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Properties

Label 18.0.15079916901...6875.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{31}\cdot 5^{12}$
Root discriminant $19.39$
Ramified primes $3, 5$
Class number $1$
Class group Trivial
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![64, -96, -144, 336, -336, 342, 231, -699, 1314, -1465, 1371, -1089, 723, -399, 207, -102, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 102*x^15 + 207*x^14 - 399*x^13 + 723*x^12 - 1089*x^11 + 1371*x^10 - 1465*x^9 + 1314*x^8 - 699*x^7 + 231*x^6 + 342*x^5 - 336*x^4 + 336*x^3 - 144*x^2 - 96*x + 64)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 - 102*x^15 + 207*x^14 - 399*x^13 + 723*x^12 - 1089*x^11 + 1371*x^10 - 1465*x^9 + 1314*x^8 - 699*x^7 + 231*x^6 + 342*x^5 - 336*x^4 + 336*x^3 - 144*x^2 - 96*x + 64, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} - 102 x^{15} + 207 x^{14} - 399 x^{13} + 723 x^{12} - 1089 x^{11} + 1371 x^{10} - 1465 x^{9} + 1314 x^{8} - 699 x^{7} + 231 x^{6} + 342 x^{5} - 336 x^{4} + 336 x^{3} - 144 x^{2} - 96 x + 64 \)

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:  \(-150799169014635498046875=-\,3^{31}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.39$
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:  $3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{4}{9} a^{2} - \frac{4}{9} a + \frac{4}{9}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{9} - \frac{2}{9} a^{3} - \frac{2}{9}$, $\frac{1}{18} a^{13} - \frac{1}{18} a^{12} - \frac{1}{18} a^{11} + \frac{1}{9} a^{10} + \frac{1}{18} a^{9} + \frac{1}{6} a^{8} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{6} a^{5} + \frac{1}{18} a^{4} + \frac{4}{9} a^{3} + \frac{5}{18} a^{2} - \frac{7}{18} a - \frac{4}{9}$, $\frac{1}{36} a^{14} - \frac{1}{36} a^{13} - \frac{1}{36} a^{12} - \frac{1}{18} a^{11} - \frac{1}{36} a^{10} + \frac{5}{36} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{7}{36} a^{5} + \frac{1}{18} a^{4} - \frac{7}{36} a^{3} - \frac{5}{36} a^{2} + \frac{1}{18} a + \frac{2}{9}$, $\frac{1}{504} a^{15} - \frac{1}{168} a^{14} + \frac{5}{504} a^{13} - \frac{2}{63} a^{12} + \frac{1}{72} a^{11} - \frac{5}{504} a^{10} + \frac{5}{504} a^{9} + \frac{47}{168} a^{8} - \frac{29}{168} a^{7} - \frac{59}{504} a^{6} - \frac{3}{7} a^{5} - \frac{67}{504} a^{4} + \frac{125}{504} a^{3} + \frac{25}{126} a^{2} - \frac{4}{63} a + \frac{13}{63}$, $\frac{1}{1008} a^{16} - \frac{1}{1008} a^{15} - \frac{1}{1008} a^{14} - \frac{1}{168} a^{13} - \frac{25}{1008} a^{12} - \frac{47}{1008} a^{11} - \frac{13}{112} a^{10} + \frac{95}{1008} a^{9} + \frac{121}{336} a^{8} + \frac{103}{1008} a^{7} + \frac{169}{504} a^{6} - \frac{163}{1008} a^{5} - \frac{115}{336} a^{4} - \frac{23}{72} a^{3} - \frac{1}{18} a^{2} - \frac{17}{42} a - \frac{1}{63}$, $\frac{1}{95929725909024} a^{17} - \frac{45314652931}{95929725909024} a^{16} + \frac{143127623}{31976575303008} a^{15} + \frac{73028974955}{5995607869314} a^{14} - \frac{25483329107}{1880975017824} a^{13} + \frac{824933221985}{31976575303008} a^{12} - \frac{613611622877}{13704246558432} a^{11} - \frac{85965497341}{806132150496} a^{10} + \frac{1997555137339}{31976575303008} a^{9} + \frac{13148318619637}{95929725909024} a^{8} - \frac{608316487985}{1713030819804} a^{7} + \frac{3435626311679}{31976575303008} a^{6} - \frac{36295132402915}{95929725909024} a^{5} + \frac{3748314384455}{7994143825752} a^{4} + \frac{223151775514}{999267978219} a^{3} + \frac{2547782675147}{11991215738628} a^{2} - \frac{1519112348213}{5995607869314} a - \frac{477267959887}{999267978219}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{33922039889}{13704246558432} a^{17} - \frac{1436804168039}{95929725909024} a^{16} + \frac{3451558400573}{95929725909024} a^{15} - \frac{589357380931}{47964862954512} a^{14} - \frac{297120867595}{5642925053472} a^{13} + \frac{10703795170823}{95929725909024} a^{12} - \frac{40313871479767}{95929725909024} a^{11} + \frac{2532578264663}{1880975017824} a^{10} - \frac{77255606189413}{31976575303008} a^{9} + \frac{354326607439829}{95929725909024} a^{8} - \frac{248543334933463}{47964862954512} a^{7} + \frac{94180413408521}{13704246558432} a^{6} - \frac{497390324798159}{95929725909024} a^{5} + \frac{39318642066941}{6852123279216} a^{4} - \frac{37615246786525}{23982431477256} a^{3} + \frac{12597955682797}{11991215738628} a^{2} + \frac{216160802231}{142752568317} a - \frac{1959864226556}{999267978219} \) (order $18$)
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:  \( 111472.114435 \)
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

\(\Q(\sqrt{-3}) \), 3.1.675.1 x3, \(\Q(\zeta_{9})^+\), 6.0.1366875.1, 6.0.12301875.1 x2, \(\Q(\zeta_{9})\), 9.3.224201671875.3 x3

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

Sibling fields

Degree 6 sibling: 6.0.12301875.1
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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
5Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.3.2t1.1c1$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
* 1.3e2.6t1.1c1$1$ $ 3^{2}$ $x^{6} - x^{3} + 1$ $C_6$ (as 6T1) $0$ $-1$
* 1.3e2.6t1.1c2$1$ $ 3^{2}$ $x^{6} - x^{3} + 1$ $C_6$ (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 2.3e3_5e2.3t2.1c1$2$ $ 3^{3} \cdot 5^{2}$ $x^{3} - 5$ $S_3$ (as 3T2) $1$ $0$
*2 2.3e4_5e2.6t5.3c1$2$ $ 3^{4} \cdot 5^{2}$ $x^{18} - 9 x^{17} + 39 x^{16} - 102 x^{15} + 207 x^{14} - 399 x^{13} + 723 x^{12} - 1089 x^{11} + 1371 x^{10} - 1465 x^{9} + 1314 x^{8} - 699 x^{7} + 231 x^{6} + 342 x^{5} - 336 x^{4} + 336 x^{3} - 144 x^{2} - 96 x + 64$ $S_3 \times C_3$ (as 18T3) $0$ $0$
*2 2.3e4_5e2.6t5.3c2$2$ $ 3^{4} \cdot 5^{2}$ $x^{18} - 9 x^{17} + 39 x^{16} - 102 x^{15} + 207 x^{14} - 399 x^{13} + 723 x^{12} - 1089 x^{11} + 1371 x^{10} - 1465 x^{9} + 1314 x^{8} - 699 x^{7} + 231 x^{6} + 342 x^{5} - 336 x^{4} + 336 x^{3} - 144 x^{2} - 96 x + 64$ $S_3 \times C_3$ (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 *.