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I'm asking about the conductor of an elliptic curve. I'm looking at the definition given as $$N_{E/\mathbb Q} = \prod_{p} p^{f_p},$$ and I understand that $f_p$ encodes information about the ramification of $p$ as follows: $$f_p = \begin{cases} 0 \text{, if $E$ has good reduction at $p$,}\\ 1 \text{, if $E$ has multiplicative reduction at $p$,}\\ 2 \text{, if $E$ has additive reduction at $p$, and $p\neq 2,3$,}\\ 2+\delta_p \text{, if $E$ has additive reduction at $p=2\ or\ 3$.} \end{cases} $$

Above definition from planet math.

I think one could say the conductor is a product which represents which primes the curve has bad reduction, and how bad it is.

What I don't understand is how we go from this construction to considering it so important that databases like Cremona's and the LMFDB are principally ordered by conductor.

I have noticed that computing with low conductor curves is faster. Is this true? Is there a reason why low conductor would imply computational convenience when doing elliptic curve arithmetic? Does low conductor imply that the $a$-invariants will be small, or even bounded in some way?

Elliptic curves of large rank and small conductor by Noam Elkies and Mark Watkins is a cool paper I've been skimming, but I wish I knew why finding curves of low conductor was so motivating.

Thanks.

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    [Math.SE post on Definition and Meaning of Conductor](https://mathoverflow.net/questions/2022/definition-and-meaning-of-the-conductor-of-an-elliptic-curve) has a lot of theory explanation, but I don't understand why the theory explained there gives the conductor such significance. – 2017-02-26
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    I think that the post you quoted gives you a lot of good reasons why the conductor is a very important invariant. Just to quote some of them: it is also the conductor of the Galois representation on the Tate module of the curve (and it therefore measure the ramification of that representation), it is the smallest level of a modular curve which admits a nontrivial map to the elliptic curve, it enters the functional equation of the $L$-function of the curve and it is divided by the places of bad reduction. To me, this looks like more than enough... Do you want to hear another reason? – 2017-02-27
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    Faltings theorem implies that there are only finitely many isomorphism classes of elliptic curves of a given conductor, so the conductor also gives you a natural way to order isomorphism classes of elliptic curves. Curves with small conductors are faster to compute because modular symbols algorithms become slow when the conductor - which corresponds to the level of the cuspidal space that is computed - becomes large. – 2017-02-27
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    Thank you, Ferra, your clarification helps. I think I understand better now, I wasn't paying much attention to the definition in terms of levels of modular curves before. – 2017-02-27

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