= Polynomial ring of level-one modular forms
{title2=$M_*(SL_2(\mathbb Z))=\mathbb C[E_4,E_6]$}
Subtract the constant term of a <modular form> times a same-weight monomial in $E_4,E_6$. The remainder is a <cusp form>, hence divisible by the <modular discriminant>. Induction on weight and $1728\Delta=E_4^3-E_6^2$ express every form as a weighted homogeneous polynomial in these two <Eisenstein series>. The number of monomials of a given weight equals the <dimension of level-one modular forms>, so they form a basis and the graded ring has no polynomial relation.
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