From 6a783c805f1f87774be3b9489862729494bb4c19 Mon Sep 17 00:00:00 2001
From: David Roe <roed.math@gmail.com>
Date: Sun, 9 Aug 2026 16:24:30 -0400
Subject: [PATCH] =?UTF-8?q?Family:=20genus-two=20=E2=80=94=20Genus=202=20c?=
 =?UTF-8?q?urve?=
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One family: **Genus 2 curve** (`genus-two`) — P1, effective.

- Standard form: `y^2 = f(x), deg f in {5, 6}`
- Parent: `hyperelliptic`
- Examples: `y^2 = x^5 - x + 1`

Contents: the registry entry, its `docs/FAMILIES.md` section and its rows in the classification corpus.

References cited (the claim each one supports):

- `Igusa1960` — the invariants identifying a genus-2 curve up to isomorphism (the classifier's genus-2 handle)
- `CasselsFlynn1996` — explicit arithmetic of genus-2 Jacobians

Independent of every other family PR: it needs only the registry backbone it is based on.

Part of the series that splits #1 into reviewable pieces:

1. `01-bibliography` — packaging, docs, annotated bibliography
2. `02-parsing` — equation strings to a term model
3. `03-registry` — the YAML family registry (3 seed families)
4. `04-matchers` — shape recognizers
5. `05-classify` — the classification pipeline
6. `06-solvers` — solver framework, two seed solvers, and the CLI
7. `07..09-backbone` — the 23 parent families of the DAG, by depth
8. one PR per remaining family (38 of them, mutually independent)
9. `99-polish` — restore the full doctests and tighten the invariants
---
 .../data/families/genus-two.yaml              | 27 +++++++++++++++++++
 docs/FAMILIES.md                              |  9 +++++++
 tests/test_classify.py                        |  1 +
 3 files changed, 37 insertions(+)
 create mode 100644 diophantine_classifier/data/families/genus-two.yaml

diff --git a/diophantine_classifier/data/families/genus-two.yaml b/diophantine_classifier/data/families/genus-two.yaml
new file mode 100644
index 0000000..54ef333
--- /dev/null
+++ b/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
diff --git a/docs/FAMILIES.md b/docs/FAMILIES.md
index 86434cf..c3bc567 100644
--- a/docs/FAMILIES.md
+++ b/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
diff --git a/tests/test_classify.py b/tests/test_classify.py
index e3d6386..1e62d03 100644
--- a/tests/test_classify.py
+++ b/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