It is my understanding that if I twist a hyperelliptic curve of genus 2 whose Jacobian has conductor $N$ by a prime $p$ with $p\nmid N$, that the conductor of the Jacobian of the twist is expected to be $Np^4$, but I am unable to find a reference for this. Is anyone aware of a proof of this "fact"?



Reference Request: Conductors of Twists of Hyperelliptic Curves
(This answer is Community Wiki and is extracted from comments by user eric, who has declined to post them as an answer. The CW is to invite others to contribute, especially to provide suitable references for these facts.) 


If the curve has good reduction, then the Jacobian has good reduction. The standard reference for this is SGA 7. Now you need that the $l$-adic representation is unramified; this is one direction of Neron-Ogg-Shaferevich. (One reference for that off the top of my head is the Serre-Tate Annals paper, although note that it's only the easy direction of NOS we use here.) Meanwhile, a quadratic twist is tamely ramified (this is for $p$ odd -- when $p=2$ the result you want is probably false in fact). If you need a reference for this, you can try the Artin-Tate notes on class field theory, but much like the previous reference, I'd regard this as overkill for what you are asking for. 


In brief: if you have an unramified representation of dimension $d$ and you twist it by a tamely ramified character, the resulting representation has conductor $p^d$. I don't really know what would be a great reference for this specific observation, but it's not a difficult calculation. 


