Generating Eisenstein polynomials

Contents

• Creation of templates
• Creation of polynomials

Creation of templates

TemplateForEisensteinPolynomials (F :: FldPadTmpl, n :: RngIntElt)

-> Tmpl_RngUPolElt_FldPadTmpl

Template for monic Eisenstein polynomials over F of degree n.

Parameters

• J: When given, restrict the discriminant valuation to n+J-1.

TemplateForEisensteinPolynomials (P :: PadRamifPgon)

-> Tmpl_RngUPolElt_FldPadTmpl

A template for Eisenstein polynomials with this ramification polygon.

Parameters

• Limit := "Best": When “Best”, make the template as small as possible; when “Disc” use the discriminant bound; otherwise it must be an integer.
• Residues := true: When true, incorporate residue information when it is available

Creation of polynomials

'#' (T :: Tmpl_FldPadTmplElt)

'#' (T :: Tmpl_RngUPolElt_FldPadTmpl)

-> RngIntElt

The number of elements in the template.

ToSequence (T :: Tmpl_FldPadTmplElt)

ToSequence (T :: Tmpl_RngUPolElt_FldPadTmpl)

-> []

The elements of T as a sequence.

Random (T :: Tmpl_FldPadTmplElt)

-> FldPadTmplElt

Random (T :: Tmpl_RngUPolElt_FldPadTmpl)

-> RngUPolElt_FldPadTmpl

A random element of T.

IndexSet (T :: Tmpl_FldPadTmplElt)

IndexSet (T :: Tmpl_RngUPolElt_FldPadTmpl)

-> Any

A set of indices for elements of T. This is quick to generate and can be iterated over.

GetIndex (T :: Tmpl_FldPadTmplElt, idx)

-> FldPadTmplElt

GetIndex (T :: Tmpl_RngUPolElt_FldPadTmpl, idx)

-> RngUPolElt_FldPadTmpl

Gets the element of T at index idx (which should be an element of the index set).