Minimal Weierstrass equation and different reduction types (good and bad) of an elliptic curve

Question

Why do we need the 'minimal' Weierstrass equation of an elliptic curve in order to study it's different reduction types (good and bad) ?

What happens if we don't start with a minimal Weierstrass equation ?

Answer 1

Consider the elliptic curve $E: y^2=x^3+15625$. The discriminant of this curve is $-105468750000=2^4\cdot 3^3\cdot 5^{12}$. Thus, if we reduce the given model modulo $5$, we obtain $y^2\equiv x^3 \bmod 5$ which is a singular curve over $\mathbb{F}_5$.

This "bad reduction" at $p=5$ is unnecessary (or removable, in the sense of removable discontinuities in Calculus) and it appeared because we didn't choose a better model for the curve. Since $15625=5^6$, it turns out that our curve $E$ is isomorphic over $\mathbb{Q}$ to the curve $E': y^2=x^3+1$ with discriminant $-2^4\cdot 3^3$ that has good reduction at $p=5$.

Choosing a minimal model from the beginning is desired so that all removable primes of bad reduction are already not present, and only essential primes of bad reduction remain.