From the Galois representation of an elliptic curve $E$ we can read the conductor of $E$, and further some information about the minimal discriminant. So is there any more information about the minimal discriminant we can read from the Galois representation?

For example, it is natural to consider the differences between isogenous (equivalently having the same Galois representation) elliptic curves , like in this paper LOCAL INVARIANTS OF ISOGENOUS ELLIPTIC CURVES, it seems there are indeed some invariance on the Kodaira types this isogeny condition, which implies some invariance on the minimal discriminant.

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