Registry backbone: the remaining parent families

diophantine_classifier/data/families/binary-qf-representation.yaml

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+# Family: Representation by a binary quadratic form
+# 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: binary-qf-representation
+name: Representation by a binary quadratic form
+priority: 1
+status: algorithmic
+class: quadratic
+form: a*x^2 + b*x*y + c*y^2 = n
+parents:
+- binary-quadratic
+matcher: true
+methods:
+- reduction theory
+- Cornacchia
+- class groups
+software:
+  sage: BinaryQF([a,b,c]).solve_integer(n)
+  pari: qfbsolve, qfbcornacchia
+code:
+  sage: BinaryQF([{a}, {b}, {c}]).solve_integer({n})
+  pari: qfbsolve(Qfb({a}, {b}, {c}), {n}, 1)
+examples:
+- 3*x^2 + 7*y^2 = 19
+references:
+- key: Cox1989
+  why: 'which n are represented: the class field theory of x^2 + ny^2'
+- key: Cornacchia1908
+  why: the classical algorithm for representations by definite forms

diophantine_classifier/data/families/elliptic-weierstrass.yaml

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+# Family: Elliptic curve in Weierstrass form
+# 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: elliptic-weierstrass
+name: Elliptic curve in Weierstrass form
+priority: 1
+status: algorithmic
+class: genus1
+form: y^2 + a1*x*y + a3*y = x^3 + a2*x^2 + a4*x + a6
+parents:
+- genus-one-curve
+matcher: true
+lmfdb: ec.q
+methods:
+- Mordell-Weil descent
+- elliptic logarithms for integral points
+software:
+  sage: EllipticCurve(...).gens(); .integral_points(); .S_integral_points()
+  pari: ellrank, ellratpoints
+  magma: MordellWeilShaInformation; IntegralPoints
+code:
+  sage: |
+    E = EllipticCurve({ainvs})
+    E.gens(); E.integral_points(both_signs=True)
+  magma: |
+    E := EllipticCurve({magma_ainvs});
+    IntegralPoints(E);
+examples:
+- y^2 + y = x^3 - x^2 - 10*x - 20
+references:
+- key: Cremona1997
+  why: descent algorithms (mwrank) for the Mordell-Weil group
+- key: StroekerTzanakis1994
+  why: the elliptic-logarithm method for integral points
+- key: GebelPethoZimmer1994
+  why: independent introduction of the elliptic-logarithm method

diophantine_classifier/data/families/hyperelliptic.yaml

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+# Family: Hyperelliptic 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: hyperelliptic
+name: Hyperelliptic curve
+priority: 1
+status: effective
+class: curve
+form: y^2 = f(x), deg f >= 5
+parents:
+- superelliptic
+matcher: true
+methods:
+- Baker (integral points)
+- Chabauty-Coleman + MW sieve (rational points)
+software:
+  magma: IntegralPoints (genus 2); Chabauty
+  pari: hyperellratpoints
+code:
+  magma: |
+    C := HyperellipticCurve(Polynomial({magma_coeffs}));
+    J := Jacobian(C); RankBound(J);
+examples:
+- y^2 = x^7 + 3
+references:
+- key: Baker1969
+  why: effective bounds for integral points
+- key: McCallumPoonen2012
+  why: the Chabauty-Coleman method for rational points

diophantine_classifier/data/families/legendre.yaml

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+# Family: Legendre equation (diagonal ternary quadratic)
+# 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: legendre
+name: Legendre equation (diagonal ternary quadratic)
+priority: 1
+status: algorithmic
+class: quadratic
+form: a*x^2 + b*y^2 + c*z^2 = 0
+parents:
+- quadratic-form-zero
+- diagonal-form
+matcher: true
+methods:
+- Legendre criterion
+- Holzer bounds
+- conic parametrization
+software:
+  sage: Conic([a,b,c]).has_rational_point(point=True)
+  pari: qfsolve
+  magma: HasRationalPoint
+code:
+  sage: Conic(QQ, [{a}, {b}, {c}]).rational_point()
+examples:
+- x^2 + y^2 = 2*z^2
+references:
+- key: CremonaRusin2003
+  why: efficient algorithms for finding a rational point on a conic
+- key: Holzer1950
+  why: existence of solutions of small height (Holzer's bound)

diophantine_classifier/data/families/pell-like.yaml

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+# Family: Generalized Pell equation
+# 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: pell-like
+name: Generalized Pell equation
+priority: 1
+status: algorithmic
+class: quadratic
+form: x^2 - D*y^2 = N
+parents:
+- binary-qf-representation
+matcher: true
+methods:
+- continued fractions
+- automorph orbits
+software:
+  pari: qfbsolve(Qfb(1,0,-D), N, 1)
+  sage: solve_diophantine
+code:
+  pari: qfbsolve(Qfb(1, 0, -{D}), {N}, 1)
+examples:
+- x^2 - 61*y^2 = 5
+references:
+- key: Matthews2000
+  why: exposition of the LMM continued-fraction algorithm for x^2 - Dy^2 = N

diophantine_classifier/data/families/thue.yaml

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+# Family: Thue equation
+# 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: thue
+name: Thue equation
+priority: 1
+status: algorithmic
+class: curve
+form: F(x, y) = m, F irreducible of degree >= 3
+parents:
+- binary-form
+matcher: true
+methods:
+- Baker bounds + LLL (Tzanakis-de Weger
+- Bilu-Hanrot)
+software:
+  pari: thue(thueinit(f), m)
+  sage: gp interface to thue
+  magma: Thue
+code:
+  pari: thue(thueinit({fx}, 1), {m})
+  magma: Solutions(Thue(Polynomial({magma_coeffs})), {m});
+examples:
+- x^3 + 2*y^3 = 11
+references:
+- key: Baker1968
+  why: effective bounds via linear forms in logarithms, making the finiteness algorithmic
+- key: Thue1909
+  why: the original finiteness theorem
+- key: TzanakisDeWeger1989
+  why: the first complete practical solution method
+- key: BiluHanrot1996
+  why: the high-degree algorithm implemented in PARI