That's very good, thank you a lot !

@WillSawin Because I know some methods to produce elliptic curve with bad reduction at a large set (using quadratic twist). I just want to make things more general, because I don't know some methods to systematically produce those general varieties (For example, we don't have good analogues of Néron–Ogg–Shafarevich criterion).

@WillSawin The main motivation is the previous question $mathoverflow.net/questions/324138/…, I am interested in how to construct abelian varieties with given ramification.

@DavidLampert Thank you. The motive of a smooth projective curve over a field belongs to the subcategory generated by motives of abelian varieties and Artin motives.

@AriyanJavanpeykar Thank you! If $K=\mathbb Q$, how to produce smooth projective varieties (which are not rational) with bad reduction only at $p$?

@DanielLoughran Thank you! I see, so my main interests lie in other interesting examples like abelian varieties, or more general varieties (can't be obtained from abelian varieties and Artin motives)...

@FrançoisBrunault I see...For something weaker, can we find an elliptic curve over $\mathbb Q$ with prescribed places of bad reduction?

Thank you, are there some good sufficient conditions on $N$ to imply it's a conductor?