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===Covariants of a binary linear form===
For linear forms ''ax''<math>F_1(x,y) = Ax + ''by''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 ''ax''<supmath>F_2(x,y) = Ax^2</sup> + 2''bxy''2Bxy + ''cy''<sup>Cy^2</supmath> is a polynomial algebra in 1 variable generated by the discriminant ''b''<supmath>B^2 - AC</supmath> − ''ac'' of degree 2. The algebra of covariants is a polynomial algebra in 2 variables generated by the discriminant together with the form ''f'' itself (of degree 1 and order 2). {{harv|Schur|1968|loc=II.8}} {{harv|Hilbert|1993|loc=XVI, XX}}
===Covariants of a binary cubic===
The algebra of invariants of the cubic form ''ax''<supmath>F_3(x,y) = Ax^3</sup> + 3''bx''<sup>2</sup>''y''3Bx^2y + 3''cxy''<sup>3Cxy^2</sup> + ''dy''<sup>Dy^3</supmath> is a polynomial algebra in 1 variable generated by the discriminant ''D''<math>\Delta = 3''b''<sup>3B^2C^2</sup>''c''<sup>2</sup> + 6''abcd''6ABCD- −4B^3D 4''b''<sup>3</sup>''d''- −4C^3A 4''c''<sup>3</sup>''a''- − ''a''<sup>2</sup>''d''<sup>A^2D^2</supmath> 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]] 4''H''<supmath>4H^3</sup>=''Df''<sup>^2</sup>-''T''<sup>^2</supmath> of degree 6 and order 6. {{harv|Schur|1968|loc=II.8}} {{harv|Hilbert|1993|loc=XVII, XX}}
===Covariants of a binary quartic===
The algebra of invariants of a quartic form is generated by invariants ''<math>i''</math>, ''<math>j''</math> of degrees 2, 3:
<math display="block">
\begin{aligned}
F_4(x, y)&=A x^4+4 B x^3 y+6 C x^2 y^2+4 D x y^3+E y^4\\
j_{F_4}&=A C E+2 B C D-C^3-B^2 E-A D^2
\end{aligned}
</math>
</math>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> – ''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}} ▼
▲</math>This ring is naturally isomorphic to the ring of modular forms of level 1, with the two generators corresponding to the Eisenstein series ''E''< submath> 4E_4</ submath> and ''E''< submath> 6E_6</ submath>. The algebra of covariants is generated by these two invariants together with the form ''<math>f ''</math> of degree 1 and order 4, the Hessian ''<math>H ''</math> of degree 2 and order 4, and a covariant ''<math>T ''</math> of degree 3 and order 6. They are related by a syzygy {{<math |1=''>jf ''<sup>^3 </sup> –- ''Hf ''<sup>2</sup>''i''^2i + 4''H''<sup>4H^3 </sup> + ''T ''<sup>^2 </sup> = 0 }}</math> of degree 6 and order 12. {{harv|Schur|1968|loc=II.8}} {{harv|Hilbert|1993|loc=XVIII, XXII}}
===Covariants of a binary quintic===
|+ (basic invariant, basic covariant)
! Degree of forms !! 1 !! 2 !! 3 !! 4 !! 5
!n
|-
| 1
|| (5, 20)
|| (23, 94)
|
|-
| 2
|| (6, 18)
|| (29, 92)
|
|-
| 3
||
|| (20, 63)
||(8, 28)
|
|-
| 4
||
||
|| (8, 28)
||
||
|}
Multiple forms:
* Covariants of several linear forms: The ring of invariants of ''<math>n ''</math> linear forms is generated by ''<math>n ''( ''n ''–1-1)/2 </math> invariants of degree 2. The ring of covariants of ''<math>n ''</math> linear forms is essentially the same as the ring of invariants of ''<math>n ''+1 </math> linear forms. ▼
▲* Covariants of several linear forms: The ring of invariants of ''n'' linear forms is generated by ''n''(''n''–1)/2 invariants of degree 2. The ring of covariants of ''n'' linear forms is essentially the same as the ring of invariants of ''n''+1 linear forms.
* Covariants of several linear and quadratic forms:
** The ring of invariants of a sum of ''<math>m''</math> linear forms and ''<math>n''</math> quadratic forms is generated by ''<math>m''(''m''–1-1)/2 + ''n''(''n''+1)/2</math> generators in degree 2, ''<math>nm''(''m''+1)/2 + ''n''(''n''–1-1)(''n''–2-2)/6</math> in degree 3, and ''<math>m''(''m''+1)''n''(''n''–1-1)/4</math> in degree 4.
** For the number of generators of the ring of covariants, change ''<math>m''</math> to ''<math>m''+1</math>.
* Covariants of many cubics or quartics: See {{harvtxt|Young|1898}}.
==See also==
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