For Gamma-number fields, the spinor genus of the integral trace form is determined exactly by the discriminant and the signature.
Mantilla-Soler, The Spinor Genus of the integral trace , Transactions of the Amer- ican Mathematical Society 369 (2017) , 1547-1577
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An introduction to $\Gamma$-number fields
For Gamma-number fields, the spinor genus of the integral trace form is determined exactly by the discriminant and the signature.