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===Covariants of a binary linear form===
For linear forms <math>F_1(x,y) = Ax + By</math> the only invariants are constants. The algebra of covariants is generated by the form itself of degree 1 and order 1.
===Covariants of a binary quadric===
The algebra of invariants of the quadratic form <math>F_2(x,y) = Ax^2 + 2Bxy + Cy^2</math> is a polynomial algebra in 1 variable generated by the discriminant <math>B^2 - AC</math> of degree 2. The algebra of covariants is a polynomial algebra in 2 variables generated by the discriminant together with the form
===Covariants of a binary cubic===
The algebra of invariants of the cubic form <math>F_3(x,y) = Ax^3 + 3Bx^2y + 3Cxy^2 + Dy^3</math> is a polynomial algebra in 1 variable generated by the discriminant <math>\Delta = 3B^2C^2 + 6ABCD- 4B^3D - 4C^3A - A^2D^2</math> of degree 4. The algebra of covariants is generated by the discriminant, the form itself (degree 1, order 3), the Hessian <math>H</math> (degree 2, order 2) and a covariant <math>T</math> of degree 3 and order 3. They are related by the [[Syzygy (mathematics)|syzygy]] <math>4H^3=Df^2-T^2</math> 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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