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# 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 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}}
# The algebra of invariants of a quartic form is generated by invariants ''i'', ''j'' of degrees 2, 3. The algebra of covariants is generated by these two invariants together with the form ''f'' of degree 1 and order 4, the Hessian ''H'' of degree 2 and order 4, and a covariant ''T'' of degree 3 and order 6. They are related by a syzygy ''jf''<sup>3</sup>−''Hf''<sup>2</sup>''i''+4''H''<sup>3</sup>+''T''<sup>2</sup>=0 of degree 6 and order 12. {{harv|Schur|1968|loc=II.8}} {{harv|Hilbert|1993|loc=XVIII, XXII}}
# The algebra of invariants of a quintic form was found by Sylvester and is generated by invariants of degree 4, 8, 12, 18. The generators of degrees 4, 8, 12 generate a polynomial ring, which contains the square of the generator of degree 18. The invariants are rather complicated to write out explicitly: Sylvester showed that the generators of degrees 4, 8, 12, 18 have 12, 59, 228, and 848 terms often with very large coefficients. {{harv|Schur|1968|loc=II.9}} {{harv|Hilbert|1993|loc=XVIII}}
# The algebra of invariants of a sextic form is generated by invariants of degree 2, 4, 6, 10, 15. The generators of degrees 2, 4, 6, 10 generate a polynomial ring, which contains the square of the generator of degree 15. {{harv|Schur|1968|loc=II.9}}
#{{harvtxt|von Gall|1888}} and {{harvtxt|Dixmier|Lazard|1986}} showed that the algebra of invariants of a degree 7 form is generated by a set with 1 invariant of degree 4, 3 of degree 8, 6 of degree 12, 4 of degree 14, 2 of degree 16, 9 of degree 18, and one of each of the degrees 20, 22, 26, 30
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