PARI Groups¶
See pari:polgalois for the PARI documentation of these objects.
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class
sage.groups.pari_group.
PariGroup
(x, degree)¶ 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
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cardinality
()¶ 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
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degree
()¶ 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
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label
()¶ 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'
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order
()¶ 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
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permutation_group
()¶ 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
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signature
()¶ 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
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transitive_number
()¶ 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
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