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.
1$\begingroup$ 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 $d=5$. $\endgroup$– algoriOct 27, 2011 at 10:41
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.]
$\begingroup$ The complete list of invariants was discovered by Hermite in "Sur la theorie des fonctions homogenes a deux indeterminees" Cambridge and Dublin Mathematical Journal, vol 9, pp. 172-217, 1854. Gordan discovered the larger list of covariants. Perhaps the quickest justification of the list of invariants is using the Sylvester canonical form ax^5+by^5-c^(x+y)^5 and writing the invariants in terms of a, b, c. It is done for instance in Salmon's higher algebra book. $\endgroup$ Feb 2, 2012 at 14:31