In this paper, we give an explicit description of the de Rham and $p$-adic polylogarithms for elliptic curves using the Kronecker theta function. In particular, consider an elliptic curve $E$ defined over an imaginary quadratic field $𝕂$ with complex multiplication by the full ring of integers ${𝒪}_{𝕂}$ of $𝕂$. Note that our condition implies that $𝕂$ has class number one. Assume in addition that $E$ has good reduction above a prime $p\ge 5$ unramified in ${𝒪}_{𝕂}$. In this case, we prove that the specializations of the $p$-adic elliptic...