# Supersingular Divisors on Modular Curves

- [Introduction](introduction.md)

  - [Categories](introduction.md#categories)

  - [Verbose Output](introduction.md#verbose-output)

- [Creation Functions](creation.md)

  - [Ambient Spaces](creation.md#ambient-spaces)

    - [`SupersingularModule(p,N : parameters): RngIntElt, RngInt → ModSS`](creation.md#function-supersingularmodule-rngintelt-rngint)

    - [`SupersingularModule(p): RngIntElt → ModForm`](creation.md#function-supersingularmodule-rngintelt)

    - [`Example: Creation Spaces`](creation.md#example-ex-ba21a6)

  - [Elements](creation.md#elements)

    - [`M . i: ModSS, RngIntElt → ModSSElt`](creation.md#operation-operation-modss-rngintelt-modsselt)

    - [`M ! x: ModSS, . → ModSSElt`](creation.md#operation-op-modss)

    - [`Example: Creation Elements`](creation.md#example-ex-876e96)

  - [Subspaces](creation.md#subspaces)

    - [`CuspidalSubspace(M): ModSS → ModSS`](creation.md#function-cuspidalsubspace-modss)

    - [`EisensteinSubspace(M): ModSS → ModSS`](creation.md#function-eisensteinsubspace-modss)

    - [`OrthogonalComplement(M): ModSS → ModSS`](creation.md#function-orthogonalcomplement-modss)

    - [`Kernel(I, M): [Tup], ModSS → ModSS`](creation.md#function-kernel-tup-modss)

    - [`Decomposition(M, n): ModSS, RngIntElt → [ModSS]`](creation.md#function-decomposition-modss-rngintelt)

    - [`Example: Creation Subspaces`](creation.md#example-ex-b832c7)

- [Basis](basis.md)

  - [`Basis(M): ModSS → SeqEnum`](basis.md#function-basis-modss)

- [Properties](properties.md)

  - [`AuxiliaryLevel(M): ModSS → RngIntElt`](properties.md#function-auxiliarylevel-modss)

  - [`BaseRing(M): ModSS → Rng`](properties.md#function-basering-modss)

  - [`Degree(P): ModSSElt → RngElt`](properties.md#function-degree-modsselt)

  - [`Dimension(M): ModSS → RngIntElt`](properties.md#function-dimension-modss)

  - [`Eltseq(P): ModSSElt → SeqEnum`](properties.md#function-eltseq-modsselt)

  - [`Level(M): ModSS → RngIntElt`](properties.md#function-level-modss)

  - [`ModularEquation(M): ModSS → RngMPolElt`](properties.md#function-modularequation-modss)

  - [`Prime(M): ModSS → RngIntElt`](properties.md#function-prime-modss)

  - [`Example: Properties`](properties.md#example-ex-b24edc)

- [Associated Spaces](associated-spaces.md)

  - [`BrandtModule(M): ModSS → ModBrdt`](associated-spaces.md#function-brandtmodule-modss)

  - [`ModularSymbols(M : parameters): ModSS → ModSym`](associated-spaces.md#function-modularsymbols-modss)

  - [`ModularSymbols(M, sign : parameters): ModSS, RngIntElt → ModSym`](associated-spaces.md#function-modularsymbols-modss-rngintelt)

  - [`RSpace(M): ModSS → ModTupRng, Map`](associated-spaces.md#function-rspace-modss)

  - [`Example: Associated`](associated-spaces.md#example-ex-16e86e)

- [Predicates](predicates.md)

  - [`IsAmbientSpace(M): ModSS → BoolElt`](predicates.md#function-isambientspace-modss)

  - [`M1 eq M2: ModSS, ModSS → BoolElt`](predicates.md#operation-op-eq-modss-modss)

  - [`P eq Q: ModSSElt, ModSSElt → BoolElt`](predicates.md#operation-op-eq-modsselt-modsselt)

  - [`M1 subset M2: ModSS, ModSS → BoolElt`](predicates.md#operation-op-subset-modss-modss)

  - [`UsesBrandt(M): ModSS → BoolElt`](predicates.md#function-usesbrandt-modss)

  - [`UsesMestre(M): ModSS → BoolElt`](predicates.md#function-usesmestre-modss)

  - [`Example: Predicates`](predicates.md#example-ex-999e2e)

- [Arithmetic](arithmetic.md)

  - [`P + Q: ModSSElt, ModSSElt → ModSSElt`](arithmetic.md#operation-op-plus-modsselt-modsselt)

  - [`P - Q: ModSSElt, ModSSElt → ModSSElt`](arithmetic.md#operation-op-minus-modsselt-modsselt)

  - [`a * P: RngElt, ModSSElt → ModSSElt`](arithmetic.md#operation-op-times-rngelt-modsselt)

  - [`M1 + M2: ModSS, ModSS → ModSS`](arithmetic.md#operation-op-plus-modss-modss)

  - [`M1 meet M2: ModSS, ModSS → ModSS`](arithmetic.md#operation-op-meet-modss-modss)

  - [`Example: Arithmetic`](arithmetic.md#example-ex-a36e45)

- [Operators](operators.md)

  - [`HeckeOperator(M, n): ModSS, RngIntElt → AlgMatElt`](operators.md#function-heckeoperator-modss-rngintelt)

  - [`AtkinLehnerOperator(M, q): ModSS, RngIntElt → AlgMatElt`](operators.md#function-atkinlehneroperator-modss-rngintelt)

  - [`Example: Operators`](operators.md#example-ex-d33253)

- [The Monodromy Pairing](monodromy-pairing.md)

  - [`MonodromyPairing(P, Q): ModSSElt, ModSSElt → RngIntElt`](monodromy-pairing.md#function-monodromypairing-modsselt-modsselt)

  - [`MonodromyWeights(M): ModSS → SeqEnum`](monodromy-pairing.md#function-monodromyweights-modss)

  - [`Example: Monodromy`](monodromy-pairing.md#example-ex-894027)