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Properties

Label 18.18.3627685868...7168.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{27}\cdot 37^{4}\cdot 229^{6}$
Root discriminant $38.61$
Ramified primes $2, 37, 229$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_3\wr S_3$ (as 18T119)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -14, 1, 588, 708, -7668, -3174, 25156, -3546, -21870, 5434, 8036, -2193, -1430, 389, 122, -32, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 - 32*x^16 + 122*x^15 + 389*x^14 - 1430*x^13 - 2193*x^12 + 8036*x^11 + 5434*x^10 - 21870*x^9 - 3546*x^8 + 25156*x^7 - 3174*x^6 - 7668*x^5 + 708*x^4 + 588*x^3 + x^2 - 14*x - 1)
 
gp: K = bnfinit(x^18 - 4*x^17 - 32*x^16 + 122*x^15 + 389*x^14 - 1430*x^13 - 2193*x^12 + 8036*x^11 + 5434*x^10 - 21870*x^9 - 3546*x^8 + 25156*x^7 - 3174*x^6 - 7668*x^5 + 708*x^4 + 588*x^3 + x^2 - 14*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} - 32 x^{16} + 122 x^{15} + 389 x^{14} - 1430 x^{13} - 2193 x^{12} + 8036 x^{11} + 5434 x^{10} - 21870 x^{9} - 3546 x^{8} + 25156 x^{7} - 3174 x^{6} - 7668 x^{5} + 708 x^{4} + 588 x^{3} + x^{2} - 14 x - 1 \)

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:  \(36276858684714282250737287168=2^{27}\cdot 37^{4}\cdot 229^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.61$
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:  $2, 37, 229$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{6092823676569708233669} a^{17} - \frac{1680558309135174226492}{6092823676569708233669} a^{16} - \frac{1499469842803088618944}{6092823676569708233669} a^{15} - \frac{467817797474040566611}{6092823676569708233669} a^{14} - \frac{655204327316759446168}{6092823676569708233669} a^{13} - \frac{2414062678844295457281}{6092823676569708233669} a^{12} + \frac{1506774914407792230845}{6092823676569708233669} a^{11} - \frac{1107401913715732173431}{6092823676569708233669} a^{10} - \frac{1501674946483096291111}{6092823676569708233669} a^{9} + \frac{12693221689010863391}{6092823676569708233669} a^{8} + \frac{1852754716931469727571}{6092823676569708233669} a^{7} + \frac{2758185102160020546291}{6092823676569708233669} a^{6} - \frac{316811402032143036836}{6092823676569708233669} a^{5} + \frac{1246907663698414414584}{6092823676569708233669} a^{4} + \frac{122083582830228346190}{6092823676569708233669} a^{3} + \frac{580203720789916832132}{6092823676569708233669} a^{2} + \frac{2817036109824314585082}{6092823676569708233669} a + \frac{1270382449429103434103}{6092823676569708233669}$

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:  \( 218477257.54 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3\wr S_3$ (as 18T119):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 324
The 44 conjugacy class representatives for $C_2\times C_3\wr S_3$
Character table for $C_2\times C_3\wr S_3$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 3.3.229.1, 6.6.26849792.2, 9.9.16440305941.1

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

Sibling fields

Degree 18 siblings: 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 $18$ $18$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.12.18.23$x^{12} + 52 x^{10} - 28 x^{8} + 8 x^{6} + 64 x^{4} - 32 x^{2} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
$37$37.6.4.2$x^{6} - 37 x^{3} + 6845$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
37.6.0.1$x^{6} - x + 20$$1$$6$$0$$C_6$$[\ ]^{6}$
37.6.0.1$x^{6} - x + 20$$1$$6$$0$$C_6$$[\ ]^{6}$
229Data not computed