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Let $E$ be an elliptic curve with a rational $p$-torsion point $P$. Then $E$ is isogenous to the elliptic curve $E' := E/\langle P \rangle$ via the mod $P$ map. I know that the conductor of $E$ and $E'$ are the same, but the discriminant might not be. Is there are relation between $\Delta_{E}$ and $\Delta_{E'}$?

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Other than the fact that they are both divisible by the conductor (and presumably satisfy Szpiro's conjecture), I think this is quite subtle. Even the case of conductor $30$ reveals complications.... –  Mike Bennett Jan 31 '13 at 3:50

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Section 2 in the preprint Local invariants of isogenous elliptic curves by T. and V. Dokchitser might be what you are looking for. It states a result of Coates that if $p>3$ prime and $E \to E'$ is a $p$-isogeny (this is weaker than $E$ having a $p$-torsion point) then $\Delta_E^p/{\Delta_{E'}}$ is a $12$th power, and also the new result that if $p=2$ and $p=3$ then this ratio is a third and fourth power, respectively.

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