An error occurred. See the output for details.
VERSION 2 29 10
Intrinsic 'CasselsTatePairing'

Signatures:

    (C::Crv, D::CrvHyp) -> RngIntElt
    [
        quadratic_point_on_D
    ]

        Pairing of a 4-covering (given as a QI) with a 2-covering (given as a
        hyperelliptic curve), returned as an element of Z/2.

    (C::CrvHyp, D::CrvHyp) -> RngIntElt

        The Cassels-Tate pairing of C and D, which should be hyperelliptic
        curves representing elements in the 2-Selmer group of some elliptic
        curve, returned as an element of Z/2.

    (cov1::MapSch, cov2::MapSch, phi::MapSch) -> RngIntElt

        Given a 2-isogeny or 3-isogeny phi : E -> E' between elliptic curves
        over a field K, and coverings cov1 : C -> E' and cov2 : D -> E' that
        correspond to elements of H^1(Q,E[phi]), this returns the Cassels-Tate
        pairing between cov1 and cov2 as an element of Z/2.


Intrinsic 'RankBounds'

Signatures:

    (E::CrvEll) -> RngIntElt, RngIntElt
    [
        Bound,
        Effort
    ]

        Upper and lower bounds on the Mordell-Weil rank of the elliptic curve E.

    (J::JacHyp) -> RngIntElt, RngIntElt
    [
        ReturnGenerators
    ]

        Lower and upper bounds on the Mordell-Weil rank of the given Jacobian
        (of some hyperelliptic curve over Q).

    (f::RngUPolElt, q::RngIntElt) -> RngIntElt, RngIntElt
    [
        ReturnGenerators
    ]

        Lower and upper bounds for the Mordell-Weil rank of the Jacobian of the
        cyclic cover defined by y^q = f.

    (E::SetPtEll) -> RngIntElt, RngIntElt
    [
        Bound,
        Effort
    ]

        Returns RankBounds(Curve(E)).


Intrinsic 'RankBound'

Signatures:

    (E::CrvEll) -> RngIntElt, BoolElt
    [
        Bound,
        Effort,
        Isogeny
    ]

        The Mordell-Weil rank of the elliptic curve E (or possibly a lower bound
        if the returned boolean is false).

    (J::JacHyp) -> RngIntElt

        An upper bound on the Mordell-Weil rank of the given Jacobian (of some
        hyperelliptic curve over a number field).

    (f::RngUPolElt, q::RngIntElt) -> RngIntElt

        An upper bound on the Mordell-Weil rank of the Jacobian of the curve
        defined by y^q = f over a number field).


Intrinsic 'TwoSelmerGroup'

Signatures:

    (E::CrvEll) -> GrpAb, Map
    [
        Bound,
        Fields,
        Hints,
        Raw
    ]

        The 2-Selmer group of an elliptic curve defined over a number field.

    (E::CrvEll[FldRat]) -> GrpAb, Map
    [
        Bound,
        Fields,
        Hints,
        Raw,
        RemoveTorsion,
        RemoveGens
    ]

        The 2-Selmer group of the elliptic curve E over Q. (Note: if old
        optional arguments are given, old code is called).

    (E::CrvEll[FldFunG]) -> GrpAb, Map
    [
        Fast
    ]

        The 2-Selmer group of an elliptic curve defined over a rational function
        field, as an abstract group, and map from this group to the standard
        etale algebra. The Fast option uses less precision in some of the local
        calculations (possibly too little!).

    (J::JacHyp) -> GrpAb, Map, ., .
    [
        Al,
        Fields,
        ReturnFakeSelmerData,
        ReturnRawData
    ]

        The 2-Selmer group of the Jacobian J (of some hyperelliptic curve over a
        number field).


Intrinsic 'RichelotIsogenousSurfaces'

Signatures:

    (C::CrvHyp) -> List, List
    [
        Kernels
    ]
    (J::JacHyp) -> List, List
    [
        Kernels
    ]

        Given a Jacobian of a genus 2 curve compute all (2,2)-isogenous abelian
        surfaces.


Intrinsic 'MordellWeilShaInformation'

Signatures:

    (E::CrvEll) -> SeqEnum, SeqEnum, SeqEnum
    [
        RankOnly,
        ShaInfo,
        Sha,
        Effort,
        HeightBound,
        Silent
    ]

        Print and return all obtainable information about the Mordell-Weil group
        and the Tate-Shafarevich group of the elliptic curve E (over Q or a
        number field). Returns: a sequence containing a lower and upper bound on
        the Mordell-Weil rank, a sequence of generators for the known subgroup
        of the Mordell-Weil group modulo torsion, and a sequence specifying all
        known lower and upper bounds on Sha[n] for integers n.



>> print RankOneTwoCoverDescent;
         ^
User error: Identifier 'RankOneTwoCoverDescent' has not been declared or
assigned
