## Why is $\mathcal{M}(\text{SL}_2(\mathbb{Z}))$ spanned by $E_4$ and $E_6$? [closed]

It is said that $$\mathcal{M}(\text{SL}_2(\mathbb{Z}))=\mathbb{C}[E_4,E_6]$$

where $\mathcal{M}(\text{SL}_2(\mathbb{Z}))$ is the graded ring of module forms over $\text{SL}_2(\mathbb{Z})$ and $E_4,E_6$ are normalized Eisenstein series.

I'm new to modular form and not quite familiar about the Eisenstein series so I fail to prove it.

Any one give some ideas?

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I'm sorry, but I don't think this is appropriate for MO (which is about research level mathematics). This is a result found in any undergraduate textbook on modular forms. – Dan Petersen Jan 11 at 16:09
I agree with D.P. and point you to Serre's "Course in arithmetic". – Julien Puydt Jan 11 at 16:41
Isn't "undergraduate textbook on modular forms" an oxymoron? Which universities are teaching modular forms to undergraduates? – wccanard Jan 11 at 20:01