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Properties

Label 18.0.15296862267...0976.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{24}\cdot 7^{9}$
Root discriminant $32.38$
Ramified primes $2, 3, 7$
Class number $12$
Class group $[12]$
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![53433, -91314, 112833, -105666, 66060, -32016, 1661, 19452, -5631, -4754, 2181, 222, -447, 42, 108, -10, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 12*x^16 - 10*x^15 + 108*x^14 + 42*x^13 - 447*x^12 + 222*x^11 + 2181*x^10 - 4754*x^9 - 5631*x^8 + 19452*x^7 + 1661*x^6 - 32016*x^5 + 66060*x^4 - 105666*x^3 + 112833*x^2 - 91314*x + 53433)
 
gp: K = bnfinit(x^18 - 12*x^16 - 10*x^15 + 108*x^14 + 42*x^13 - 447*x^12 + 222*x^11 + 2181*x^10 - 4754*x^9 - 5631*x^8 + 19452*x^7 + 1661*x^6 - 32016*x^5 + 66060*x^4 - 105666*x^3 + 112833*x^2 - 91314*x + 53433, 1)
 

Normalized defining polynomial

\( x^{18} - 12 x^{16} - 10 x^{15} + 108 x^{14} + 42 x^{13} - 447 x^{12} + 222 x^{11} + 2181 x^{10} - 4754 x^{9} - 5631 x^{8} + 19452 x^{7} + 1661 x^{6} - 32016 x^{5} + 66060 x^{4} - 105666 x^{3} + 112833 x^{2} - 91314 x + 53433 \)

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:  \(-1529686226742116542605950976=-\,2^{27}\cdot 3^{24}\cdot 7^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.38$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{5}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{10} - \frac{1}{2} a^{8} - \frac{1}{6} a^{6} + \frac{1}{6} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{11} - \frac{1}{6} a^{9} - \frac{1}{6} a^{7} + \frac{1}{6} a^{5} + \frac{1}{6} a^{3} - \frac{1}{2} a$, $\frac{1}{18} a^{14} - \frac{1}{9} a^{11} - \frac{2}{9} a^{8} + \frac{1}{3} a^{7} + \frac{1}{9} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{2}$, $\frac{1}{1404} a^{15} - \frac{1}{156} a^{14} - \frac{7}{468} a^{13} - \frac{23}{351} a^{12} + \frac{77}{468} a^{11} - \frac{2}{39} a^{10} - \frac{229}{1404} a^{9} - \frac{7}{117} a^{8} - \frac{59}{468} a^{7} - \frac{76}{351} a^{6} + \frac{11}{156} a^{5} - \frac{1}{78} a^{4} + \frac{163}{468} a^{3} + \frac{17}{39} a^{2} + \frac{4}{13} a - \frac{5}{52}$, $\frac{1}{23868} a^{16} + \frac{1}{7956} a^{15} + \frac{217}{7956} a^{14} - \frac{320}{5967} a^{13} + \frac{215}{2652} a^{12} + \frac{134}{1989} a^{11} - \frac{3433}{23868} a^{10} - \frac{197}{1989} a^{9} - \frac{2371}{7956} a^{8} - \frac{2713}{5967} a^{7} - \frac{19}{612} a^{6} + \frac{47}{306} a^{5} - \frac{245}{612} a^{4} - \frac{28}{663} a^{3} + \frac{47}{663} a^{2} + \frac{239}{884} a + \frac{63}{221}$, $\frac{1}{26288187056359947970313336883443484} a^{17} + \frac{270401429070271416759489050083}{13144093528179973985156668441721742} a^{16} - \frac{3727816112018567538548055269603}{26288187056359947970313336883443484} a^{15} + \frac{21270040473509650630974494873959}{773181972245880822656274614218926} a^{14} - \frac{135531147364531563558369815758642}{6572046764089986992578334220860871} a^{13} + \frac{1202427368230443635593798100440483}{26288187056359947970313336883443484} a^{12} - \frac{73925050731310838471693434637659}{505542058776152845582948786220067} a^{11} + \frac{111532020856908904516762091962895}{1546363944491761645312549228437852} a^{10} - \frac{87234023816642598654922022405437}{6572046764089986992578334220860871} a^{9} - \frac{12926112080525688796994488528679389}{26288187056359947970313336883443484} a^{8} + \frac{906595962976342013841239868156664}{6572046764089986992578334220860871} a^{7} + \frac{7737542714440985029400691222718193}{26288187056359947970313336883443484} a^{6} + \frac{185679936155380350115268904119774}{730227418232220776953148246762319} a^{5} - \frac{292589124527298384597001549305991}{8762729018786649323437778961147828} a^{4} - \frac{1250709905095825280697733454900459}{8762729018786649323437778961147828} a^{3} + \frac{52881623210633908205482805099827}{324545519214320345312510331894364} a^{2} - \frac{140364859520143651941122577863151}{324545519214320345312510331894364} a + \frac{67369888723737588829003149644399}{973636557642961035937530995683092}$

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

Class group and class number

$C_{12}$, which has order $12$

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:  \( 793654.483575 \)
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

\(\Q(\sqrt{-14}) \), 3.1.4536.1 x3, \(\Q(\zeta_{9})^+\), 6.0.1152216576.1, 6.0.1152216576.2, 6.0.14224896.1 x2, 9.3.93329542656.3 x3

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

Sibling fields

Degree 6 sibling: 6.0.14224896.1
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 R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$2$2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
$3$3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
$7$7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.2e3_7.2t1.2c1$1$ $ 2^{3} \cdot 7 $ $x^{2} + 14$ $C_2$ (as 2T1) $1$ $-1$
* 1.2e3_3e2_7.6t1.12c1$1$ $ 2^{3} \cdot 3^{2} \cdot 7 $ $x^{6} + 36 x^{4} - 2 x^{3} + 597 x^{2} + 90 x + 4047$ $C_6$ (as 6T1) $0$ $-1$
* 1.2e3_3e2_7.6t1.12c2$1$ $ 2^{3} \cdot 3^{2} \cdot 7 $ $x^{6} + 36 x^{4} - 2 x^{3} + 597 x^{2} + 90 x + 4047$ $C_6$ (as 6T1) $0$ $-1$
* 1.3e2.3t1.1c1$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 1.3e2.3t1.1c2$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
*2 2.2e3_3e4_7.3t2.1c1$2$ $ 2^{3} \cdot 3^{4} \cdot 7 $ $x^{3} - 3 x - 26$ $S_3$ (as 3T2) $1$ $0$
*2 2.2e3_3e2_7.6t5.3c1$2$ $ 2^{3} \cdot 3^{2} \cdot 7 $ $x^{18} - 12 x^{16} - 10 x^{15} + 108 x^{14} + 42 x^{13} - 447 x^{12} + 222 x^{11} + 2181 x^{10} - 4754 x^{9} - 5631 x^{8} + 19452 x^{7} + 1661 x^{6} - 32016 x^{5} + 66060 x^{4} - 105666 x^{3} + 112833 x^{2} - 91314 x + 53433$ $S_3 \times C_3$ (as 18T3) $0$ $0$
*2 2.2e3_3e2_7.6t5.3c2$2$ $ 2^{3} \cdot 3^{2} \cdot 7 $ $x^{18} - 12 x^{16} - 10 x^{15} + 108 x^{14} + 42 x^{13} - 447 x^{12} + 222 x^{11} + 2181 x^{10} - 4754 x^{9} - 5631 x^{8} + 19452 x^{7} + 1661 x^{6} - 32016 x^{5} + 66060 x^{4} - 105666 x^{3} + 112833 x^{2} - 91314 x + 53433$ $S_3 \times C_3$ (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 *.