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Properties

Label 18.18.3735121995...9701.1
Degree $18$
Signature $[18, 0]$
Discriminant $3^{9}\cdot 7^{17}\cdot 13^{8}$
Root discriminant $34.02$
Ramified primes $3, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_9:C_3$ (as 18T14)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-167, -5034, -20493, 52230, 54518, -119105, -37793, 114292, -1627, -52953, 10019, 12270, -3542, -1470, 529, 87, -37, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 37*x^16 + 87*x^15 + 529*x^14 - 1470*x^13 - 3542*x^12 + 12270*x^11 + 10019*x^10 - 52953*x^9 - 1627*x^8 + 114292*x^7 - 37793*x^6 - 119105*x^5 + 54518*x^4 + 52230*x^3 - 20493*x^2 - 5034*x - 167)
 
gp: K = bnfinit(x^18 - 2*x^17 - 37*x^16 + 87*x^15 + 529*x^14 - 1470*x^13 - 3542*x^12 + 12270*x^11 + 10019*x^10 - 52953*x^9 - 1627*x^8 + 114292*x^7 - 37793*x^6 - 119105*x^5 + 54518*x^4 + 52230*x^3 - 20493*x^2 - 5034*x - 167, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 37 x^{16} + 87 x^{15} + 529 x^{14} - 1470 x^{13} - 3542 x^{12} + 12270 x^{11} + 10019 x^{10} - 52953 x^{9} - 1627 x^{8} + 114292 x^{7} - 37793 x^{6} - 119105 x^{5} + 54518 x^{4} + 52230 x^{3} - 20493 x^{2} - 5034 x - 167 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3735121995389959988293999701=3^{9}\cdot 7^{17}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.02$
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 not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{222225347305043503} a^{17} - \frac{19189462771559126}{222225347305043503} a^{16} - \frac{60758850364608325}{222225347305043503} a^{15} - \frac{83102730652766375}{222225347305043503} a^{14} - \frac{84449810906900368}{222225347305043503} a^{13} + \frac{7118753871159514}{222225347305043503} a^{12} - \frac{21703286539719843}{222225347305043503} a^{11} + \frac{70978366457228090}{222225347305043503} a^{10} + \frac{78494716856250861}{222225347305043503} a^{9} - \frac{79943355886716584}{222225347305043503} a^{8} + \frac{96069511366188757}{222225347305043503} a^{7} + \frac{47046614999400649}{222225347305043503} a^{6} + \frac{54940732463379612}{222225347305043503} a^{5} - \frac{106560484063197334}{222225347305043503} a^{4} - \frac{27407176794006610}{222225347305043503} a^{3} + \frac{61926252210689528}{222225347305043503} a^{2} + \frac{51738669835992098}{222225347305043503} a - \frac{66673293263060753}{222225347305043503}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $17$
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:  \( 83610666.1939 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_9:C_3$ (as 18T14):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 22 conjugacy class representatives for $C_2\times C_9:C_3$
Character table for $C_2\times C_9:C_3$ is not computed

Intermediate fields

\(\Q(\sqrt{21}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{21})^+\), 9.9.164648481361.1

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 $18$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R $18$ R ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ $18$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
7Data not computed
$13$13.6.4.1$x^{6} + 39 x^{3} + 676$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.4.2$x^{6} - 13 x^{3} + 338$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$