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Properties

Label 18.18.1742009952...9232.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{26}\cdot 3^{37}\cdot 7^{8}$
Root discriminant $61.83$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times D_9:C_3$ (as 18T45)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1058, 5508, 378, -34644, 12150, 93636, -13302, -126036, -24462, 67486, 24408, -15606, -7440, 1566, 972, -54, -54, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 54*x^16 - 54*x^15 + 972*x^14 + 1566*x^13 - 7440*x^12 - 15606*x^11 + 24408*x^10 + 67486*x^9 - 24462*x^8 - 126036*x^7 - 13302*x^6 + 93636*x^5 + 12150*x^4 - 34644*x^3 + 378*x^2 + 5508*x - 1058)
 
gp: K = bnfinit(x^18 - 54*x^16 - 54*x^15 + 972*x^14 + 1566*x^13 - 7440*x^12 - 15606*x^11 + 24408*x^10 + 67486*x^9 - 24462*x^8 - 126036*x^7 - 13302*x^6 + 93636*x^5 + 12150*x^4 - 34644*x^3 + 378*x^2 + 5508*x - 1058, 1)
 

Normalized defining polynomial

\( x^{18} - 54 x^{16} - 54 x^{15} + 972 x^{14} + 1566 x^{13} - 7440 x^{12} - 15606 x^{11} + 24408 x^{10} + 67486 x^{9} - 24462 x^{8} - 126036 x^{7} - 13302 x^{6} + 93636 x^{5} + 12150 x^{4} - 34644 x^{3} + 378 x^{2} + 5508 x - 1058 \)

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:  \(174200995291297962325510541279232=2^{26}\cdot 3^{37}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.83$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} - \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a$, $\frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{4}{9} a^{2} - \frac{4}{9} a - \frac{4}{9}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{9} - \frac{1}{9} a^{3} + \frac{1}{9}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{10} - \frac{1}{9} a^{4} + \frac{1}{9} a$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{2}{9} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{4}{9} a - \frac{4}{9}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{9} - \frac{1}{9} a^{6} + \frac{1}{9}$, $\frac{1}{27} a^{16} - \frac{1}{27} a^{15} + \frac{1}{27} a^{13} - \frac{1}{27} a^{12} - \frac{2}{27} a^{10} + \frac{2}{27} a^{9} - \frac{1}{27} a^{7} + \frac{1}{27} a^{6} + \frac{8}{27} a^{4} - \frac{8}{27} a^{3} - \frac{7}{27} a + \frac{7}{27}$, $\frac{1}{308456904977780031} a^{17} + \frac{383401783405502}{308456904977780031} a^{16} - \frac{160511798953342}{11424329813991853} a^{15} - \frac{14852656166363342}{308456904977780031} a^{14} + \frac{12567669986195648}{308456904977780031} a^{13} + \frac{1113254653250222}{34272989441975559} a^{12} - \frac{775315429169420}{308456904977780031} a^{11} - \frac{41093662300080514}{308456904977780031} a^{10} - \frac{4703826449803513}{34272989441975559} a^{9} - \frac{41324244732079723}{308456904977780031} a^{8} - \frac{5195920100239862}{308456904977780031} a^{7} - \frac{195207400594464}{11424329813991853} a^{6} - \frac{62507439896505058}{308456904977780031} a^{5} - \frac{99608966837438741}{308456904977780031} a^{4} + \frac{91484186660998}{34272989441975559} a^{3} - \frac{116767281437405332}{308456904977780031} a^{2} - \frac{119410088913171227}{308456904977780031} a - \frac{8669617639805741}{34272989441975559}$

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:  \( 65797975081.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_9:C_3$ (as 18T45):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times D_9:C_3$
Character table for $C_2\times D_9:C_3$

Intermediate fields

\(\Q(\sqrt{3}) \), 3.3.756.1, 6.6.27433728.1, 9.9.238130328086784.1

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

Sibling fields

Degree 18 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 R R $18$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
2Data not computed
3Data not computed
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.12.6.1$x^{12} + 294 x^{8} + 3430 x^{6} + 21609 x^{4} + 487403 x^{2} + 2941225$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$