I know what is the $j$-invariant but I am asking about a general definition in sence of classical invariant theory. The following possible definition seems to be wrong: Consider an elliptic curve $C: y^2=x^3+ax+b.$ Acting by the transformation $$ \begin{array}{c} x=a_{1,1}x'+a_{1,2}y'+c_1,\\ y=a_{2,1}x'+a_{2,2}y'+c_2, \end{array} $$ we get new curve $C'.$ The invariant is a function of coefficients of the curves which is stable under above transformation. This definition is wrong becouse $C'$ now consists $y'^3$ and it isn't elliptic curve.

So, what is correct definition of an invariant of elliptic curve?

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    $\begingroup$ You are using a definition of "elliptic curve" which complots against you. $\endgroup$ Apr 7, 2011 at 18:08
  • $\begingroup$ The equation of $C'$ IS an elliptic curve, but it is not in the Weierstrass form. So the formula of $j$ is more complicated in this case. In Silvermann's book, transformation doesn't involve $y'$ in $x'$. $\endgroup$ Apr 7, 2011 at 18:14
  • $\begingroup$ Oh, so the transformation must be of the restricted form x=a1,2x′+c1.? $\endgroup$
    – Melania
    Apr 7, 2011 at 18:33
  • $\begingroup$ @Mariano, Auguste. I understood, the definition of el.curve should be more general: $a_0y^2+a_1y=b_0 x^3+b_1 x^2 +b_2 x+b_3,$ and the transformation should be restricted $x=a_{1,1}x'+c_1,\\ y=a_{2,1}x'+a_{2,2}y'+c_2.$ Then we may define invariants as usual. Thanks $\endgroup$
    – Melania
    Apr 7, 2011 at 19:18
  • $\begingroup$ books.google.com/books?id=CDP1zxFJLucC $\endgroup$ Apr 7, 2011 at 23:04


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