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Properties

Label 18.0.70867390469...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 5^{9}\cdot 7^{12}$
Root discriminant $16.36$
Ramified primes $2, 5, 7$
Class number $2$
Class group $[2]$
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -16, 133, -672, 2254, -5202, 8521, -10268, 9523, -7178, 4639, -2684, 1429, -696, 296, -102, 28, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 28*x^16 - 102*x^15 + 296*x^14 - 696*x^13 + 1429*x^12 - 2684*x^11 + 4639*x^10 - 7178*x^9 + 9523*x^8 - 10268*x^7 + 8521*x^6 - 5202*x^5 + 2254*x^4 - 672*x^3 + 133*x^2 - 16*x + 1)
 
gp: K = bnfinit(x^18 - 6*x^17 + 28*x^16 - 102*x^15 + 296*x^14 - 696*x^13 + 1429*x^12 - 2684*x^11 + 4639*x^10 - 7178*x^9 + 9523*x^8 - 10268*x^7 + 8521*x^6 - 5202*x^5 + 2254*x^4 - 672*x^3 + 133*x^2 - 16*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 28 x^{16} - 102 x^{15} + 296 x^{14} - 696 x^{13} + 1429 x^{12} - 2684 x^{11} + 4639 x^{10} - 7178 x^{9} + 9523 x^{8} - 10268 x^{7} + 8521 x^{6} - 5202 x^{5} + 2254 x^{4} - 672 x^{3} + 133 x^{2} - 16 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-7086739046912000000000=-\,2^{18}\cdot 5^{9}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.36$
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:  $2, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a$, $\frac{1}{6} a^{16} + \frac{1}{6} a^{15} + \frac{1}{6} a^{14} + \frac{1}{6} a^{12} - \frac{1}{3} a^{11} + \frac{1}{6} a^{10} - \frac{1}{3} a^{9} + \frac{1}{6} a^{8} - \frac{1}{3} a^{7} + \frac{1}{6} a^{6} - \frac{1}{3} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{663355841298966} a^{17} - \frac{20960532270032}{331677920649483} a^{16} - \frac{66852520941845}{331677920649483} a^{15} + \frac{4366415597510}{331677920649483} a^{14} - \frac{124240802820083}{663355841298966} a^{13} - \frac{18827701525133}{331677920649483} a^{12} + \frac{17217999776699}{663355841298966} a^{11} + \frac{11389760561359}{331677920649483} a^{10} - \frac{319778808102805}{663355841298966} a^{9} + \frac{112089790505170}{331677920649483} a^{8} + \frac{322976050415777}{663355841298966} a^{7} - \frac{32180936527754}{331677920649483} a^{6} - \frac{91410028779871}{663355841298966} a^{5} - \frac{106502554051585}{331677920649483} a^{4} - \frac{23862472482941}{110559306883161} a^{3} + \frac{45179512904146}{110559306883161} a^{2} + \frac{258077287973341}{663355841298966} a - \frac{119440817515009}{331677920649483}$

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

Class group and class number

$C_{2}$, which has order $2$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $8$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1403.6050065 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-5}) \), 3.1.980.1 x3, \(\Q(\zeta_{7})^+\), 6.0.19208000.2, 6.0.19208000.1, 6.0.392000.1 x2, 9.3.941192000.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.392000.1
Degree 9 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 R ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.2e2_5.2t1.1c1$1$ $ 2^{2} \cdot 5 $ $x^{2} + 5$ $C_2$ (as 2T1) $1$ $-1$
* 1.2e2_5_7.6t1.2c1$1$ $ 2^{2} \cdot 5 \cdot 7 $ $x^{6} - 2 x^{5} + 12 x^{4} - 14 x^{3} + 87 x^{2} - 64 x + 281$ $C_6$ (as 6T1) $0$ $-1$
* 1.2e2_5_7.6t1.2c2$1$ $ 2^{2} \cdot 5 \cdot 7 $ $x^{6} - 2 x^{5} + 12 x^{4} - 14 x^{3} + 87 x^{2} - 64 x + 281$ $C_6$ (as 6T1) $0$ $-1$
* 1.7.3t1.1c1$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
* 1.7.3t1.1c2$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
*2 2.2e2_5_7e2.3t2.1c1$2$ $ 2^{2} \cdot 5 \cdot 7^{2}$ $x^{3} - x^{2} + 5 x - 13$ $S_3$ (as 3T2) $1$ $0$
*2 2.2e2_5_7.6t5.3c1$2$ $ 2^{2} \cdot 5 \cdot 7 $ $x^{18} - 6 x^{17} + 28 x^{16} - 102 x^{15} + 296 x^{14} - 696 x^{13} + 1429 x^{12} - 2684 x^{11} + 4639 x^{10} - 7178 x^{9} + 9523 x^{8} - 10268 x^{7} + 8521 x^{6} - 5202 x^{5} + 2254 x^{4} - 672 x^{3} + 133 x^{2} - 16 x + 1$ $S_3 \times C_3$ (as 18T3) $0$ $0$
*2 2.2e2_5_7.6t5.3c2$2$ $ 2^{2} \cdot 5 \cdot 7 $ $x^{18} - 6 x^{17} + 28 x^{16} - 102 x^{15} + 296 x^{14} - 696 x^{13} + 1429 x^{12} - 2684 x^{11} + 4639 x^{10} - 7178 x^{9} + 9523 x^{8} - 10268 x^{7} + 8521 x^{6} - 5202 x^{5} + 2254 x^{4} - 672 x^{3} + 133 x^{2} - 16 x + 1$ $S_3 \times C_3$ (as 18T3) $0$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.