Let $C$ be a projective curve defined by $5y^2=x^4-17$ over $\Bbb{Q}$ and $C'$ be its normalization(N.B. $C$ is singular at infinity).
I want to determine in which prime $p$, $C'$ has bad reduction. $C’$ should be elliptic curve, but I don't have explicit equation of $C'$, so cannot calculate its discriminant. But to determine bad primes, I need to calculate discriminant some how.
What is standard approach of this kind of problem ? Thank you for your help.
P.S.
The curve $C'$ is binational to the elliptic curve $y^2=x^3-\frac{68}{25} x$.So $C'$ is genus $1$ curve. But binational map does not keep discriminant, so this elliptic curve isn't useful for calculating bad primes.