short vectors: 0 0 0 4 0 20 0 62 0 170 0 316 0 638 0 1036 &Gram (#13 <- H20) 4 2 6 0 -2 6 0 -2 -1 6 0 2 -2 0 6 0 0 2 -2 -2 6 0 0 0 -2 -2 3 6 0 -2 0 2 1 -2 0 6 2 1 -2 -1 -2 0 2 1 8 1 3 0 -2 2 0 0 0 2 8 |Aut| = 2^4 #short vectors: 0 0 0 2 0 18 0 68 0 180 0 318 0 614 0 1034 &Gram (#14 <- H21) 2 0 2 0 0 2 0 0 0 2 0 0 0 0 2 1 0 0 0 0 10 0 1 0 0 0 0 10 0 0 1 0 0 0 0 10 0 0 0 1 0 0 0 0 10 0 0 0 0 1 0 0 0 0 10 |Aut| = 2^13*3*5 #short vectors: 0 10 0 40 0 80 0 90 0 132 0 400 0 820 0 1000 &Gram (#15 <- H22) 2 0 2 0 0 2 0 0 0 4 0 0 0 0 4 0 0 0 1 2 6 0 0 0 2 1 1 6 1 0 0 0 0 0 0 10 0 1 0 0 0 0 0 0 10 0 0 1 0 0 0 0 0 0 10 |Aut| = 2^10*3 #short vectors: 0 6 0 16 0 40 0 114 0 240 0 344 0 504 0 1024 &Gram (#16 <- H23) 2 0 2 0 0 4 0 0 -2 4 0 0 2 -2 6 0 0 2 0 -1 6 0 0 1 1 -1 0 8 0 0 1 -2 2 1 2 8 1 0 0 0 0 0 0 0 10 0 -1 0 0 0 0 0 0 0 10 |Aut| = 2^8*3 #short vectors: 0 4 0 10 0 38 0 96 0 180 0 370 0 658 0 1006 &Gram (#17 <- H24,H27) 2 0 4 0 -2 4 0 -2 0 4 0 -2 2 0 4 0 0 0 1 0 8 0 0 -1 0 0 -2 8 0 1 -1 0 -1 2 2 8 0 0 0 0 -1 -2 -2 2 8 1 0 0 0 0 0 0 0 0 10 |Aut| = 2^6*3*5 #short vectors: 0 2 0 20 0 40 0 42 0 124 0 320 0 764 0 1040 &Gram (#18 <- H25) 2 0 4 0 0 4 0 0 0 4 0 0 0 0 4 0 0 -1 0 2 6 0 1 0 2 0 0 6 0 0 2 0 -1 -1 0 6 0 -2 0 1 0 0 0 0 6 1 0 0 0 0 0 0 0 0 10 |Aut| = 2^9 #short vectors: 0 2 0 8 0 32 0 74 0 156 0 352 0 700 0 1016 &Gram (#19 <- H26) 2 0 4 0 0 4 0 -2 2 6 0 -2 -2 0 6 0 2 0 -1 0 6 0 0 2 0 -1 0 6 0 1 0 -2 0 3 -2 8 0 0 1 0 1 -2 3 -2 8 1 0 0 0 0 0 0 0 0 10 |Aut| = 2^7 #short vectors: 0 2 0 4 0 32 0 82 0 148 0 376 0 732 0 1000 &Gram (#20 <- H28) 2 0 4 0 2 4 0 0 0 4 0 0 0 2 4 0 -2 -2 -1 -1 8 0 -2 0 -1 0 4 8 0 -1 -1 -2 0 1 1 8 0 1 0 -2 0 0 0 -2 8 1 0 0 0 0 0 0 0 0 10 |Aut| = 2^6*3^2 #short vectors: 0 2 0 12 0 24 0 74 0 220 0 288 0 508 0 1056 &Gram (#21 <- H29,H30) 2 0 4 0 0 4 0 0 2 4 0 -2 -2 0 6 0 -2 -2 -2 3 8 0 1 -2 -2 -1 3 8 0 0 -1 0 -1 -2 0 8 0 0 0 1 2 2 0 2 8 1 0 0 0 0 0 0 0 0 10 |Aut| = 2^5*3 #short vectors: 0 2 0 8 0 28 0 78 0 184 0 332 0 620 0 1028 &Gram (#22 <- H31) 2 0 6 0 2 6 0 1 2 6 0 2 0 -2 6 0 2 0 -1 -1 6 0 0 -2 0 -2 0 6 0 0 -2 0 2 -2 -1 6 0 -3 1 -2 -2 0 -2 0 8 1 0 0 0 0 0 0 0 0 10 |Aut| = 2^5*5 #short vectors: 0 2 0 0 0 20 0 102 0 224 0 340 0 524 0 1020 &Gram (#23 <- H32) 2 0 6 0 3 6 0 2 0 6 0 0 -2 3 6 0 0 -2 0 0 6 0 2 2 0 0 -3 6 0 0 0 2 0 0 -2 6 0 0 0 0 -2 2 -2 3 6 1 0 0 0 0 0 0 0 0 10 |Aut| = 2^7*3 #short vectors: 0 2 0 0 0 24 0 98 0 196 0 360 0 604 0 1008