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===Covariants of a binary quartic===
The algebra of invariants of a quartic form is generated by invariants ''i'', ''j'' of degrees 2, 3. This ring is naturally isomorphic to the ring of modular forms of level 1, with the two generators corresponding to the Eisenstein series ''E''<sub>4</sub> and ''E''<sub>6</sub>. 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 {{math|1=''jf''<sup>3</sup>
===Covariants of a binary quintic===
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