What does carminum mean in Latin?

carminum, a noun, means “song, poem”.

What is the genitive of carminum?

The genitive singular is carminī. carminum is neuter, second declension.

In classical Latin, [ˈkar.mɪ.nũː]; in ecclesiastical (church) Latin, [ˈkar.mi.num].

noun · neutersecond declension

carminum n (genitive carminī)

Classical[ˈkar.mɪ.nũː]Ecclesiastical[ˈkar.mi.num]

New Latinneutersong, poem
In nuptiis iuris et medicinae doctorum / Omnia sunt plena carminorum.1670, Antonius De Arena Provençalis, De Bragardissima Villa de Soleriis. Ad suos compagnones studiantes, qui sunt de persona friantes, bassas Dansas & Branlos practicantes, nouuellos quamplurimos mandat. Nova Novorum Novissima, Sive Poemata Stylo Macaronico., page 155:
1600s, Cornelius a Lapide, Commentarii in Sacram Scripturam, Tomus X: Acts, James, Epistles of John, the Apocalypse, p 1360: Et Sybilla lib. 6. Carminum:

Back-formation from carmina, the nominative/accusative plural of the third-declension noun carmen.