Analysis Tools

Analysis begins after a curve has earned more attention.

The right tool depends on what is missing: points, independence, subgroup quality, alternative search geometry, or an upper bound.

Decision guide

Question	Tool
Need another rational point?	Target, Quartics, Family geometry
Have points but do not know whether they add rank?	Independence
Need to improve/check subgroup index?	Saturation
Want canonical-height geometry / reduced basis guidance?	Lattices
Want an upper bound or exact rank?	Descent
Want alternate point-search models?	Quartics & Coverings

Start from the live curve state

Before running anything expensive, open the curve and record:

• rigorous lower bound;
• rigorous upper bound;
• exact rank if already known;
• number of exact points;
• complete rigorous witness basis status;
• prior timeout/error history.

Do not repeat an expensive method that already hit a deterministic engine limit on the identical model unless you have changed something meaningful.

Independence

Use Independence when new exact points are present but the rigorous lower bound has not risen.

The standard path tests candidates against the authoritative rigorous basis.

Hard-case strategies can precondition or saturate the basis, then retry exact certification.

See Independence.

Saturation

Saturation asks whether the current subgroup is missing divisible points.

A nontrivial index can replace the basis with a better/saturated basis without increasing rank.

That can improve later independence, lattice, or descent work.

See Saturation.

Lattices and heights

Height-pairing data helps answer:

• Is the current basis badly conditioned?
• Which point combinations are short/long?
• Are there numerically suspicious near-relations?
• Which combinations are useful anchors for further search?

These are powerful diagnostics but numerical geometry alone is not proof of independence.

See Lattices & Heights.

Quartics and coverings

Pointed quartics and exact coverings give alternative rational-point search spaces.

A successful quartic hit matters only after exact map-back to the elliptic curve.

Timeouts in one quartic model do not imply the original curve has no additional points.

See Quartics & Coverings.

Descent and rank bounds

Use Descent when you want a rigorous upper bound.

Normal automatic rank-bounds behavior is PARI-first, with bounded fallback/escalation behavior implemented by the release.

If the upper bound meets the rigorous lower bound, exact rank is closed.

See Descent & Rank Bounds.

Suggested workflow for a promising high-rank curve

A reasonable order is:

1. verify all newly discovered points exactly;
2. run Independence;
3. if the basis is awkward, try bounded Saturation;
4. build a height lattice for geometry;
5. use Target/Quartics if you still want more points;
6. run Descent when closing the interval is worth the cost.

This is a workflow suggestion, not a theorem. Some families have specialized geometry that should run earlier.

Avoiding wasted compute

Before increasing a timeout by 10×, ask:

• Did the previous run time out or actually complete?
• Was it searching the right model?
• Was the dominant difficulty height, denominator, covering size, or engine limit?
• Does another exact transform make the same problem smaller?
• Is the current rigorous basis complete?

Longer compute is useful only when it attacks the actual bottleneck.

Evidence rule

Every Analysis page is still subject to What Counts as Proof?.