Constructing a dual pair of algebras from a dihedral field

Constructing dual pairs from dihedral number fields

dual_pairs.dual_pair_from_dihedral_field.dual_pair_from_dihedral_field(L, F)

Return a dual pair encoding a dihedral Galois representation.

INPUT:

  • L – either a Galois extension of \(\mathbf{Q}\) with dihedral Galois group, or a polynomial over \(\mathbf{Q}\) whose splitting field is such an extension

  • F – a finite field such that the Galois group of L over \(\mathbf{Q}\) can be embedded into \(\mathrm{GL}_2(F)\)

EXAMPLES:

sage: from dual_pairs.dual_pair_from_dihedral_field import dual_pair_from_dihedral_field
sage: R.<x> = QQ[]

An example of level 23 over \(\mathbf{F}_2\):

sage: f = x^3 - x - 1
sage: dual_pair_from_dihedral_field(f.splitting_field('a'), GF(2))
Dual pair of algebras over Rational Field
A = Finite flat algebra of degree 4 over Rational Field, product of:
Number Field in a0 with defining polynomial x
Number Field in a1 with defining polynomial x^3 - x^2 + 1
B = Finite flat algebra of degree 4 over Rational Field, product of:
Number Field in a0 with defining polynomial x
Number Field in a1 with defining polynomial x^3 - x^2 + 1

An example of level 13 over \(\mathbf{F}_3\):

sage: f = x^4 + x^2 - 3
sage: L = f.splitting_field('w')
sage: dual_pair_from_dihedral_field(L, GF(3))
Dual pair of algebras over Rational Field
A = Finite flat algebra of degree 9 over Rational Field, product of:
Number Field in a0 with defining polynomial x
Number Field in a1 with defining polynomial x^4 - x^3 - x^2 - x + 1
Number Field in a2 with defining polynomial x^4 - x^3 - x^2 + x + 1
B = Finite flat algebra of degree 9 over Rational Field, product of:
Number Field in a0 with defining polynomial x
Number Field in a1 with defining polynomial x^4 - x^3 - x^2 - x + 1
Number Field in a2 with defining polynomial x^4 - x^3 - x^2 + x + 1

An example of level 16 over \(\mathbf{F}_3\) (cf. Serre, Divisibilité de certaines fonctions arithmétiques, exemple (4.4)):

sage: dual_pair_from_dihedral_field(x^4 - 12, GF(3))
Dual pair of algebras over Rational Field
A = Finite flat algebra of degree 9 over Rational Field, product of:
Number Field in a0 with defining polynomial x
Number Field in a1 with defining polynomial x^4 - 2*x^3 - 2*x + 1
Number Field in a2 with defining polynomial x^4 - 3*x^2 + 3
B = Finite flat algebra of degree 9 over Rational Field, product of:
Number Field in a0 with defining polynomial x
Number Field in a1 with defining polynomial x^4 - 2*x^3 - 2*x + 1
Number Field in a2 with defining polynomial x^4 - 3*x^2 + 3

An example of level 23 over \(\mathbf{F}_5\):

sage: dual_pair_from_dihedral_field(x^3 - x - 1, GF(5))
Dual pair of algebras over Rational Field
A = Finite flat algebra of degree 25 over Rational Field, product of:
Number Field in a0 with defining polynomial x
Number Field in a1 with defining polynomial x^3 - x^2 + 1
Number Field in a2 with defining polynomial x^3 - x^2 + 1
Number Field in a3 with defining polynomial x^3 - x^2 + 1
Number Field in a4 with defining polynomial x^6 - 3*x^5 + 5*x^4 - 5*x^3 + 5*x^2 - 3*x + 1
Number Field in a5 with defining polynomial x^6 - 3*x^5 + 5*x^4 - 5*x^3 + 5*x^2 - 3*x + 1
Number Field in a6 with defining polynomial x^3 - x^2 + 1
B = Finite flat algebra of degree 25 over Rational Field, product of:
Number Field in a0 with defining polynomial x
Number Field in a1 with defining polynomial x^12 - x^11 + x^10 - x^8 - 2*x^7 + 2*x^6 - 3*x^5 + x^4 + x^3 + x^2 + 1
Number Field in a2 with defining polynomial x^12 - 2*x^11 + 3*x^10 - 5*x^9 - 6*x^8 + 6*x^7 - 51*x^6 + 87*x^5 - 104*x^4 + 172*x^3 - 181*x^2 + 94*x - 19