Serre's Theorem states (according to this textbook):
Let C be an elliptic curve given by a Weierstrass equation with rational coefficients. Assume that C does not have complex multiplication. Then there is an integer N 1, depending on the curve C, such that for any relatively prime integer n the Galois representation is an isomorphism
What the text doesn't explain is how
N depends on C, only that it does. So effectively the text is telling me that, for some mystical number N,
is an isomorphism, but finding N is down to black magic.
So I was wondering: if it isn't too difficult, how can I find the integer N? Though saying that, if it requires pages and pages of incredibly advanced mathematics, I'll just assume the reason it was only mentioned in passing was because it is beyond anything I should be able to do.
"It's never crowded along the extra mile"
Graduated, and done with maths forever