# History of the Coefficients of Elliptic Curves — Why $a_6$? [duplicate]

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I would like to know what is the motivation behind the naming convention of the Weierstrass form of elliptic curves given as $$E:y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6.$$

I can see that $a_1,a_2,a_3,a_4$ are named by sorting monomials by lex order with $x<y$. But why go from $a_5$ to $a_6$?

## marked as duplicate by Zhen Lin, Fabian, user91500, Claude Leibovici, HakimJun 5 '14 at 6:54

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• In confirmation of @zyx’s response, the sorting by lex order has nothing to do with it. – Lubin Jun 5 '14 at 3:07

## 1 Answer

The convention is from having $x$ of degree $2$, $y$ of degree $3$, and giving degree $i$ to $a_i$ so that, under this assignment of degrees, every term has degree $6$. Equivalently, $a_i$ is the coefficient of the degree $6-i$ term, under the same weighting of $x$ and $y$.

• I see! Thanks for your answer, but why adopt this convention? – BlackAdder Jun 5 '14 at 2:32
• The $a_i$ are, in a suitable sense, weight $i$ generators of rings of modular forms. Classically, over the complex numbers, $a_4$ and $a_6$ are proportional to Eisenstein series of weights $4$ and $6$. In more general situations (see Tate's Formulaire written up by Deligne) the weight $n$ modular forms are certain degree $n$ polynomials in the $a_i$, again with $a_i$ having weight $i$. – zyx Jun 5 '14 at 3:01
• Thanks for the clarification. Alas, I have not yet gone into modular forms, but when I do, I will look out for what you have said. Thanks again! – BlackAdder Jun 5 '14 at 3:29