Family: plane-cubic — Plane cubic curve (ternary cubic)

diophantine_classifier/data/families/plane-cubic.yaml

@@ -0,0 +1,26 @@
+# Family: Plane cubic curve (ternary cubic)
+# Schema: see data/families/README.md; prose documentation in docs/FAMILIES.md.
+# References are keys into data/references.bib, each with a 'why' annotation.
+slug: plane-cubic
+name: Plane cubic curve (ternary cubic)
+priority: 2
+status: algorithmic
+class: genus1
+form: C(x, y, z) = 0 homogeneous cubic
+parents:
+- genus-one-curve
+matcher: true
+lmfdb: ec.q
+methods:
+- point search
+- Nagell reduction to Weierstrass form
+- descent on torsors
+software:
+  sage: EllipticCurve_from_cubic; Curve.rational_points(bound)
+  magma: MinimalModel; ThreeDescent
+notes: 'May fail the Hasse principle (Selmer: 3x^3 + 4y^3 + 5z^3 = 0).'
+references:
+- key: Nagell1928
+  why: the classical reduction of a cubic with a rational point to Weierstrass form
+- key: Selmer1951
+  why: '3x^3 + 4y^3 + 5z^3 = 0: failure of the Hasse principle for plane cubics'

docs/FAMILIES.md

@@ -146,6 +146,16 @@ by the elliptic-logarithm method once generators are known.
 **References.** Mordell 1922; Siegel 1929; Baker 1968; Gebel–Pethő–Zimmer 1994;
 Stroeker–Tzanakis 1994; Cremona, *Algorithms for Modular Elliptic Curves*.
 
+### `plane-cubic` — Ternary cubic / plane cubic curve — P2, algorithmic*
+**Form.** C(x, y, z) = 0 homogeneous cubic (smooth).
+**Status.** Genus 1 torsor; may fail the Hasse principle (Selmer's 3x³ + 4y³ + 5z³ = 0);
+with a known rational point, Nagell's algorithm gives a birational map to Weierstrass
+form. Finding the first point is the hard step (descent, Brauer–Manin, heuristics) —
+mirrored in the classifier design.
+**Software.** Sage: `EllipticCurve_from_cubic`, `Curve.rational_points(bound)`;
+Magma: `MinimalModel`, `FourDescent`/`ThreeDescent` for point search.
+**References.** Nagell 1928; Selmer 1951; Poonen, *Rational Points on Varieties*.
+
 ### `cubic-surface` — Cubic surfaces / del Pezzo — P3, research
 **Form.** F(x, y, z, w) = 0 cubic (e.g. diagonal ax³+by³+cz³+dw³ = 0).
 **Status.** Rational points conjecturally dense once one exists (unirationality);