William C. Jagy May 2004 Integer coefficient positive ternary quadratic forms that are spinor regular but are NOT regular. In each case the coefficients of the form are the Schiemann-reduced representative for the equivalence class. As to the order, the integer sextuple {a, b, c, d, e, f} refers to the map (x, y, z) --> a x^2 + b y^2 + c z^2 + d y z + e z x + f x y. Symmetric Matrix of double the form: 2 a f e f 2 b d e d 2 c The discriminant "Disc" is as favored by Watson and by Brandt and Intrau, half the determinant of the matrix: Disc = 4 a b c + d e f - a d^2 - b e^2 - c f^2. In particular, 2 a b c <= Disc <= 4 a b c; see Watson's book, p. 29. ------------------------------------------------------------- {Disc, 64, Coeff, {2, 2, 5, 2, 2, 0}, Spinor Genus Size, 1} {Disc, 108, Coeff, {3, 3, 4, 0, 0, 3}, Spinor Genus Size, 1} {Disc, 108, Coeff, {3, 4, 4, 4, 3, 3}, Spinor Genus Size, 1} {Disc, 128, Coeff, {1, 4, 9, 4, 0, 0}, Spinor Genus Size, 1} {Disc, 256, Coeff, {2, 5, 8, 4, 0, 2}, Spinor Genus Size, 1} {Disc, 256, Coeff, {4, 4, 5, 0, 4, 0}, Spinor Genus Size, 1} {Disc, 324, Coeff, {1, 7, 12, 0, 0, 1}, Spinor Genus Size, 1} {Disc, 343, Coeff, {2, 7, 8, 7, 1, 0}, Spinor Genus Size, 1} {Disc, 432, Coeff, {3, 7, 7, 5, 3, 3}, Spinor Genus Size, 2} {Disc, 432, Coeff, {4, 4, 9, 0, 0, 4}, Spinor Genus Size, 1} {Disc, 432, Coeff, {3, 4, 9, 0, 0, 0}, Spinor Genus Size, 1} {Disc, 1024, Coeff, {4, 9, 9, 2, 4, 4}, Spinor Genus Size, 1} {Disc, 1024, Coeff, {4, 5, 13, 2, 0, 0}, Spinor Genus Size, 1} {Disc, 1024, Coeff, {5, 8, 8, 0, 4, 4}, Spinor Genus Size, 1} {Disc, 1372, Coeff, {7, 8, 9, 6, 7, 0}, Spinor Genus Size, 1} {Disc, 1728, Coeff, {4, 9, 12, 0, 0, 0}, Spinor Genus Size, 1} {Disc, 2048, Coeff, {4, 8, 17, 0, 4, 0}, Spinor Genus Size, 1} {Disc, 3888, Coeff, {4, 9, 28, 0, 4, 0}, Spinor Genus Size, 1} {Disc, 4096, Coeff, {9, 9, 16, 8, 8, 2}, Spinor Genus Size, 1} {Disc, 4096, Coeff, {4, 9, 32, 0, 0, 4}, Spinor Genus Size, 1} {Disc, 4096, Coeff, {5, 13, 16, 0, 0, 2}, Spinor Genus Size, 1} {Disc, 5488, Coeff, {8, 9, 25, 2, 4, 8}, Spinor Genus Size, 1} {Disc, 6912, Coeff, {9, 16, 16, 16, 0, 0}, Spinor Genus Size, 1} {Disc, 6912, Coeff, {13, 13, 16, -8, 8, 10}, Spinor Genus Size, 1} {Disc, 16384, Coeff, {9, 17, 32, -8, 8, 6}, Spinor Genus Size, 1} {Disc, 16384, Coeff, {9, 16, 36, 16, 4, 8}, Spinor Genus Size, 1} {Disc, 27648, Coeff, {9, 16, 48, 0, 0, 0}, Spinor Genus Size, 2} {Disc, 62208, Coeff, {9, 16, 112, 16, 0, 0}, Spinor Genus Size, 1} {Disc, 87808, Coeff, {29, 32, 36, 32, 12, 24}, Spinor Genus Size, 1} ------------------------------------------------------------- CONJECTURED complete. I have checked up to Disc 575000 as of May 2004 References: W.K. Chan and A.G.Earnest: Discriminant Bounds for Spinor Regular Ternary Quadratic Lattices. (preprint) A. Schiemann: Ternary positive definite quadratic forms are determined by their theta series. Math. Ann. 308(1997) 507-517 W.C.Jagy, I.Kaplansky, A.Schiemann: There are 913 regular ternary forms. Mathematika 44(1997) 332-341. G.L. Watson: Integral Quadratic Forms (Cambridge, 1960).