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Properties

Label 18.6.19961325392...3321.1
Degree $18$
Signature $[6, 6]$
Discriminant $7^{12}\cdot 229^{6}$
Root discriminant $22.39$
Ramified primes $7, 229$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times S_4$ (as 18T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -6, -11, -21, 9, 151, -5, -116, 41, -50, -6, 53, 4, -7, -4, -5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^16 - 4*x^15 - 7*x^14 + 4*x^13 + 53*x^12 - 6*x^11 - 50*x^10 + 41*x^9 - 116*x^8 - 5*x^7 + 151*x^6 + 9*x^5 - 21*x^4 - 11*x^3 - 6*x^2 + 1)
 
gp: K = bnfinit(x^18 - 5*x^16 - 4*x^15 - 7*x^14 + 4*x^13 + 53*x^12 - 6*x^11 - 50*x^10 + 41*x^9 - 116*x^8 - 5*x^7 + 151*x^6 + 9*x^5 - 21*x^4 - 11*x^3 - 6*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{16} - 4 x^{15} - 7 x^{14} + 4 x^{13} + 53 x^{12} - 6 x^{11} - 50 x^{10} + 41 x^{9} - 116 x^{8} - 5 x^{7} + 151 x^{6} + 9 x^{5} - 21 x^{4} - 11 x^{3} - 6 x^{2} + 1 \)

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:  \(1996132539284958146953321=7^{12}\cdot 229^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.39$
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:  $7, 229$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3}$, $\frac{1}{3} a^{14} + \frac{1}{3} a$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{2}$, $\frac{1}{397113} a^{16} + \frac{4742}{397113} a^{15} - \frac{58565}{397113} a^{14} + \frac{62153}{397113} a^{13} - \frac{63014}{397113} a^{12} - \frac{35533}{397113} a^{11} - \frac{21938}{132371} a^{10} + \frac{3129}{132371} a^{9} - \frac{99206}{397113} a^{8} + \frac{113518}{397113} a^{7} + \frac{51357}{132371} a^{6} + \frac{123271}{397113} a^{5} - \frac{1743}{132371} a^{4} + \frac{32226}{132371} a^{3} + \frac{26139}{132371} a^{2} - \frac{54275}{132371} a + \frac{14092}{397113}$, $\frac{1}{151300053} a^{17} - \frac{106}{151300053} a^{16} + \frac{1043741}{151300053} a^{15} - \frac{1880825}{50433351} a^{14} - \frac{6458170}{151300053} a^{13} + \frac{870668}{151300053} a^{12} + \frac{1671199}{50433351} a^{11} + \frac{64640}{50433351} a^{10} + \frac{9988438}{151300053} a^{9} - \frac{41139389}{151300053} a^{8} + \frac{7485251}{50433351} a^{7} + \frac{41853481}{151300053} a^{6} + \frac{7239840}{16811117} a^{5} + \frac{12585791}{50433351} a^{4} - \frac{2297007}{16811117} a^{3} + \frac{36110725}{151300053} a^{2} - \frac{74860567}{151300053} a + \frac{43535479}{151300053}$

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:  \( 143286.989817 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_4$ (as 18T33):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 15 conjugacy class representatives for $C_3\times S_4$
Character table for $C_3\times S_4$

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.3.229.1, 6.2.52441.1, 9.9.1412845546861.1

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

Sibling fields

Degree 12 sibling: data not computed
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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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
$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}$
229Data not computed