Return to Answer

-------------------------------------------------
    55296 :     1   24  576    0    0    0
    55296 :    24   25   25   14    0    0 
------------------------------------------------
    55296 :     4   24  145    0    4    0
    55296 :     9   24   64    0    0    0
-----------------------------------------------

---------------------------------------
 g0  :      55296 :     1   24  576    0    0    0 auto 8
 g1  :      55296 :    24   25   25   14    0    0 auto 8
---------------------------------------
 g2  :      55296 :     4   24  145    0    4    0 auto 8
 g3  :      55296 :     9   24   64    0    0    0 auto 8
---------------------------------------


jagy@phobeusjunior:~$ sage
┌────────────────────────────────────────────────────────────────────┐
│ SageMath Version 6.9, Release Date: 2015-10-10                     │
│ Type "notebook()" for the browser-based notebook interface.        │
│ Type "help()" for help.                                            │
└────────────────────────────────────────────────────────────────────┘
sage:  q1 = QuadraticForm(ZZ,3,[1,0,0, 24,0, 576] )
sage: q1
Quadratic form in 3 variables over Integer Ring with coefficients: 
[ 1 0 0 ]
[ * 24 0 ]
[ * * 576 ]
sage: q1.det()
110592
sage: q1.number_of_automorphisms()
8
sage: q1.conway_mass()
1/2
sage: 
sage: quit
Exiting Sage (CPU time 0m0.41s, Wall time 1m5.23s).
jagy@phobeusjunior:~$

added: I do not know of any software that deals with indefinite ternaries. What follows is about positive forms

Now that I think of it, people may not realize that the entirety of the tables by Brandt and Intrau are available, compiled by Alexander Schiemann TABLE 1 and TABLE 2. As far as the words "odd" and "even," Alexander followed usage from integral lattices, as in SPLAG by Conway and Sloane

added: I do not know of any software that deals with indefinite ternaries. What follows is about positive forms

as long as the power of two dividing the discriminant is not too large, you will get the genus correctly split into spinor genera with my Magma program. I guess I will put samples first. Note that it correctly says the genus of $x^2 + 24 y^2 + 576 z^2$ has four classes, but it prints out four spinor genera with repeat of the one spinor genus, which is nonsense. The trouble first came to my attention about 1996 when Manjul Bhargava was corresponding with Irving Kaplansky, Manjul asked Magma to find all forms alone in a genus, and it gave the wrong answer for $x^2 + 8 y^2 + 64 z^2 \; . \;$ I put good deal of relevant material at TERNARY

as long as the power of two dividing the discriminant is not too large, you will get the genus correctly split into spinor genera with my Magma program. I guess I will put samples first. Note that it correctly says the genus of $x^2 + 24 y^2 + 576 z^2$ has four classes, but it prints out four spinor genera with repeat of the one spinor genus, which is nonsense. The actual genus has two spinor genera,

-------------------------------------------------
    55296 :     1   24  576    0    0    0
    55296 :    24   25   25   14    0    0 
------------------------------------------------
    55296 :     4   24  145    0    4    0
    55296 :     9   24   64    0    0    0
-----------------------------------------------

The trouble first came to my attention about 1996 when Manjul Bhargava was corresponding with Irving Kaplansky, Manjul asked Magma to find all forms alone in a genus, and it gave the wrong answer for $x^2 + 8 y^2 + 64 z^2 \; . \;$ I put a good deal of relevant material at TERNARY