Standardise the model, then extract the arithmetic
The first step with any curve over ℚ is to replace it by its minimal model, so that invariants are canonical and comparable. Tate's algorithm then determines the reduction type, Tamagawa number and conductor exponent at each bad prime. Torsion is settled by a short search constrained by Mazur's theorem. Rank requires descent, which produces an upper bound and a point search, and may leave a gap attributable to the Tate–Shafarevich group.
Learning objectives
- Compute a minimal Weierstrass model.
- Apply Tate's algorithm to obtain the reduction type and conductor.
- Determine the torsion subgroup using Mazur's constraint.
- Explain the structure of a descent computation.
- Use canonical heights to test independence of points.
Section 01Minimal models
Weierstrass equations related by the admissible substitutions form an isomorphism class; the minimal model is the one whose discriminant has the smallest absolute value. Over ℚ it exists and is unique up to the remaining trivial transformations.
The conductor, the Tamagawa numbers and the real period are all defined relative to the minimal model. Computing them from a non-minimal equation gives wrong values — and they will still look like plausible values, so the error will not announce itself. Minimising first is not optional.
The minimal model is found by computing the invariants c4, c6, Δ, then removing twelfth powers from Δ consistently with fourth and sixth powers in c4 and c6 — a prime-by-prime reduction requiring the factorisation of the discriminant.
Section 02Tate's algorithm
At each prime dividing the minimal discriminant, Tate's algorithm classifies the reduction by a sequence of tests on the coefficients, outputting the Kodaira symbol, the Tamagawa number and the conductor exponent.
| Kodaira type | Reduction | Tamagawa number cp | Conductor exponent |
|---|---|---|---|
| I0 | Good | 1 | 0 |
| In (n ≥ 1) | Multiplicative | n or gcd(n, 2), by splitting | 1 |
| II, II* | Additive | 1 | ≥ 2 |
| III, III* | Additive | 2 | ≥ 2 |
| IV, IV* | Additive | 1 or 3 | ≥ 2 |
| In* | Additive | 2 or 4 | ≥ 2 |
The conductor is the level of the modular form attached to the curve, so it is the key into modular form databases and is how curves are indexed in standard tables. The Tamagawa numbers enter the strong BSD formula directly.
Section 03Torsion
- Compute the minimal model and its discriminant.
- Apply Nagell–Lutz: a torsion point has integral coordinates with y = 0 or y2 dividing the discriminant. A finite list of candidates.
- Alternatively and usually faster, reduce modulo several primes of good reduction; the torsion injects into E(Fp), so gcd of the group orders bounds it.
- Test each candidate by computing its multiples until the identity or an excessively large coordinate appears.
- Confirm the resulting group is one of the fifteen allowed by Mazur's theorem.
Reducing modulo a handful of good primes and taking the gcd of the group orders usually pins down the torsion immediately and costs almost nothing. Nagell–Lutz is the certificate; reduction is the search.
Section 04Rank and descent
- Stage 012-descentMap E(ℚ)/2E(ℚ) into a Selmer group computed from local conditions, giving an upper bound on the rank.
- Stage 02Search for pointsLook for rational points of bounded height, giving a lower bound. Sieving and lattice methods extend the reachable height.
- Stage 03Compare boundsIf the upper and lower bounds agree, the rank is determined.
- Stage 04Handle the gapA discrepancy is attributed to the Tate–Shafarevich group; higher descents or Heegner point constructions may close it.
Independence of found points is tested by the canonical height pairing: the height regulator matrix is non-singular exactly when the points are independent, and its determinant enters the strong BSD formula.
Rank computation is not a guaranteed-terminating algorithm. A 2-descent may leave a gap that higher descents fail to close, and generators of very large height may escape any feasible search. Reported ranks should carry their method and any conditionality.
ReferenceFrequently asked questions
Why do curves need to be minimised before comparison?
Because the same curve has infinitely many Weierstrass equations with different discriminants. Without minimisation, two equations for the same curve appear to have different invariants and database lookups fail.
How large can the coordinates of a generator be?
There is no known effective bound in general, and curves are known whose smallest generator has coordinates with hundreds of digits. This is precisely why rank computation is hard and why the analytic prediction is valuable as a guide.
What is the canonical height for?
It is a quadratic form on the Mordell-Weil group that measures the arithmetic size of a point and is compatible with the group law. It is used to test independence of points, to bound searches, and it appears in the strong BSD formula through the regulator.
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