Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $\Delta$ be the global minimal discriminant. By A. Wiles et al. $E$ is modular so that there is a corresponding modular form $f$. I am wondering if the invariant $\Delta$ appears in the context of $f$. Anything would be helpful for me. Literature, result, reference etc.
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4$\begingroup$ $\Delta$ can't really be said to appear in the context of $f$ - in fact two elliptic curves associated to the same modular form can have different discriminants (e.g. lmfdb.org/EllipticCurve/Q/11/a/2 and lmfdb.org/EllipticCurve/Q/11/a/3). However one can always give a lower bound for $\Delta$ in terms of $f$ (e.g. the conductor, but there are sharper lower bounds as well). $\endgroup$– Will SawinCommented Apr 22 at 15:30
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1$\begingroup$ To add to Will's comment, the modular form $f$ only picks out the isogeny class of $E$, not the isomorphism class. Then $f$ determines the conductor $N$ of $E$, which places local restrictions on $\Delta$ (e.g., you know the set of primes dividing $\Delta$). Look up discriminant-conductor relations. $\endgroup$– KimballCommented Apr 22 at 20:18
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