AlgebraicFunctionField(F, UP, UPUP, modulus)ΒΆ

curve.spad line 781 [edit on github]

Function field defined by f(x, y) = 0.

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from Magma

*: (%, Fraction Integer) -> %

from RightModule Fraction Integer

*: (%, Fraction UP) -> %

from RightModule Fraction UP

*: (%, Integer) -> % if Fraction UP has LinearlyExplicitOver Integer

from RightModule Integer

*: (Fraction Integer, %) -> %

from LeftModule Fraction Integer

*: (Fraction UP, %) -> %

from LeftModule Fraction UP

*: (Integer, %) -> %

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

/: (%, %) -> %

from Field

=: (%, %) -> Boolean

from BasicType

^: (%, Integer) -> %

from DivisionRing

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

absolutelyIrreducible?: () -> Boolean

from FunctionFieldCategory(F, UP, UPUP)

algSplitSimple: (%, UP -> UP) -> Record(num: %, den: UP, derivden: UP, gd: UP)

from FunctionFieldCategory(F, UP, UPUP)

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %

from NonAssociativeSemiRng

associates?: (%, %) -> Boolean

from EntireRing

associator: (%, %, %) -> %

from NonAssociativeRng

basis: () -> Vector %

from FramedModule Fraction UP

branchPoint?: F -> Boolean

from FunctionFieldCategory(F, UP, UPUP)

branchPoint?: UP -> Boolean

from FunctionFieldCategory(F, UP, UPUP)

branchPointAtInfinity?: () -> Boolean

from FunctionFieldCategory(F, UP, UPUP)

characteristic: () -> NonNegativeInteger

from NonAssociativeRing

characteristicPolynomial: % -> UPUP

from FiniteRankAlgebra(Fraction UP, UPUP)

charthRoot: % -> % if Fraction UP has FiniteFieldCategory

from FiniteFieldCategory

charthRoot: % -> Union(%, failed) if Fraction UP has CharacteristicNonZero

from CharacteristicNonZero

coerce: % -> %

from Algebra %

coerce: % -> OutputForm

from CoercibleTo OutputForm

coerce: Fraction Integer -> %

from Algebra Fraction Integer

coerce: Fraction UP -> %

from Algebra Fraction UP

coerce: Integer -> %

from NonAssociativeRing

commutator: (%, %) -> %

from NonAssociativeRng

complementaryBasis: Vector % -> Vector %

from FunctionFieldCategory(F, UP, UPUP)

conditionP: Matrix % -> Union(Vector %, failed) if Fraction UP has FiniteFieldCategory

from PolynomialFactorizationExplicit

convert: % -> InputForm if Fraction UP has Finite

from ConvertibleTo InputForm

convert: % -> UPUP

from ConvertibleTo UPUP

convert: % -> Vector Fraction UP

from FramedModule Fraction UP

convert: UPUP -> %

from MonogenicAlgebra(Fraction UP, UPUP)

convert: Vector Fraction UP -> %

from FramedModule Fraction UP

coordinates: % -> Vector Fraction UP

from FramedModule Fraction UP

coordinates: (%, Vector %) -> Vector Fraction UP

from FiniteRankAlgebra(Fraction UP, UPUP)

coordinates: (Vector %, Vector %) -> Matrix Fraction UP

from FiniteRankAlgebra(Fraction UP, UPUP)

coordinates: Vector % -> Matrix Fraction UP

from FramedModule Fraction UP

createPrimitiveElement: () -> % if Fraction UP has FiniteFieldCategory

from FiniteFieldCategory

D: % -> % if Fraction UP has DifferentialRing

from DifferentialRing

D: (%, Fraction UP -> Fraction UP) -> %

from DifferentialExtension Fraction UP

D: (%, Fraction UP -> Fraction UP, NonNegativeInteger) -> %

from DifferentialExtension Fraction UP

D: (%, List Symbol) -> % if Fraction UP has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

D: (%, List Symbol, List NonNegativeInteger) -> % if Fraction UP has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

D: (%, NonNegativeInteger) -> % if Fraction UP has DifferentialRing

from DifferentialRing

D: (%, Symbol) -> % if Fraction UP has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

D: (%, Symbol, NonNegativeInteger) -> % if Fraction UP has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

definingPolynomial: () -> UPUP

from MonogenicAlgebra(Fraction UP, UPUP)

derivationCoordinates: (Vector %, Fraction UP -> Fraction UP) -> Matrix Fraction UP

from MonogenicAlgebra(Fraction UP, UPUP)

differentiate: % -> % if Fraction UP has DifferentialRing

from DifferentialRing

differentiate: (%, Fraction UP -> Fraction UP) -> %

from DifferentialExtension Fraction UP

differentiate: (%, Fraction UP -> Fraction UP, NonNegativeInteger) -> %

from DifferentialExtension Fraction UP

differentiate: (%, List Symbol) -> % if Fraction UP has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

differentiate: (%, List Symbol, List NonNegativeInteger) -> % if Fraction UP has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

differentiate: (%, NonNegativeInteger) -> % if Fraction UP has DifferentialRing

from DifferentialRing

differentiate: (%, Symbol) -> % if Fraction UP has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

differentiate: (%, Symbol, NonNegativeInteger) -> % if Fraction UP has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

differentiate: (%, UP -> UP) -> %

from FunctionFieldCategory(F, UP, UPUP)

discreteLog: % -> NonNegativeInteger if Fraction UP has FiniteFieldCategory

from FiniteFieldCategory

discreteLog: (%, %) -> Union(NonNegativeInteger, failed) if Fraction UP has FiniteFieldCategory

from FieldOfPrimeCharacteristic

discriminant: () -> Fraction UP

from FramedAlgebra(Fraction UP, UPUP)

discriminant: Vector % -> Fraction UP

from FiniteRankAlgebra(Fraction UP, UPUP)

divide: (%, %) -> Record(quotient: %, remainder: %)

from EuclideanDomain

elliptic: () -> Union(UP, failed)

from FunctionFieldCategory(F, UP, UPUP)

elt: (%, F, F) -> F

from FunctionFieldCategory(F, UP, UPUP)

enumerate: () -> List % if Fraction UP has Finite

from Finite

euclideanSize: % -> NonNegativeInteger

from EuclideanDomain

expressIdealMember: (List %, %) -> Union(List %, failed)

from PrincipalIdealDomain

exquo: (%, %) -> Union(%, failed)

from EntireRing

extendedEuclidean: (%, %) -> Record(coef1: %, coef2: %, generator: %)

from EuclideanDomain

extendedEuclidean: (%, %, %) -> Union(Record(coef1: %, coef2: %), failed)

from EuclideanDomain

factor: % -> Factored %

from UniqueFactorizationDomain

factorPolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if Fraction UP has FiniteFieldCategory

from PolynomialFactorizationExplicit

factorsOfCyclicGroupSize: () -> List Record(factor: Integer, exponent: NonNegativeInteger) if Fraction UP has FiniteFieldCategory

from FiniteFieldCategory

factorSquareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if Fraction UP has FiniteFieldCategory

from PolynomialFactorizationExplicit

gcd: (%, %) -> %

from GcdDomain

gcd: List % -> %

from GcdDomain

gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial %

from GcdDomain

generator: () -> %

from MonogenicAlgebra(Fraction UP, UPUP)

genus: () -> NonNegativeInteger

from FunctionFieldCategory(F, UP, UPUP)

hash: % -> SingleInteger if Fraction UP has Hashable

from Hashable

hashUpdate!: (HashState, %) -> HashState if Fraction UP has Hashable

from Hashable

hyperelliptic: () -> Union(UP, failed)

from FunctionFieldCategory(F, UP, UPUP)

index: PositiveInteger -> % if Fraction UP has Finite

from Finite

init: % if Fraction UP has FiniteFieldCategory

from StepThrough

integral?: % -> Boolean

from FunctionFieldCategory(F, UP, UPUP)

integral?: (%, F) -> Boolean

from FunctionFieldCategory(F, UP, UPUP)

integral?: (%, UP) -> Boolean

from FunctionFieldCategory(F, UP, UPUP)

integralAtInfinity?: % -> Boolean

from FunctionFieldCategory(F, UP, UPUP)

integralBasis: () -> Vector %

from FunctionFieldCategory(F, UP, UPUP)

integralBasisAtInfinity: () -> Vector %

from FunctionFieldCategory(F, UP, UPUP)

integralCoordinates: % -> Record(num: Vector UP, den: UP)

from FunctionFieldCategory(F, UP, UPUP)

integralDerivationMatrix: (UP -> UP) -> Record(num: Matrix UP, den: UP)

from FunctionFieldCategory(F, UP, UPUP)

integralMatrix: () -> Matrix Fraction UP

from FunctionFieldCategory(F, UP, UPUP)

integralMatrixAtInfinity: () -> Matrix Fraction UP

from FunctionFieldCategory(F, UP, UPUP)

integralRepresents: (Vector UP, UP) -> %

from FunctionFieldCategory(F, UP, UPUP)

inv: % -> %

from DivisionRing

inverseIntegralMatrix: () -> Matrix Fraction UP

from FunctionFieldCategory(F, UP, UPUP)

inverseIntegralMatrixAtInfinity: () -> Matrix Fraction UP

from FunctionFieldCategory(F, UP, UPUP)

knownInfBasis: NonNegativeInteger -> Void

knownInfBasis(n) is undocumented.

latex: % -> String

from SetCategory

lcm: (%, %) -> %

from GcdDomain

lcm: List % -> %

from GcdDomain

lcmCoef: (%, %) -> Record(llcm_res: %, coeff1: %, coeff2: %)

from LeftOreRing

leftPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

leftRecip: % -> Union(%, failed)

from MagmaWithUnit

lift: % -> UPUP

from MonogenicAlgebra(Fraction UP, UPUP)

lookup: % -> PositiveInteger if Fraction UP has Finite

from Finite

minimalPolynomial: % -> UPUP

from FiniteRankAlgebra(Fraction UP, UPUP)

multiEuclidean: (List %, %) -> Union(List %, failed)

from EuclideanDomain

nextItem: % -> Union(%, failed) if Fraction UP has FiniteFieldCategory

from StepThrough

nonSingularModel: Symbol -> List Polynomial F

from FunctionFieldCategory(F, UP, UPUP)

norm: % -> Fraction UP

from FiniteRankAlgebra(Fraction UP, UPUP)

normalizeAtInfinity: Vector % -> Vector %

from FunctionFieldCategory(F, UP, UPUP)

numberOfComponents: () -> NonNegativeInteger

from FunctionFieldCategory(F, UP, UPUP)

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

order: % -> OnePointCompletion PositiveInteger if Fraction UP has FiniteFieldCategory

from FieldOfPrimeCharacteristic

order: % -> PositiveInteger if Fraction UP has FiniteFieldCategory

from FiniteFieldCategory

plenaryPower: (%, PositiveInteger) -> %

from NonAssociativeAlgebra %

prime?: % -> Boolean

from UniqueFactorizationDomain

primeFrobenius: % -> % if Fraction UP has FiniteFieldCategory

from FieldOfPrimeCharacteristic

primeFrobenius: (%, NonNegativeInteger) -> % if Fraction UP has FiniteFieldCategory

from FieldOfPrimeCharacteristic

primitive?: % -> Boolean if Fraction UP has FiniteFieldCategory

from FiniteFieldCategory

primitiveElement: () -> % if Fraction UP has FiniteFieldCategory

from FiniteFieldCategory

primitivePart: % -> %

from FunctionFieldCategory(F, UP, UPUP)

principalIdeal: List % -> Record(coef: List %, generator: %)

from PrincipalIdealDomain

quo: (%, %) -> %

from EuclideanDomain

ramified?: F -> Boolean

from FunctionFieldCategory(F, UP, UPUP)

ramified?: UP -> Boolean

from FunctionFieldCategory(F, UP, UPUP)

ramifiedAtInfinity?: () -> Boolean

from FunctionFieldCategory(F, UP, UPUP)

random: () -> % if Fraction UP has Finite

from Finite

rank: () -> PositiveInteger

from FiniteRankAlgebra(Fraction UP, UPUP)

rationalPoint?: (F, F) -> Boolean

from FunctionFieldCategory(F, UP, UPUP)

rationalPoints: () -> List List F if F has Finite

from FunctionFieldCategory(F, UP, UPUP)

recip: % -> Union(%, failed)

from MagmaWithUnit

reduce: Fraction UPUP -> Union(%, failed)

from MonogenicAlgebra(Fraction UP, UPUP)

reduce: UPUP -> %

from MonogenicAlgebra(Fraction UP, UPUP)

reduceBasisAtInfinity: Vector % -> Vector %

from FunctionFieldCategory(F, UP, UPUP)

reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix Fraction UP, vec: Vector Fraction UP)

from LinearlyExplicitOver Fraction UP

reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix Integer, vec: Vector Integer) if Fraction UP has LinearlyExplicitOver Integer

from LinearlyExplicitOver Integer

reducedSystem: Matrix % -> Matrix Fraction UP

from LinearlyExplicitOver Fraction UP

reducedSystem: Matrix % -> Matrix Integer if Fraction UP has LinearlyExplicitOver Integer

from LinearlyExplicitOver Integer

regularRepresentation: % -> Matrix Fraction UP

from FramedAlgebra(Fraction UP, UPUP)

regularRepresentation: (%, Vector %) -> Matrix Fraction UP

from FiniteRankAlgebra(Fraction UP, UPUP)

rem: (%, %) -> %

from EuclideanDomain

representationType: () -> Union(prime, polynomial, normal, cyclic) if Fraction UP has FiniteFieldCategory

from FiniteFieldCategory

represents: (Vector Fraction UP, Vector %) -> %

from FiniteRankAlgebra(Fraction UP, UPUP)

represents: (Vector UP, UP) -> %

from FunctionFieldCategory(F, UP, UPUP)

represents: Vector Fraction UP -> %

from FramedModule Fraction UP

retract: % -> Fraction Integer if Fraction UP has RetractableTo Fraction Integer

from RetractableTo Fraction Integer

retract: % -> Fraction UP

from RetractableTo Fraction UP

retract: % -> Integer if Fraction UP has RetractableTo Integer

from RetractableTo Integer

retractIfCan: % -> Union(Fraction Integer, failed) if Fraction UP has RetractableTo Fraction Integer

from RetractableTo Fraction Integer

retractIfCan: % -> Union(Fraction UP, failed)

from RetractableTo Fraction UP

retractIfCan: % -> Union(Integer, failed) if Fraction UP has RetractableTo Integer

from RetractableTo Integer

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

sample: %

from AbelianMonoid

singular?: F -> Boolean

from FunctionFieldCategory(F, UP, UPUP)

singular?: UP -> Boolean

from FunctionFieldCategory(F, UP, UPUP)

singularAtInfinity?: () -> Boolean

from FunctionFieldCategory(F, UP, UPUP)

size: () -> NonNegativeInteger if Fraction UP has Finite

from Finite

sizeLess?: (%, %) -> Boolean

from EuclideanDomain

smaller?: (%, %) -> Boolean if Fraction UP has Finite

from Comparable

solveLinearPolynomialEquation: (List SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> Union(List SparseUnivariatePolynomial %, failed) if Fraction UP has FiniteFieldCategory

from PolynomialFactorizationExplicit

special_order: (%, List UP) -> Integer

from FunctionFieldCategory(F, UP, UPUP)

squareFree: % -> Factored %

from UniqueFactorizationDomain

squareFreePart: % -> %

from UniqueFactorizationDomain

squareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if Fraction UP has FiniteFieldCategory

from PolynomialFactorizationExplicit

subtractIfCan: (%, %) -> Union(%, failed)

from CancellationAbelianMonoid

tableForDiscreteLogarithm: Integer -> Table(PositiveInteger, NonNegativeInteger) if Fraction UP has FiniteFieldCategory

from FiniteFieldCategory

trace: % -> Fraction UP

from FiniteRankAlgebra(Fraction UP, UPUP)

traceMatrix: () -> Matrix Fraction UP

from FramedAlgebra(Fraction UP, UPUP)

traceMatrix: Vector % -> Matrix Fraction UP

from FiniteRankAlgebra(Fraction UP, UPUP)

unit?: % -> Boolean

from EntireRing

unitCanonical: % -> %

from EntireRing

unitNormal: % -> Record(unit: %, canonical: %, associate: %)

from EntireRing

yCoordinates: % -> Record(num: Vector UP, den: UP)

from FunctionFieldCategory(F, UP, UPUP)

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra %

Algebra Fraction Integer

Algebra Fraction UP

BasicType

BiModule(%, %)

BiModule(Fraction Integer, Fraction Integer)

BiModule(Fraction UP, Fraction UP)

CancellationAbelianMonoid

canonicalsClosed

canonicalUnitNormal

CharacteristicNonZero if Fraction UP has CharacteristicNonZero

CharacteristicZero if Fraction UP has CharacteristicZero

CoercibleFrom Fraction Integer if Fraction UP has RetractableTo Fraction Integer

CoercibleFrom Fraction UP

CoercibleFrom Integer if Fraction UP has RetractableTo Integer

CoercibleTo OutputForm

CommutativeRing

CommutativeStar

Comparable if Fraction UP has Finite

ConvertibleTo InputForm if Fraction UP has Finite

ConvertibleTo UPUP

DifferentialExtension Fraction UP

DifferentialRing if Fraction UP has DifferentialRing

DivisionRing

EntireRing

EuclideanDomain

Field

FieldOfPrimeCharacteristic if Fraction UP has FiniteFieldCategory

Finite if Fraction UP has Finite

FiniteFieldCategory if Fraction UP has FiniteFieldCategory

FiniteRankAlgebra(Fraction UP, UPUP)

FramedAlgebra(Fraction UP, UPUP)

FramedModule Fraction UP

FullyLinearlyExplicitOver Fraction UP

FullyRetractableTo Fraction UP

FunctionFieldCategory(F, UP, UPUP)

GcdDomain

Hashable if Fraction UP has Hashable

IntegralDomain

LeftModule %

LeftModule Fraction Integer

LeftModule Fraction UP

LeftOreRing

LinearlyExplicitOver Fraction UP

LinearlyExplicitOver Integer if Fraction UP has LinearlyExplicitOver Integer

Magma

MagmaWithUnit

Module %

Module Fraction Integer

Module Fraction UP

MonogenicAlgebra(Fraction UP, UPUP)

Monoid

NonAssociativeAlgebra %

NonAssociativeAlgebra Fraction Integer

NonAssociativeAlgebra Fraction UP

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

PartialDifferentialRing Symbol if Fraction UP has PartialDifferentialRing Symbol

PolynomialFactorizationExplicit if Fraction UP has FiniteFieldCategory

PrincipalIdealDomain

RetractableTo Fraction Integer if Fraction UP has RetractableTo Fraction Integer

RetractableTo Fraction UP

RetractableTo Integer if Fraction UP has RetractableTo Integer

RightModule %

RightModule Fraction Integer

RightModule Fraction UP

RightModule Integer if Fraction UP has LinearlyExplicitOver Integer

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

StepThrough if Fraction UP has FiniteFieldCategory

TwoSidedRecip

UniqueFactorizationDomain

unitsKnown