A sequence containing the places where the given model of E has bad reduction, for an elliptic curve E defined over a function field.
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 < f, e > of places (specified by field elements f) and multiplicities e.
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 < Pl, vp(d), fp, cp, K, split > and Emin where Pl is the place, vp(d) is the valuation of the local minimal discriminant, fp is the valuation of the conductor, cp 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 Emin is a model of E (integral and) minimal at Pl.
Returns a sequence of tuples as described above for all places of bad reduction of the elliptic curve E.
A sequence of tuples < K, n >, 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.
The number of components of a fibre with the Kodaira symbol K.
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.
A model of the elliptic curve E (defined over a rational function field) which minimises the quantity Max([Degree(ai)/i]), where a1, a2, a3, a4, a6 are the Weierstrass coefficients.
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).
The trace of Frobenius ap for the reduction of E at the place p, specified as an element of the base field.