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).