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===Covariants of a binary cubic===
The algebra of invariants of the cubic form ''ax''<sup>3</sup> + 3''bx''<sup>2</sup>''y'' + 3''cxy''<sup>2</sup> + ''dy''<sup>3</sup> is a polynomial algebra in 1 variable generated by the discriminant ''D'' = 3''b''<sup>2</sup>''c''<sup>2</sup> + 6''abcd'' − 4''b''<sup>3</sup>''d'' − 4''c''<sup>3</sup>''a'' − ''a''<sup>2</sup>''d''<sup>2</sup> of degree 4. The algebra of covariants is generated by the discriminant, the form itself (degree 1, order 3), the Hessian ''H'' (degree 2, order 2) and a covariant ''T'' of degree 3 and order 3. They are related by the [[Syzygy (mathematics)|syzygy]] 4''h''<sup>3</sup>=''Df''<sup>2</sup>-''T''<sup>2</sup> of degree 6 and order 6. {{harv|Schur|1968|loc=II.8}} {{harv|Hilbert|1993|loc=XVII, XX}}
===Covariants of a binary quartic===
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