In the computation of the conductor of an elliptic curve usually we use the Tate algorithm to determine the singular fibre of an elliptic curve, but what should we do when we have a variable in the discriminant for example this elliptic curve:
E: $y^{2}+xy=x^{3}+\frac{2c-1}{4} x^{2}+\frac{2^{\alpha}+2D^{p}}{64}x$ And discriminant $\Delta$ = $2^{2\alpha-6}MD^{2p}$ Where $M=m^{2}(m^{2}-2)$ and c is even When M is even from Tate algorithm $E$ has multiplicative reduction at 2 . And the conductor $N =2\prod l $, but when $M$ is odd, how we can determine the conductor of this elliptic curve ? Second question: asume that the Galois representation is irreducible for some prime $p>11$ and it is modular .what is the level $N$ of the cusp form of Galois representation of this elliptic curve? Can we ignore the $M$ from the level of the cusp form and take it just 2 ?