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Database artin
Description Artin representations
Status production
Contact John Jones
Code artin_representations
Collections representations, field_data

Todo: Add index information

Collection representations

  • Dim (int): dimension
  • Indicator (int): Frobenius-Schur indicator, 1 for orthogonal, -1 for symplectic, and 0 for other
  • BadPrimes (list of int-as-string): list of bad primes, i.e., primes dividing the conductor. Stored as strings since they may get too big
  • HardPrimes (list of int-as-string): primes dividing the polynomial discriminant for the defining field. These include the Bad Primes.
  • Conductor (int-as-string): conductor
  • Conductor_key (string): encoding of the conductor for searching. The first four characters give the number of decimal digits minus 1 (padded with zeros), followed by the conductor itself.
  • Hide (int): 0 if we should show it when searching for Artin rep'ns, 1 if not. The representations are invariants of the Galois closure of the given field. More than one field can have the same Galois closure. We pick a best/minimal one and show that. We have data for others for linking to the number field database.
  • NFGal (string): comma-separated list of integers giving the coefficients of the polynomial for the corresponding number field after polredabs, starting with the constant coefficient.
  • CharacterField (int): the n for writing the character
  • Baselabel (string): our label in the form a.b.c.e.f.g where a=the dimension, b=conductor in factored form (using e for ^ and t form *), c=|Galois group|, d=size of the Galois orbit of characters, e= is the Galois group in the form ntm where n is the degree and m is the "t-number", g is a counter to distinguish representations with the same data.
  • GaloisConjugates (list): list of Galois conjugate character information.

Each entry in the GaloisConjugates list is a dictionary with the following entries

  • LocalFactors (list of list of int-as-strings): local factors for the L-function
  • Character (list of list of ints): character for this representation. Each sublist are coefficients for the character value written on a power basis for Z[zetan]
  • Sign (int): sign of the functional equation when we know it is 1 or -1, otherwise we give 0.
  • HardFactors: local factors for bad primes
  • GalOrbIndex (ints): an index assigned to the given character

Index information for representations collection (to be added)

  • ...
  • representations.rand (auxilliary collection used for random objection access)

Collection field_data

  • Polynomial (string): coefficients of a polynomial defining this field, the comma-separated list of coefficients as a string. This is the main identifier for this field from the representations collection, and also matches entries in the number field database.
  • TransitiveDegree (int): degree of the polynomial
  • Size (int-as-string): order of the Galois group
  • DBIndex (int): counter to distinguish those with the same degree and size, used for crossreferencing from artrep table
  • ArtinReps (list of pairs, [string, int]): the string is the baselabel of an entry from the Artin representation database, and the int is the GalOrbIndex for a particular character with that Baselabel
  • ConjClasses (list of dicts): for each conjugacy class of the group: its Order (int), Representative (list of ints giving a permutation), and Size (int)
  • Frobs (list of ints): if the i-th entry is j, then the Frobenius for the i-th prime lies in the j-th conjugacy class
  • G-Gens (list of list of ints): inner lists are permutations given as lists which generate the Galois group
  • QpRts-p (int): the prime p used for computing the roots p-adicly
  • QpRts-minpoly (list of ints): coefficients for a defining polynomial over Qp used for explicitly writing roots. The first coefficient is the constant term.
  • QpRts-prec (int): p-adic roots are computed up to (p^prec)
  • QpRts (list of list of string): each entry is a p-adic root, where entries in the list give the coefficients of powers of p in the p-adic approximation. They are themselves polynomials in a, where a is a root of the QpRts-minpoly
  • ComplexConjugation (int): index for ConjClasses to say where complex conjugation lies
  • FrobResolvents (list of dicts):
  • G-Name (string): name for the Galois group, but we usually substitute a latex'ed name from the Galois group database, but this is a fallback