MAGMA Computational Algebra System

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, given as a divisor of the function field.

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

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

The 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 < Pl, v_p(d), f_p, c_p, K, split > 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 > ]

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 < KS, n >, corresponding to the places of bad reduction of the elliptic curve E. Here KS is the Kodaira symbol, and n is the degree of the corresponding place.

NumberOfComponents(KS) : SymKod -> RngIntElt

The number of components of a fiber with the given Kodaira symbol.

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 the 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 a₁, a₂, a₃, a₄, a₆ 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).