Skip to main content
4 of 10
added 197 characters in body
Will Jagy
  • 25.7k
  • 2
  • 65
  • 121

Thought I recognized this. It appears in Benham, Earnest, Hsia, and Hung (1990), Theorem 1, item (3.4). Page 6.

=====Discriminant     ==Genus Size==    3

[ 1, 1, 32, 0, 0, 0 ]
[ 2, 2, 9, 2, -2, 0 ]

---**----- end of  spinor genus  1    --------

[ 1, 4, 9, -4, 0, 0 ]

---**----- end of  spinor genus  2    --------

The loner is one of the very few spinor regular forms that are not regular. The following constitutes a proof that $\langle 1,4,9,4,0,0 \rangle$ does not lie in the same spinor genus as $\langle 1,1,32,0,0,0 \rangle:$ we have $$ x^2 + 4 y^2 + 9 z^2 + 4yz \neq 2 m^2, $$ where all prime factors $p$ of $m$ satisfy $p \equiv 1 \pmod 4.$ There are often elementary proofs of spinor exceptional integers, once they have been noticed. In this case, $$ x^2 + (2y+z)^2 \neq 2 (m-2z)(m+2z), $$ where a crucial detail is that $z$ is odd.

Will Jagy
  • 25.7k
  • 2
  • 65
  • 121