# PARI Groups#

See pari:polgalois for the PARI documentation of these objects.

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

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

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

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'

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

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

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

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