# Plane Quartics

- [Invariants](quartic_invariants.md)

  - [`DixmierOhnoInvariants(C): Crv → SeqEnum, SeqEnum`](quartic_invariants.md#function-dixmierohnoinvariants-crv)

  - [`DixmierOhnoInvariants(f): RngMPolElt → SeqEnum, SeqEnum`](quartic_invariants.md#function-dixmierohnoinvariants-rngmpolelt)

  - [`Example: Dixmierohno Ex`](quartic_invariants.md#example-ex-11474d)

  - [`DiscriminantOfTernaryQuartic(f): RngMPolElt → Any`](quartic_invariants.md#function-discriminantofternaryquartic-rngmpolelt)

  - [`Example: Discriminant Ex`](quartic_invariants.md#example-ex-a1e8bd)

  - [`DiscriminantFromDixmierOhnoInvariants(DO): SeqEnum → Any`](quartic_invariants.md#function-discriminantfromdixmierohnoinvariants-seqenum)

  - [`DixmierOhnoInvariantsEqual(DO1, DO2): SeqEnum, SeqEnum → BoolElt`](quartic_invariants.md#function-dixmierohnoinvariantsequal-seqenum-seqenum)

  - [`DixmierOhnoAlgebraicRelations(DOinv): SeqEnum → SeqEnum`](quartic_invariants.md#function-dixmierohnoalgebraicrelations-seqenum)

  - [`CovariantHessian(Phi): RngMPolElt → RngMPolElt`](quartic_invariants.md#function-covarianthessian-rngmpolelt)

  - [`ContravariantSigmaAndPsi(Phi): RngMPolElt → RngMPolElt, RngMPolElt`](quartic_invariants.md#function-contravariantsigmaandpsi-rngmpolelt)

  - [`QuarticCovariantsAndContravariants(Phi): RngMPolElt → SeqEnum`](quartic_invariants.md#function-quarticcovariantsandcontravariants-rngmpolelt)

- [Creation from Invariants](quartic_from_invariants.md)

  - [`PlaneQuarticFromDixmierOhnoInvariants(DO): SeqEnum → Crv, SeqEnum`](quartic_from_invariants.md#function-planequarticfromdixmierohnoinvariants-seqenum)

  - [`TernaryQuarticFromDixmierOhnoInvariants(DO): SeqEnum → RngMPolElt, SeqEnum`](quartic_from_invariants.md#function-ternaryquarticfromdixmierohnoinvariants-seqenum)

  - [`Example: Reconstruction Ex`](quartic_from_invariants.md#example-ex-dbee6a)

- [Isomorphisms](quartic_isomorphisms.md)

  - [`IsIsomorphicPlaneQuartics(X1, X2): CrvPln, CrvPln → BoolElt, SeqEnum`](quartic_isomorphisms.md#function-isisomorphicplanequartics-crvpln-crvpln)

  - [`IsIsomorphicTernaryQuartics(f1, f2): RngMPolElt, RngMPolElt → BoolElt, SeqEnum`](quartic_isomorphisms.md#function-isisomorphicternaryquartics-rngmpolelt-rngmpolelt)

  - [`IsomorphismsOfPlaneQuartics(X1, X2): CrvPln, CrvPln → SeqEnum`](quartic_isomorphisms.md#function-isomorphismsofplanequartics-crvpln-crvpln)

  - [`IsomorphismsOfTernaryQuartics(f1, f2): RngMPolElt, RngMPolElt → SeqEnum`](quartic_isomorphisms.md#function-isomorphismsofternaryquartics-rngmpolelt-rngmpolelt)

  - [`AutomorphismsOfPlaneQuartic(X): CrvPln → SeqEnum`](quartic_isomorphisms.md#function-automorphismsofplanequartic-crvpln)

  - [`AutomorphismsOfTernaryQuartic(f): RngMPolElt → SeqEnum`](quartic_isomorphisms.md#function-automorphismsofternaryquartic-rngmpolelt)

  - [`AutomorphismGroupOfPlaneQuartic(X): CrvPln → GrpPerm, Map`](quartic_isomorphisms.md#function-automorphismgroupofplanequartic-crvpln)

  - [`AutomorphismGroupOfTernaryQuartic(f): RngMPolElt → GrpPerm, Map`](quartic_isomorphisms.md#function-automorphismgroupofternaryquartic-rngmpolelt)

  - [`Example: Auts Ex`](quartic_isomorphisms.md#example-ex-153f49)

  - [`AutomorphismGroupOfPlaneQuartic(X, Autos): CrvPln, SeqEnum → GrpPerm, Map`](quartic_isomorphisms.md#function-automorphismgroupofplanequartic-crvpln-seqenum)

  - [`AutomorphismGroupOfTernaryQuartic(f, Autos): RngMPolElt, SeqEnum → GrpPerm, Map`](quartic_isomorphisms.md#function-automorphismgroupofternaryquartic-rngmpolelt-seqenum)

  - [`GeometricAutomorphismGroup(C): Crv → GrpPerm`](quartic_isomorphisms.md#function-geometricautomorphismgroup-crv)

- [Twists](quartic_twists.md)

  - [`TwistsOfPlaneQuartic(C, Autos): Crv, SeqEnum → SeqEnum[Crv], GrpPerm`](quartic_twists.md#function-twistsofplanequartic-crv-seqenum)

  - [`Twists(C): Crv → SeqEnum, GrpPerm`](quartic_twists.md#function-twists-crv)

  - [`Example: Twists Ex`](quartic_twists.md#example-ex-ae635c)