Estimating the output size of a query is a fundamental yet longstanding problem in database query processing. Traditional cardinality estimators used by database systems can routinely underestimate the true output size by orders of magnitude, which leads to significant system performance penalty. Recently, upper bounds have been proposed that are based on information inequalities and incorporate sizes and max-degrees from input relations, yet they their main benefit is limited to cyclic queries, because they degenerate to rather trivial formulas on acyclic queries. We introduce a significant extension of the upper bounds, by incorporating $\ell_p$-norms of the degree sequences of join attributes. Our bounds are significantly lower than previously known bounds, even when applied to acyclic queries. These bounds are also based on information theory, they come with a matching query evaluation algorithm, are computable in exponential time in the query size, and are provably tight when all degrees are "simple".