The seven root-free admissible rank-19 lattices at det 160..224 (Section 3.3; SM Section S1.2): Gram matrices and invariants. == P160: det 160, even, minimum 4, roots 0, kissing number 9480, Smith invariants [40, 2, 2], |Aut| = 92160, |O_0| = 960, rank fixed by O_0 = 0 4 1 1 2 0 0 0 -1 1 3 0 0 2 0 -2 2 2 4 -1 1 4 2 2 0 1 1 1 2 4 2 -1 3 -1 -2 2 1 4 -2 1 2 4 2 1 1 0 2 2 3 2 -2 2 0 -2 4 1 5 -2 2 2 2 4 -1 0 0 1 2 4 3 -1 3 -1 -2 2 3 4 -3 0 0 1 -1 4 0 0 1 0 -1 0 -2 -2 2 0 3 0 3 0 0 1 1 0 0 4 2 1 2 0 -1 0 0 -3 0 1 0 -1 2 0 1 0 0 0 2 4 2 2 -1 1 0 1 -1 1 1 -1 -2 0 -1 1 2 1 1 1 2 4 2 0 3 -2 1 0 0 3 0 1 -2 1 2 2 2 0 2 2 2 4 2 2 -1 2 -1 -1 3 1 2 -1 3 4 3 4 -1 0 -1 0 2 8 3 -2 5 0 -5 4 3 7 -3 0 2 2 3 0 -1 1 3 2 3 6 -3 3 1 -2 4 2 3 -4 0 -1 -2 -1 -2 0 0 -2 -1 -2 -3 4 -1 -3 3 -6 -1 -3 1 2 3 2 3 -2 0 1 1 2 5 3 -1 6 0 -3 3 0 3 -3 0 -1 0 -1 2 -3 -1 0 -1 0 1 -3 0 8 -2 5 -2 1 0 -2 -2 -2 -2 0 0 1 0 -1 -5 -2 3 -3 -2 6 -5 -3 -5 1 2 2 4 2 3 1 1 3 3 4 4 -6 3 5 -5 12 0 6 -1 2 1 1 3 0 0 -1 0 1 3 2 -1 0 -2 -3 0 8 5 -2 4 4 5 4 3 -1 -2 1 2 7 3 -3 3 1 -5 6 5 14 -5 -1 -2 -2 -3 0 2 0 -2 -1 -3 -4 1 -3 0 1 -1 -2 -5 6 == P180: det 180, even, minimum 4, roots 0, kissing number 8910, Smith invariants [60, 3], |Aut| = 11520, |O_0| = 360, rank fixed by O_0 = 0 4 2 2 -1 -2 1 -2 1 0 1 -3 0 2 -3 1 3 3 -3 2 2 4 0 1 1 0 -2 2 1 2 -3 2 3 -3 0 5 3 0 3 2 0 4 0 0 1 -2 0 -2 0 0 -1 -1 1 1 -1 -1 -2 0 -1 1 0 4 4 -1 -2 0 0 1 0 1 -1 2 -2 0 -3 4 0 -2 1 0 4 8 -2 -2 0 -2 1 2 2 0 3 -3 -1 -4 5 0 1 0 1 -1 -2 4 2 0 -1 -2 0 -1 -1 2 1 1 1 -3 -2 -2 -2 -2 -2 -2 2 8 1 0 0 3 0 -1 4 0 -2 2 -2 -5 1 2 0 0 0 0 1 4 1 3 -1 1 2 -2 1 3 4 -1 0 0 1 -2 0 -2 -1 0 1 4 2 -2 0 1 -4 1 3 3 1 2 1 2 0 1 1 -2 0 3 2 6 -2 2 2 -3 0 2 3 1 2 -3 -3 0 0 2 0 3 -1 -2 -2 6 -2 -3 5 0 -5 -3 0 -4 0 2 -1 1 2 -1 0 1 0 2 -2 4 3 0 -2 2 1 2 2 2 3 -1 -1 0 -1 -1 2 1 2 -3 3 6 -5 0 5 5 -1 4 -3 -3 1 2 3 2 4 -2 -4 -3 5 0 -5 12 -3 -7 -7 1 -7 1 0 1 -2 -3 1 0 1 1 0 0 -2 0 -3 4 1 3 -3 1 3 5 -1 0 -1 1 -2 3 3 2 -5 2 5 -7 1 10 7 -1 5 3 3 -1 -3 -4 1 2 4 3 3 -3 1 5 -7 3 7 10 -4 3 -3 0 -2 4 5 -3 -2 -1 1 1 0 2 -1 1 -3 -1 -4 8 1 2 3 0 0 0 -2 -5 0 2 2 -4 2 4 -7 1 5 3 1 8 == P196: det 196, even, minimum 4, roots 0, kissing number 8526, Smith invariants [28, 7], |Aut| = 5376, |O_0| = 168, rank fixed by O_0 = 0 4 0 -2 0 -3 -2 0 0 0 -1 0 0 -2 1 1 1 0 -1 -3 0 4 2 -2 -1 0 1 -1 0 1 0 0 0 -3 0 -1 -2 0 0 -2 2 8 0 -3 -1 1 -2 6 -3 -2 0 -3 1 0 -1 -1 4 2 0 -2 0 8 -5 -3 1 1 3 -4 -4 3 -6 1 -1 3 0 -1 -2 -3 -1 -3 -5 12 7 -3 0 -4 6 6 -3 9 -4 -2 -2 2 -1 6 -2 0 -1 -3 7 6 -2 -1 -1 3 3 -2 5 -3 -2 -2 0 0 5 0 1 1 1 -3 -2 4 3 -2 1 -2 1 -1 1 4 2 1 0 -3 0 -1 -2 1 0 -1 3 6 -5 3 0 0 1 1 5 4 4 -2 -4 0 0 6 3 -4 -1 -2 -5 10 -7 -3 0 -6 1 -4 -2 -3 4 4 -1 1 -3 -4 6 3 1 3 -7 8 5 0 7 -3 2 1 3 -3 -1 0 0 -2 -4 6 3 -2 0 -3 5 8 1 6 -3 -2 0 3 -3 -1 0 0 0 3 -3 -2 1 0 0 0 1 6 -1 0 -2 1 0 -2 -4 -2 0 -3 -6 9 5 -1 1 -6 7 6 -1 10 -3 0 -2 2 -1 3 1 -3 1 1 -4 -3 1 1 1 -3 -3 0 -3 8 3 1 1 2 -3 1 0 0 -1 -2 -2 4 5 -4 2 -2 -2 0 3 8 3 3 1 -3 1 -1 -1 3 -2 -2 2 4 -2 1 0 1 -2 1 3 6 4 -3 -5 0 -2 -1 0 2 0 1 4 -3 3 3 0 2 1 3 4 6 -2 -3 -1 0 4 -1 -1 0 0 -2 4 -3 -3 -2 -1 2 1 -3 -2 6 5 -3 0 2 -2 6 5 -3 -4 4 -1 -1 -4 3 -3 -3 -5 -3 5 12 == P208: det 208, even, minimum 4, roots 0, kissing number 8292, Smith invariants [52, 2, 2], |Aut| = 2304, |O_0| = 96, rank fixed by O_0 = 1 4 0 -2 -1 0 2 0 1 0 1 1 1 -1 0 1 0 1 1 0 0 4 -2 1 0 1 0 1 -3 -1 -1 1 2 0 -1 2 1 1 -2 -2 -2 4 1 0 0 0 0 4 1 1 -2 0 0 0 -3 2 0 3 -1 1 1 6 -1 0 2 -3 2 -2 -3 -2 5 4 0 -5 5 -4 3 0 0 0 -1 4 1 0 4 0 2 4 3 2 0 -2 1 2 1 -1 2 1 0 0 1 6 2 3 0 4 4 2 1 0 0 -1 3 2 1 0 0 0 2 0 2 4 -1 -1 2 0 0 1 0 -1 -2 0 -3 3 1 1 0 -3 4 3 -1 8 -1 4 6 3 -1 -3 -2 4 3 4 -1 0 -3 4 2 0 0 -1 -1 8 0 1 -2 2 3 2 -6 5 0 3 1 -1 1 -2 2 4 2 4 0 6 5 1 -2 -3 -1 1 0 1 2 1 -1 1 -3 4 4 0 6 1 5 8 4 0 -1 -1 1 2 4 -1 1 1 -2 -2 3 2 0 3 -2 1 4 6 2 1 -1 1 0 3 -4 -1 2 0 5 2 1 1 -1 2 -2 0 2 10 7 0 -5 7 -2 -1 0 0 0 4 0 0 0 -3 3 -3 -1 1 7 8 2 -6 6 -2 -1 1 -1 0 0 -2 0 -1 -2 2 -1 -1 -1 0 2 4 -2 1 0 0 0 2 -3 -5 1 -1 -2 4 -6 1 1 1 -5 -6 -2 10 -6 3 -4 1 1 2 5 2 3 0 3 5 0 2 0 7 6 1 -6 14 -1 3 1 1 0 -4 1 2 -3 4 0 1 4 3 -2 -2 0 3 -1 8 -4 0 -2 3 3 -1 1 3 -1 3 2 -1 -4 -1 -1 0 -4 3 -4 8 == P216: det 216, even, minimum 4, roots 0, kissing number 8118, Smith invariants [72, 3], |Aut| = 1728, |O_0| = 72, rank fixed by O_0 = 1 4 0 2 -2 2 -2 4 0 3 -1 -2 2 0 5 1 -5 -2 1 4 0 4 2 -2 -3 1 -1 0 0 0 0 0 3 0 1 1 1 -2 -3 2 2 4 -1 -1 -1 2 0 1 -1 -1 1 3 2 1 -3 0 0 0 -2 -2 -1 8 -1 -1 -3 0 -4 1 4 0 -1 -5 -3 1 0 1 0 2 -3 -1 -1 6 -2 2 -1 2 1 -3 0 -4 5 1 -4 -2 1 4 -2 1 -1 -1 -2 4 -4 1 -1 1 1 -1 0 -4 1 4 2 -3 -4 4 -1 2 -3 2 -4 10 -1 5 -4 -2 2 1 8 0 -6 -3 6 7 0 0 0 0 -1 1 -1 4 -1 -1 1 1 2 -5 -1 1 1 -3 -1 3 0 1 -4 2 -1 5 -1 6 -2 -2 1 -1 7 2 -3 -2 3 4 -1 0 -1 1 1 1 -4 -1 -2 4 0 0 -4 0 0 1 1 -2 -2 -2 0 -1 4 -3 1 -2 1 -2 0 6 1 -1 -5 -3 4 2 1 -2 2 0 1 0 0 -1 2 1 1 0 1 4 -1 1 -1 -2 -1 1 3 0 3 3 -1 -4 0 1 2 -1 -4 -1 -1 10 -4 1 0 0 -2 -2 5 0 2 -5 5 -4 8 -5 7 0 -5 1 -4 16 3 -8 -4 6 7 1 1 1 -3 1 1 0 -1 2 0 -3 -1 1 3 4 -1 -1 -1 0 -5 1 -3 1 -4 4 -6 1 -3 1 4 -2 0 -8 -1 10 4 -2 -7 -2 1 0 0 -2 2 -3 1 -2 1 2 -1 0 -4 -1 4 4 -2 -5 1 -2 0 1 1 -3 6 -3 3 -2 1 1 -2 6 -1 -2 -2 8 5 4 -3 0 0 4 -4 7 -1 4 -2 -2 3 -2 7 0 -7 -5 5 10 == P220: det 220, even, minimum 4, roots 0, kissing number 8040, Smith invariants [220], |Aut| = 480, |O_0| = 60, rank fixed by O_0 = 1 4 1 -1 -2 0 -2 -2 0 -2 -2 -1 -2 1 0 0 0 0 0 1 1 4 -2 -1 2 -4 2 -3 -1 -1 -2 3 -2 1 -2 -1 2 3 -1 -1 -2 6 1 1 3 1 3 2 -1 -1 0 4 1 4 1 -3 -4 5 -2 -1 1 4 0 2 2 0 3 2 1 -1 2 -1 1 2 1 -2 2 0 2 1 0 4 -2 3 -2 0 -1 -3 4 -1 2 -1 2 0 0 4 -2 -4 3 2 -2 8 -3 4 2 0 0 -2 2 -2 2 2 -2 -4 1 -2 2 1 2 3 -3 8 -5 4 2 2 2 2 0 1 0 1 1 4 0 -3 3 0 -2 4 -5 8 -2 -2 -3 0 1 1 2 1 -3 -3 -2 -2 -1 2 3 0 2 4 -2 6 2 4 -2 5 -3 3 1 0 -1 4 -2 -1 -1 2 -1 0 2 -2 2 4 4 -2 0 -1 0 0 1 0 0 -1 -2 -1 1 -3 0 2 -3 4 4 10 -6 4 -5 2 -1 0 1 1 -2 3 0 -1 4 -2 2 0 -2 -2 -6 10 -5 3 -3 1 0 3 -2 1 -2 4 2 -1 2 2 1 5 0 4 -5 10 -4 6 1 -2 -3 6 0 1 1 -1 2 -2 0 1 -3 -1 -5 3 -4 6 -1 -1 0 -1 0 0 -2 4 1 -1 2 1 2 3 0 2 -3 6 -1 6 0 -2 -3 4 0 -1 1 2 2 2 0 1 1 0 -1 1 1 -1 0 6 -1 -2 5 0 2 -3 1 0 -2 1 -3 0 1 0 0 -2 0 -2 -1 4 2 -2 0 3 -4 -2 0 -4 1 -3 -1 0 1 3 -3 -1 -3 -2 2 6 -5 1 -1 5 2 4 1 4 -2 4 0 1 -2 6 0 4 5 -2 -5 14 == P224: det 224, even, minimum 4, roots 0, kissing number 7986, Smith invariants [56, 2, 2], |Aut| = 4608, |O_0| = 48, rank fixed by O_0 = 2 4 2 0 1 1 2 -1 -1 2 -1 3 3 -3 4 3 -2 -2 1 1 2 4 0 1 0 2 -1 -2 3 -1 3 3 -2 4 3 -3 -2 2 1 0 0 4 2 2 2 -1 -2 1 2 1 2 1 2 3 -3 0 0 1 1 1 2 4 2 2 0 -2 1 2 2 2 1 2 3 -3 0 -1 0 1 0 2 2 4 2 -1 -2 1 2 2 2 0 1 3 -3 1 -2 0 2 2 2 2 2 4 -1 -3 3 1 3 3 -1 4 4 -4 -1 1 1 -1 -1 -1 0 -1 -1 4 -2 0 1 -2 -2 2 0 -3 2 0 2 -1 -1 -2 -2 -2 -2 -3 -2 8 -5 -2 -2 -2 0 -6 -3 5 2 -4 -1 2 3 1 1 1 3 0 -5 6 0 4 3 -1 6 4 -6 -2 3 0 -1 -1 2 2 2 1 1 -2 0 4 -1 -2 3 -1 1 1 3 -1 -1 3 3 1 2 2 3 -2 -2 4 -1 6 5 -2 5 5 -7 -2 0 0 3 3 2 2 2 3 -2 -2 3 -2 5 8 -3 6 4 -7 -4 1 2 -3 -2 1 1 0 -1 2 0 -1 3 -2 -3 6 -3 -2 1 3 -1 -3 4 4 2 2 1 4 0 -6 6 -1 5 6 -3 10 5 -7 -5 5 2 3 3 3 3 3 4 -3 -3 4 1 5 4 -2 5 8 -7 -1 -1 1 -2 -3 -3 -3 -3 -4 2 5 -6 1 -7 -7 1 -7 -7 14 4 0 0 -2 -2 0 0 1 -1 0 2 -2 3 -2 -4 3 -5 -1 4 6 -4 -2 1 2 0 -1 -2 1 2 -4 3 -1 0 1 -1 5 -1 0 -4 8 2 1 1 1 0 0 1 -1 -1 0 -1 0 2 -3 2 1 0 -2 2 4 Isometries (U^T B U = A): A = HM_no_65, B = Martinet L19, det U = 1 A = HM_no_69, B = Nebe-Sloane dim19kis9480min4det160 (Kallus), det U = 1 A = HM_no_76, B = Martinet LL19c, det U = 1 A = HM_no_76, B = Nebe-Sloane dim19kis8634min4det192 (Kallus), det U = 1 A = HM_no_84, B = Martinet O'19f even (dual norm 54), det U = 1 A = P160, B = HM no. 69 = Lambda_M20 (row j=5), det U = 1 A = P160, B = Martinet LL19b, det U = -1 A = P160, B = Martinet O'19a, even part (section of O'20 at dual norm 40), det U = 1 A = P160, B = Nebe-Sloane dim19kis9480min4det160 (Kallus), det U = 1 A = P180, B = HM no. 72 = Lambda_A6 (row j=8), det U = -1 A = P180, B = Martinet KK'19b, det U = -1 A = P180, B = Martinet O'19b even (dual norm 45), det U = -1 A = P196, B = HM no. 77 = Lambda_L2(7) (row j=13), det U = -1 A = P196, B = Martinet O'19d even (as printed), det U = 1 A = P208, B = Martinet O'19e even (dual norm 52), det U = -1 A = P208, B = Nebe-Sloane dim19kis8292min4det208 (Kallus), det U = -1 A = P216, B = Martinet KK'19c, det U = 1 A = P220, B = Martinet O'19g even (dual norm 55), det U = -1