PARI Groups¶
See pari:polgalois for the PARI documentation of these objects.
- class sage.groups.pari_group.PariGroup(x, degree)[source]¶
- Bases: - object- EXAMPLES: - sage: PariGroup([6, -1, 2, "S3"], 3) PARI group [6, -1, 2, S3] of degree 3 sage: R.<x> = PolynomialRing(QQ) sage: f = x^4 - 17*x^3 - 2*x + 1 sage: G = f.galois_group(pari_group=True); G PARI group [24, -1, 5, "S4"] of degree 4 - >>> from sage.all import * >>> PariGroup([Integer(6), -Integer(1), Integer(2), "S3"], Integer(3)) PARI group [6, -1, 2, S3] of degree 3 >>> R = PolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1) >>> f = x**Integer(4) - Integer(17)*x**Integer(3) - Integer(2)*x + Integer(1) >>> G = f.galois_group(pari_group=True); G PARI group [24, -1, 5, "S4"] of degree 4 - cardinality()[source]¶
- Return the order of - self.- EXAMPLES: - sage: R.<x> = PolynomialRing(QQ) sage: f1 = x^4 - 17*x^3 - 2*x + 1 sage: G1 = f1.galois_group(pari_group=True) sage: G1.order() 24 - >>> from sage.all import * >>> R = PolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1) >>> f1 = x**Integer(4) - Integer(17)*x**Integer(3) - Integer(2)*x + Integer(1) >>> G1 = f1.galois_group(pari_group=True) >>> G1.order() 24 
 - degree()[source]¶
- Return the degree of this group. - EXAMPLES: - sage: R.<x> = PolynomialRing(QQ) sage: f1 = x^4 - 17*x^3 - 2*x + 1 sage: G1 = f1.galois_group(pari_group=True) sage: G1.degree() 4 - >>> from sage.all import * >>> R = PolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1) >>> f1 = x**Integer(4) - Integer(17)*x**Integer(3) - Integer(2)*x + Integer(1) >>> G1 = f1.galois_group(pari_group=True) >>> G1.degree() 4 
 - label()[source]¶
- Return the human readable description for this group generated by Pari. - EXAMPLES: - sage: R.<x> = QQ[] sage: f1 = x^4 - 17*x^3 - 2*x + 1 sage: G1 = f1.galois_group(pari_group=True) sage: G1.label() 'S4' - >>> from sage.all import * >>> R = QQ['x']; (x,) = R._first_ngens(1) >>> f1 = x**Integer(4) - Integer(17)*x**Integer(3) - Integer(2)*x + Integer(1) >>> G1 = f1.galois_group(pari_group=True) >>> G1.label() 'S4' 
 - order()[source]¶
- Return the order of - self.- EXAMPLES: - sage: R.<x> = PolynomialRing(QQ) sage: f1 = x^4 - 17*x^3 - 2*x + 1 sage: G1 = f1.galois_group(pari_group=True) sage: G1.order() 24 - >>> from sage.all import * >>> R = PolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1) >>> f1 = x**Integer(4) - Integer(17)*x**Integer(3) - Integer(2)*x + Integer(1) >>> G1 = f1.galois_group(pari_group=True) >>> G1.order() 24 
 - permutation_group()[source]¶
- Return the corresponding GAP transitive group. - EXAMPLES: - sage: R.<x> = QQ[] sage: f = x^8 - x^5 + x^4 - x^3 + 1 sage: G = f.galois_group(pari_group=True) sage: G.permutation_group() Transitive group number 44 of degree 8 - >>> from sage.all import * >>> R = QQ['x']; (x,) = R._first_ngens(1) >>> f = x**Integer(8) - x**Integer(5) + x**Integer(4) - x**Integer(3) + Integer(1) >>> G = f.galois_group(pari_group=True) >>> G.permutation_group() Transitive group number 44 of degree 8 
 - signature()[source]¶
- Return 1 if contained in the alternating group, -1 otherwise. - EXAMPLES: - sage: R.<x> = QQ[] sage: f1 = x^4 - 17*x^3 - 2*x + 1 sage: G1 = f1.galois_group(pari_group=True) sage: G1.signature() -1 - >>> from sage.all import * >>> R = QQ['x']; (x,) = R._first_ngens(1) >>> f1 = x**Integer(4) - Integer(17)*x**Integer(3) - Integer(2)*x + Integer(1) >>> G1 = f1.galois_group(pari_group=True) >>> G1.signature() -1 
 - transitive_number()[source]¶
- If the transitive label is nTk, return \(k\). - EXAMPLES: - sage: R.<x> = QQ[] sage: f1 = x^4 - 17*x^3 - 2*x + 1 sage: G1 = f1.galois_group(pari_group=True) sage: G1.transitive_number() 5 - >>> from sage.all import * >>> R = QQ['x']; (x,) = R._first_ngens(1) >>> f1 = x**Integer(4) - Integer(17)*x**Integer(3) - Integer(2)*x + Integer(1) >>> G1 = f1.galois_group(pari_group=True) >>> G1.transitive_number() 5