In 1987, Barry Mazur and John Tate formulated refined conjectures of the ``Birch and Swinnerton-Dyer type'', and one of these conjectures was essentially proved in the prime conductor case by Ehud de Shalit in 1995. One of the main objects in de Shalit's work is the so-called refined L-invariant, which happens to be a Hecke operator. We apply some results of the theory of Mazur's Eisenstein ideal to study in which power of the Eisenstein ideal L belongs. As a corollary of our study, we give a surprising elementary formula on supersingular j-invariants.