Properties

Label 18.0.295...643.2
Degree $18$
Signature $[0, 9]$
Discriminant $-2.954\times 10^{21}$
Root discriminant \(15.59\)
Ramified prime see page
Class number $1$
Class group trivial
Galois group $C_9\times S_3$ (as 18T16)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^15 + 36*x^12 - 53*x^9 + 27*x^6 + 1)
 
gp: K = bnfinit(x^18 - 9*x^15 + 36*x^12 - 53*x^9 + 27*x^6 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, 0, 0, 27, 0, 0, -53, 0, 0, 36, 0, 0, -9, 0, 0, 1]);
 

\( x^{18} - 9x^{15} + 36x^{12} - 53x^{9} + 27x^{6} + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $18$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:   \(-2954312706550833698643\) \(\medspace = -\,3^{45}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  \(15.59\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:   \(3\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$\card{ \Aut(K/\Q) }$:  $9$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2071}a^{15}-\frac{393}{2071}a^{12}-\frac{235}{2071}a^{9}-\frac{937}{2071}a^{6}-\frac{519}{2071}a^{3}+\frac{480}{2071}$, $\frac{1}{2071}a^{16}-\frac{393}{2071}a^{13}-\frac{235}{2071}a^{10}-\frac{937}{2071}a^{7}-\frac{519}{2071}a^{4}+\frac{480}{2071}a$, $\frac{1}{2071}a^{17}-\frac{393}{2071}a^{14}-\frac{235}{2071}a^{11}-\frac{937}{2071}a^{8}-\frac{519}{2071}a^{5}+\frac{480}{2071}a^{2}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $8$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:   \( \frac{350}{2071} a^{15} - \frac{2935}{2071} a^{12} + \frac{10945}{2071} a^{9} - \frac{13158}{2071} a^{6} + \frac{6811}{2071} a^{3} - \frac{1822}{2071} \)  (order $18$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:   $\frac{350}{2071}a^{16}-\frac{2935}{2071}a^{13}+\frac{10945}{2071}a^{10}-\frac{13158}{2071}a^{7}+\frac{6811}{2071}a^{4}-\frac{3893}{2071}a$, $\frac{6}{109}a^{16}-\frac{69}{109}a^{13}+\frac{334}{109}a^{10}-\frac{717}{109}a^{7}+\frac{483}{109}a^{4}+\frac{46}{109}a$, $\frac{6}{109}a^{17}-\frac{604}{2071}a^{16}-\frac{75}{2071}a^{15}-\frac{69}{109}a^{14}+\frac{5420}{2071}a^{13}+\frac{481}{2071}a^{12}+\frac{334}{109}a^{11}-\frac{21669}{2071}a^{10}-\frac{1014}{2071}a^{9}-\frac{717}{109}a^{8}+\frac{31630}{2071}a^{7}-\frac{2210}{2071}a^{6}+\frac{483}{109}a^{5}-\frac{15813}{2071}a^{4}+\frac{3718}{2071}a^{3}+\frac{46}{109}a^{2}+\frac{20}{2071}a-\frac{793}{2071}$, $\frac{975}{2071}a^{17}+\frac{726}{2071}a^{16}+\frac{451}{2071}a^{15}-\frac{8324}{2071}a^{14}-\frac{5733}{2071}a^{13}-\frac{3279}{2071}a^{12}+\frac{31821}{2071}a^{11}+\frac{19922}{2071}a^{10}+\frac{9991}{2071}a^{9}-\frac{41684}{2071}a^{8}-\frac{17542}{2071}a^{7}-\frac{2174}{2071}a^{6}+\frac{24151}{2071}a^{5}+\frac{4270}{2071}a^{4}-\frac{6259}{2071}a^{3}-\frac{6259}{2071}a^{2}-\frac{1519}{2071}a-\frac{975}{2071}$, $\frac{544}{2071}a^{17}+\frac{1418}{2071}a^{16}+\frac{275}{2071}a^{15}-\frac{4621}{2071}a^{14}-\frac{12601}{2071}a^{13}-\frac{2454}{2071}a^{12}+\frac{17130}{2071}a^{11}+\frac{49905}{2071}a^{10}+\frac{9931}{2071}a^{9}-\frac{18901}{2071}a^{8}-\frac{71569}{2071}a^{7}-\frac{15368}{2071}a^{6}-\frac{680}{2071}a^{5}+\frac{36541}{2071}a^{4}+\frac{10529}{2071}a^{3}+\frac{10529}{2071}a^{2}+\frac{1352}{2071}a-\frac{544}{2071}$, $\frac{653}{2071}a^{17}-\frac{303}{2071}a^{16}-\frac{425}{2071}a^{15}-\frac{6038}{2071}a^{14}+\frac{3103}{2071}a^{13}+\frac{3416}{2071}a^{12}+\frac{24651}{2071}a^{11}-\frac{13706}{2071}a^{10}-\frac{11959}{2071}a^{9}-\frac{38194}{2071}a^{8}+\frac{25036}{2071}a^{7}+\frac{10948}{2071}a^{6}+\frac{19376}{2071}a^{5}-\frac{12565}{2071}a^{4}-\frac{1022}{2071}a^{3}-\frac{1352}{2071}a^{2}-\frac{2541}{2071}a+\frac{1029}{2071}$, $\frac{140}{2071}a^{15}-\frac{1174}{2071}a^{12}+\frac{4378}{2071}a^{9}-\frac{4849}{2071}a^{6}-\frac{175}{2071}a^{3}-a^{2}-a+\frac{928}{2071}$, $\frac{6}{109}a^{17}-\frac{6}{109}a^{16}+\frac{140}{2071}a^{15}-\frac{69}{109}a^{14}+\frac{69}{109}a^{13}-\frac{1174}{2071}a^{12}+\frac{334}{109}a^{11}-\frac{334}{109}a^{10}+\frac{4378}{2071}a^{9}-\frac{717}{109}a^{8}+\frac{717}{109}a^{7}-\frac{4849}{2071}a^{6}+\frac{483}{109}a^{5}-\frac{483}{109}a^{4}-\frac{175}{2071}a^{3}+\frac{46}{109}a^{2}+\frac{63}{109}a+\frac{928}{2071}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 23143.3269777 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{9}\cdot 23143.3269777 \cdot 1}{18\sqrt{2954312706550833698643}}\approx 0.361030481189$

Galois group

$C_9\times S_3$ (as 18T16):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 54
The 27 conjugacy class representatives for $C_9\times S_3$
Character table for $C_9\times S_3$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{9})\)

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

Sibling fields

Degree 27 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 $18$ R $18$ ${\href{/padicField/7.9.0.1}{9} }^{2}$ $18$ ${\href{/padicField/13.9.0.1}{9} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }^{3}$ ${\href{/padicField/19.3.0.1}{3} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{9}$ $18$ $18$ ${\href{/padicField/31.9.0.1}{9} }^{2}$ ${\href{/padicField/37.3.0.1}{3} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{9}$ $18$ ${\href{/padicField/43.9.0.1}{9} }^{2}$ $18$ ${\href{/padicField/53.2.0.1}{2} }^{9}$ $18$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display Deg $18$$18$$1$$45$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.3.2t1.a.a$1$ $ 3 $ \(\Q(\sqrt{-3}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.9.3t1.a.a$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
* 1.9.6t1.a.a$1$ $ 3^{2}$ \(\Q(\zeta_{9})\) $C_6$ (as 6T1) $0$ $-1$
* 1.9.6t1.a.b$1$ $ 3^{2}$ \(\Q(\zeta_{9})\) $C_6$ (as 6T1) $0$ $-1$
* 1.9.3t1.a.b$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
1.27.18t1.a.a$1$ $ 3^{3}$ \(\Q(\zeta_{27})\) $C_{18}$ (as 18T1) $0$ $-1$
1.27.18t1.a.b$1$ $ 3^{3}$ \(\Q(\zeta_{27})\) $C_{18}$ (as 18T1) $0$ $-1$
1.27.18t1.a.c$1$ $ 3^{3}$ \(\Q(\zeta_{27})\) $C_{18}$ (as 18T1) $0$ $-1$
1.27.9t1.a.a$1$ $ 3^{3}$ \(\Q(\zeta_{27})^+\) $C_9$ (as 9T1) $0$ $1$
1.27.18t1.a.d$1$ $ 3^{3}$ \(\Q(\zeta_{27})\) $C_{18}$ (as 18T1) $0$ $-1$
1.27.9t1.a.b$1$ $ 3^{3}$ \(\Q(\zeta_{27})^+\) $C_9$ (as 9T1) $0$ $1$
1.27.18t1.a.e$1$ $ 3^{3}$ \(\Q(\zeta_{27})\) $C_{18}$ (as 18T1) $0$ $-1$
1.27.9t1.a.c$1$ $ 3^{3}$ \(\Q(\zeta_{27})^+\) $C_9$ (as 9T1) $0$ $1$
1.27.9t1.a.d$1$ $ 3^{3}$ \(\Q(\zeta_{27})^+\) $C_9$ (as 9T1) $0$ $1$
1.27.9t1.a.e$1$ $ 3^{3}$ \(\Q(\zeta_{27})^+\) $C_9$ (as 9T1) $0$ $1$
1.27.9t1.a.f$1$ $ 3^{3}$ \(\Q(\zeta_{27})^+\) $C_9$ (as 9T1) $0$ $1$
1.27.18t1.a.f$1$ $ 3^{3}$ \(\Q(\zeta_{27})\) $C_{18}$ (as 18T1) $0$ $-1$
2.243.3t2.b.a$2$ $ 3^{5}$ 3.1.243.1 $S_3$ (as 3T2) $1$ $0$
2.243.6t5.b.a$2$ $ 3^{5}$ 6.0.177147.1 $S_3\times C_3$ (as 6T5) $0$ $0$
2.243.6t5.b.b$2$ $ 3^{5}$ 6.0.177147.1 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.729.18t16.a.a$2$ $ 3^{6}$ 18.0.2954312706550833698643.2 $C_9\times S_3$ (as 18T16) $0$ $0$
* 2.729.18t16.a.b$2$ $ 3^{6}$ 18.0.2954312706550833698643.2 $C_9\times S_3$ (as 18T16) $0$ $0$
* 2.729.18t16.a.c$2$ $ 3^{6}$ 18.0.2954312706550833698643.2 $C_9\times S_3$ (as 18T16) $0$ $0$
* 2.729.18t16.a.d$2$ $ 3^{6}$ 18.0.2954312706550833698643.2 $C_9\times S_3$ (as 18T16) $0$ $0$
* 2.729.18t16.a.e$2$ $ 3^{6}$ 18.0.2954312706550833698643.2 $C_9\times S_3$ (as 18T16) $0$ $0$
* 2.729.18t16.a.f$2$ $ 3^{6}$ 18.0.2954312706550833698643.2 $C_9\times S_3$ (as 18T16) $0$ $0$

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 *.