Build the idea in three steps
- Five coefficients are a lot to carry. The library first condenses them into four helper quantities — b₂, b₄, b₆ and b₈ — each a short polynomial in the coefficients.
- The c-quantities combine the helpers further. The one our proof leans on is c₄ = b₂² − 24·b₄: a single summary number computed straight from the curve’s coefficients.
- The two summary numbers divide the work: Δ records where the curve breaks, and c₄ tells how badly. Semistability demands that every prime dividing Δ spare c₄ — the tame kind of breakdown.
A small example
Take y² = x³ − x, i.e. a₄ = −1 with the other coefficients 0. Then b₂ = 0 and b₄ = 2·(−1) = −2, so c₄ = 0² − 24·(−2) = 48 — a nonzero fingerprint.