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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.
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
< 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.
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