Content deleted Content added
→Invariants of several binary forms: simpler table |
→Covariants of a binary quartic: added quartic |
||
Line 58:
===Covariants of a binary quartic===
The algebra of invariants of a quartic form is generated by invariants ''i'', ''j'' of degrees 2, 3
\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\\ i_{F_4}&=A E-4 B D+3 C^2 \\ j_{F_4}&=A C E+2 B C D-C^3-B^2 E-A D^2 \end{aligned} </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}} ===Covariants of a binary quintic===
| |||