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Does anyone have a reference for the invariant binary forms of a quintic? That is, what are the $SL_2(C)$ invariant polynomial functions on the space of binary quintics.

flag	asked Oct 26 2011 at 23:08 Neil 31●1

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Complementing Noam's answer: for $d\leq 6$ the algebras of $SL_2$-invariants of degree $d$ forms in 2 variables can be found in 10.2 of Dolgachev's Lectures on invariant theory, as well as a reference in English for $d=5$. – algori Oct 27 2011 at 10:41

Noam D. Elkies · Answer 1 · 2011-10-27 04:11:29Z UTC

According to this page there are independent invariants of degree 4, 8, and 12, plus a degree-18 invariant whose square is a polynomial in the first three. This is attributed to Gordan (1868):

P. Gordan, Beweis, dass jede Covariante und Invariante einer binären Form eine ganze Funktion mit numerischen Coeffizienten einer endlichen Anzahl solcher Formen ist, Journ. f. Math. 69 (1868), 323–354.

That page also describes the invariants, and even the covariants, for several other degrees, and also variations such as multiple forms.

[All I did was Google it; the present MO question already turned up in the first page of search results only a few hours after it was posted.]

Invariant forms of a binary quintic

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