Family: genus-two — Genus 2 curve

diophantine_classifier/data/families/genus-two.yaml

@@ -0,0 +1,27 @@
+# Family: Genus 2 curve
+# 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: genus-two
+name: Genus 2 curve
+priority: 1
+status: effective
+class: curve
+form: y^2 = f(x), deg f in {5, 6}
+parents:
+- hyperelliptic
+matcher: true
+lmfdb: g2c.q
+methods:
+- Igusa invariants for identification
+- Chabauty machinery
+software:
+  magma: full pipeline (Jacobian, RankBound, Chabauty)
+  sage: HyperellipticCurve; igusa_clebsch_invariants
+examples:
+- y^2 = x^5 - x + 1
+references:
+- key: Igusa1960
+  why: the invariants identifying a genus-2 curve up to isomorphism (the classifier's
+    genus-2 handle)
+- key: CasselsFlynn1996
+  why: explicit arithmetic of genus-2 Jacobians

docs/FAMILIES.md

@@ -189,6 +189,15 @@ PARI: `hyperellratpoints`.
 **References.** Baker 1969; Faltings 1983; Chabauty 1941, Coleman 1985;
 McCallum–Poonen survey 2012; Balakrishnan–Dogra–Müller–Tuitman–Vonk 2019.
 
+### `genus-two` — Genus 2 curves — P1, as above with rich tooling
+**Form.** y² = f(x), deg f ∈ {5, 6}.
+**Status.** The best-supported genus ≥ 2 case: Igusa invariants identify the curve
+up to ℚ̄-isomorphism (the classifier's identification handle in genus 2, per the
+Library design), BSD-style databases exist (LMFDB), Chabauty machinery mature.
+**Software.** Magma: full pipeline (`Jacobian`, `RankBound`, `Chabauty`);
+Sage: `HyperellipticCurve`, `igusa_clebsch_invariants`; LMFDB genus 2 database.
+**References.** Igusa 1960; Cassels–Flynn; Stoll's surveys.
+
 ### `superelliptic` — Superelliptic curves — P1, effective (integral)
 **Form.** yᵐ = f(x), m ≥ 2, deg f ≥ 2 (genus ≥ 1 cases).
 **Status.** Integral points finite and effective (Baker); reduction to Thue

tests/test_classify.py

@@ -27,6 +27,7 @@
     # higher-genus curves and binary forms
     ("x^3 + 2*y^3 = 11", "", "thue"),
     ("x^4 - 2*y^4 = 1", "", "thue"),
+    ("y^2 = x^5 - x + 1", "", "genus-two"),
     ("y^2 = x^7 + 3", "", "hyperelliptic"),
     ("y^3 = x^4 + 2", "", "superelliptic"),
     ("x^3*y + y^3*z + z^3*x = 0", "", "general-curve"),     # Klein quartic