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We study elliptic surfaces over $\mathbb{Q}(T)$ with coefficients of aWeierstrass model being polynomials in $\mathbb{Q}[T]$ with degree at most 2.We derive an explicit expression for their rank over $\mathbb{Q}(T)$ dependingon the factorization and other simple properties of certain polynomials.Finally, we give sharp estimates for the ranks of the considered families andwe present several applications, among which there are lists of rationalpoints, generic families with maximal rank and generalizations of formerresults.

Ranks of elliptic curves over $\mathbb{Q}(T)$ of small degree in $T$