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Properties

Label 18.6.19042319699...0621.1
Degree $18$
Signature $[6, 6]$
Discriminant $3^{9}\cdot 7^{12}\cdot 107^{6}\cdot 167^{3}$
Root discriminant $70.61$
Ramified primes $3, 7, 107, 167$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T176

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![118777, -210290, 13362, 75884, -7212, 18796, -36963, 30688, 3728, -7434, 956, -274, -43, 14, -12, 31, -4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 - 4*x^16 + 31*x^15 - 12*x^14 + 14*x^13 - 43*x^12 - 274*x^11 + 956*x^10 - 7434*x^9 + 3728*x^8 + 30688*x^7 - 36963*x^6 + 18796*x^5 - 7212*x^4 + 75884*x^3 + 13362*x^2 - 210290*x + 118777)
 
gp: K = bnfinit(x^18 - 4*x^17 - 4*x^16 + 31*x^15 - 12*x^14 + 14*x^13 - 43*x^12 - 274*x^11 + 956*x^10 - 7434*x^9 + 3728*x^8 + 30688*x^7 - 36963*x^6 + 18796*x^5 - 7212*x^4 + 75884*x^3 + 13362*x^2 - 210290*x + 118777, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} - 4 x^{16} + 31 x^{15} - 12 x^{14} + 14 x^{13} - 43 x^{12} - 274 x^{11} + 956 x^{10} - 7434 x^{9} + 3728 x^{8} + 30688 x^{7} - 36963 x^{6} + 18796 x^{5} - 7212 x^{4} + 75884 x^{3} + 13362 x^{2} - 210290 x + 118777 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1904231969930704953529379769840621=3^{9}\cdot 7^{12}\cdot 107^{6}\cdot 167^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.61$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 7, 107, 167$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{15624590178938399} a^{16} + \frac{6786896033846793}{15624590178938399} a^{15} + \frac{4941770758781193}{15624590178938399} a^{14} - \frac{3959479321545963}{15624590178938399} a^{13} - \frac{187019214635090}{15624590178938399} a^{12} + \frac{6987974068469567}{15624590178938399} a^{11} - \frac{7001360791515203}{15624590178938399} a^{10} + \frac{7623778316786124}{15624590178938399} a^{9} + \frac{5173720455988879}{15624590178938399} a^{8} + \frac{6736088496166602}{15624590178938399} a^{7} + \frac{6996319848372186}{15624590178938399} a^{6} - \frac{6489045093808236}{15624590178938399} a^{5} + \frac{1533675332976392}{15624590178938399} a^{4} + \frac{6538858792932013}{15624590178938399} a^{3} - \frac{1964938923570226}{15624590178938399} a^{2} - \frac{929995518035755}{15624590178938399} a - \frac{5033709085379463}{15624590178938399}$, $\frac{1}{9228125978304653767647813962539} a^{17} + \frac{62673439232972}{9228125978304653767647813962539} a^{16} + \frac{2496257221712119529949946859482}{9228125978304653767647813962539} a^{15} + \frac{3376608229882214742961791359962}{9228125978304653767647813962539} a^{14} + \frac{107195166544811620996958665425}{9228125978304653767647813962539} a^{13} - \frac{2149672901226926373212557571515}{9228125978304653767647813962539} a^{12} - \frac{4559124383896294025256187203890}{9228125978304653767647813962539} a^{11} + \frac{2629420375946215213033427307772}{9228125978304653767647813962539} a^{10} + \frac{4345515842047064149427881115947}{9228125978304653767647813962539} a^{9} + \frac{848728938022662947543660962583}{9228125978304653767647813962539} a^{8} - \frac{1331557631111406873222035149519}{9228125978304653767647813962539} a^{7} + \frac{479609346206026006169962140489}{9228125978304653767647813962539} a^{6} - \frac{4484600633796610247094476005727}{9228125978304653767647813962539} a^{5} - \frac{42854712645036270849243658389}{318211240631194957505097033191} a^{4} + \frac{863885687680091492849806588737}{9228125978304653767647813962539} a^{3} - \frac{2466068199263152842126457776210}{9228125978304653767647813962539} a^{2} - \frac{4364543495977154165598230882189}{9228125978304653767647813962539} a + \frac{4235524735468636502159693570361}{9228125978304653767647813962539}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3933666991.41 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T176:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 576
The 40 conjugacy class representatives for t18n176
Character table for t18n176 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.3.321.1, 9.9.3891377265489.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 18 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.12.9.2$x^{12} - 9 x^{4} + 27$$4$$3$$9$$D_4 \times C_3$$[\ ]_{4}^{6}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$107$107.6.0.1$x^{6} - x + 6$$1$$6$$0$$C_6$$[\ ]^{6}$
107.6.3.1$x^{6} - 214 x^{4} + 11449 x^{2} - 99228483$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
107.6.3.1$x^{6} - 214 x^{4} + 11449 x^{2} - 99228483$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$167$167.3.0.1$x^{3} - x + 11$$1$$3$$0$$C_3$$[\ ]^{3}$
167.3.0.1$x^{3} - x + 11$$1$$3$$0$$C_3$$[\ ]^{3}$
167.6.0.1$x^{6} - x + 23$$1$$6$$0$$C_6$$[\ ]^{6}$
167.6.3.2$x^{6} - 27889 x^{2} + 51232093$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$