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Properties

Label 18.0.26552945316...5127.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 7^{9}\cdot 13^{12}$
Root discriminant $63.29$
Ramified primes $3, 7, 13$
Class number $2128$ (GRH)
Class group $[4, 532]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1068904, -619836, 1606332, -776387, 994161, -431268, 306560, -108612, 45528, -4424, -1398, 3414, -208, -996, 264, 80, -24, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 24*x^16 + 80*x^15 + 264*x^14 - 996*x^13 - 208*x^12 + 3414*x^11 - 1398*x^10 - 4424*x^9 + 45528*x^8 - 108612*x^7 + 306560*x^6 - 431268*x^5 + 994161*x^4 - 776387*x^3 + 1606332*x^2 - 619836*x + 1068904)
 
gp: K = bnfinit(x^18 - 3*x^17 - 24*x^16 + 80*x^15 + 264*x^14 - 996*x^13 - 208*x^12 + 3414*x^11 - 1398*x^10 - 4424*x^9 + 45528*x^8 - 108612*x^7 + 306560*x^6 - 431268*x^5 + 994161*x^4 - 776387*x^3 + 1606332*x^2 - 619836*x + 1068904, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 24 x^{16} + 80 x^{15} + 264 x^{14} - 996 x^{13} - 208 x^{12} + 3414 x^{11} - 1398 x^{10} - 4424 x^{9} + 45528 x^{8} - 108612 x^{7} + 306560 x^{6} - 431268 x^{5} + 994161 x^{4} - 776387 x^{3} + 1606332 x^{2} - 619836 x + 1068904 \)

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:  \(-265529453168247662686860141155127=-\,3^{24}\cdot 7^{9}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.29$
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:  $3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(819=3^{2}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{819}(1,·)$, $\chi_{819}(391,·)$, $\chi_{819}(328,·)$, $\chi_{819}(139,·)$, $\chi_{819}(274,·)$, $\chi_{819}(211,·)$, $\chi_{819}(22,·)$, $\chi_{819}(664,·)$, $\chi_{819}(601,·)$, $\chi_{819}(412,·)$, $\chi_{819}(547,·)$, $\chi_{819}(484,·)$, $\chi_{819}(295,·)$, $\chi_{819}(685,·)$, $\chi_{819}(757,·)$, $\chi_{819}(118,·)$, $\chi_{819}(55,·)$, $\chi_{819}(568,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{6} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{22243748} a^{15} + \frac{1004067}{11121874} a^{14} - \frac{715819}{11121874} a^{13} + \frac{1226263}{22243748} a^{12} - \frac{1596301}{22243748} a^{11} + \frac{582454}{5560937} a^{10} + \frac{470119}{22243748} a^{9} - \frac{4700767}{22243748} a^{8} - \frac{4988187}{11121874} a^{7} - \frac{120011}{11121874} a^{6} - \frac{4150567}{22243748} a^{5} - \frac{39243}{22243748} a^{4} - \frac{1408896}{5560937} a^{3} - \frac{6585997}{22243748} a^{2} - \frac{1600653}{5560937} a - \frac{1055139}{5560937}$, $\frac{1}{1017406789772} a^{16} + \frac{13689}{1017406789772} a^{15} + \frac{17051343497}{254351697443} a^{14} - \frac{99779688375}{1017406789772} a^{13} + \frac{22356036815}{1017406789772} a^{12} - \frac{19155950632}{254351697443} a^{11} - \frac{116216271849}{1017406789772} a^{10} - \frac{36430499363}{508703394886} a^{9} - \frac{73699521837}{1017406789772} a^{8} - \frac{97114391847}{254351697443} a^{7} + \frac{74871282283}{1017406789772} a^{6} + \frac{117797340099}{1017406789772} a^{5} + \frac{47961126944}{254351697443} a^{4} + \frac{508693178503}{1017406789772} a^{3} - \frac{253176268391}{1017406789772} a^{2} - \frac{93800158956}{254351697443} a + \frac{1765381148}{4799088631}$, $\frac{1}{3605740643098168570381530266924} a^{17} - \frac{363152581552164835}{3605740643098168570381530266924} a^{16} + \frac{26312321225855997661809}{1802870321549084285190765133462} a^{15} - \frac{61727641092556297899187284021}{901435160774542142595382566731} a^{14} - \frac{95950869363296683555392496108}{901435160774542142595382566731} a^{13} - \frac{74561565477732864520875660638}{901435160774542142595382566731} a^{12} + \frac{9684425359481368075166133011}{3605740643098168570381530266924} a^{11} + \frac{181578257504583774219540216695}{3605740643098168570381530266924} a^{10} + \frac{186284120438229304205345050531}{1802870321549084285190765133462} a^{9} + \frac{37112215215764390495547960074}{901435160774542142595382566731} a^{8} + \frac{334148565210247635390938379013}{1802870321549084285190765133462} a^{7} + \frac{371617854459289100671520949431}{901435160774542142595382566731} a^{6} + \frac{314887359435647824340300657642}{901435160774542142595382566731} a^{5} - \frac{547302521479651103486448574385}{3605740643098168570381530266924} a^{4} - \frac{385602419556619403065948066703}{901435160774542142595382566731} a^{3} + \frac{937318856350107032170393222517}{3605740643098168570381530266924} a^{2} + \frac{804771211584090872633527510367}{1802870321549084285190765133462} a - \frac{4367385334633465950057924122}{17008210580651738539535520127}$

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

Class group and class number

$C_{4}\times C_{532}$, which has order $2128$ (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:  $8$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 400417.136445 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-7}) \), 3.3.13689.2, 3.3.13689.1, 3.3.169.1, \(\Q(\zeta_{9})^+\), 6.0.64274331303.6, 6.0.64274331303.4, 6.0.9796423.1, 6.0.2250423.1, 9.9.2565164201769.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data not computed
$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}$
13Data not computed