Local Computations

BadPlaces(E): CrvEll -> [ PlcFunElt ]

A sequence containing the places where the given model of \(E\) has bad reduction, for an elliptic curve \(E\) defined over a function field.

Conductor(E): CrvEll -> DivFunElt

The conductor of an elliptic curve \(E\) defined over a function field \(F\). In general this is returned as a divisor of \(F\). When \(F\) is a rational function field it is returned as a sequence of tuples \(\langle f, e \rangle\) of places (specified by field elements \(f\)) and multiplicities \(e\).

LocalInformation(E, Pl): CrvEll[FldFun], PlcFunElt -> Tup, CrvEll

LocalInformation(E, f): CrvEll[FldFunRat], FldFunRatUElt -> Tup, CrvEll

This function performs Tate’s algorithm for an elliptic curve \(E\) over a function field to determine the reduction type and a minimal model at the given place \(Pl\). When \(E\) is defined over a rational function field \(F(t)\) the place is simply given as a field element \(f\) (which must either be \(1/t\) or an irreducible polynomial in \(t\).)

The model is not required to be integral on input. The output is of the form \(\langle Pl, v_p(d), f_p, c_p, K, split \rangle\) and \(E_{min}\) where \(Pl\) is the place, \(v_p(d)\) is the valuation of the local minimal discriminant, \(f_p\) is the valuation of the conductor, \(c_p\) is the Tamagawa number, \(K\) is the Kodaira Symbol, \(split\) is a boolean that is false if reduction is of nonsplit multiplicative type and true otherwise, and \(E_{min}\) is a model of \(E\) (integral and) minimal at \(Pl\).

LocalInformation(E): CrvEll -> [ < Tup > ]

Returns a sequence of tuples as described above for all places of bad reduction of the elliptic curve \(E\).

KodairaSymbols(E): CrvEll -> [ <SymKod, RngIntElt> ]

A sequence of tuples \(\left< K, n \right>\), corresponding to the places of bad reduction of the elliptic curve \(E\). Here \(K\) is the Kodaira symbol and \(n\) is the degree of the corresponding place.

NumberOfComponents(K): SymKod -> RngIntElt

The number of components of a fibre with the Kodaira symbol \(K\).

MinimalModel(E): CrvEll[FldFunG] -> CrvEll, MapIsoSch

A model of the elliptic curve \(E\) (defined over a function field, which must have genus \(0\)) that is minimal at all finite places, together with a map from \(E\) to this minimal model.

MinimalDegreeModel(E): CrvEll[FldFunRat] -> CrvEll, Map, Map

A model of the elliptic curve \(E\) (defined over a rational function field) which minimises the quantity Max([Degree\((a_i)/i\)]), where \(a_1, a_2, a_3, a_4, a_6\) are the Weierstrass coefficients.

IsConstantCurve(E): CrvEll[FldFunRat] -> BoolElt, CrvEll

For an elliptic curve \(E\) defined over a rational function field \(F(t)\), the function returns true if and only if \(E\) is isomorphic over \(F(t)\) to an elliptic curve with coefficients in \(F\) (and also returns such a curve in that case).

TraceOfFrobenius(E, p): CrvEll[FldFunRat], RngElt -> BoolElt, CrvEll

The trace of Frobenius \(a_p\) for the reduction of \(E\) at the place \(p\), specified as an element of the base field.