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In this paper we consider the question of when all Seshadri constants on aproduct of two isogenous elliptic curves $E_1\times E_2$ without complexmultiplication are integers. By studying elliptic curves on $E_1\times E_2$ wetranslate this question into a purely numerical problem expressed by quadraticforms. By solving that problem, we show that all Seshadri constants on$E_1\times E_2$ are integers if and only if the minimal degree of an isogeny$E_1\to E_2$ equals 1 or 2. Furthermore, this method enables a characterizationof irreducible principal polarizations on $E_1\times E_2$.

Integrality of Seshadri constants and irreducibility of principal polarizations on products of two isogenous elliptic curves