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.
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{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}
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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).