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Properties

Label 18.0.35577987625...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{33}\cdot 5^{6}$
Root discriminant $20.34$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois group $C_3\times S_3^2$ (as 18T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 12, 75, 195, 381, 543, 567, 507, 309, 214, 132, 69, 75, -21, 33, -18, 15, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 15*x^16 - 18*x^15 + 33*x^14 - 21*x^13 + 75*x^12 + 69*x^11 + 132*x^10 + 214*x^9 + 309*x^8 + 507*x^7 + 567*x^6 + 543*x^5 + 381*x^4 + 195*x^3 + 75*x^2 + 12*x + 1)
 
gp: K = bnfinit(x^18 - 3*x^17 + 15*x^16 - 18*x^15 + 33*x^14 - 21*x^13 + 75*x^12 + 69*x^11 + 132*x^10 + 214*x^9 + 309*x^8 + 507*x^7 + 567*x^6 + 543*x^5 + 381*x^4 + 195*x^3 + 75*x^2 + 12*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 15 x^{16} - 18 x^{15} + 33 x^{14} - 21 x^{13} + 75 x^{12} + 69 x^{11} + 132 x^{10} + 214 x^{9} + 309 x^{8} + 507 x^{7} + 567 x^{6} + 543 x^{5} + 381 x^{4} + 195 x^{3} + 75 x^{2} + 12 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-355779876259553472000000=-\,2^{12}\cdot 3^{33}\cdot 5^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.34$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a$, $\frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{4}{9} a^{2} - \frac{4}{9} a - \frac{4}{9}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{9} - \frac{1}{3} a^{6} + \frac{2}{9} a^{3} + \frac{1}{9}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{10} - \frac{1}{3} a^{7} + \frac{2}{9} a^{4} + \frac{1}{9} a$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{4}{9} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{4}{9} a - \frac{4}{9}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{9} - \frac{1}{9} a^{6} + \frac{1}{3} a^{3} - \frac{2}{9}$, $\frac{1}{63} a^{16} - \frac{1}{21} a^{14} + \frac{1}{63} a^{13} - \frac{1}{63} a^{12} - \frac{5}{63} a^{10} - \frac{2}{63} a^{9} + \frac{3}{7} a^{8} + \frac{23}{63} a^{7} - \frac{8}{21} a^{6} + \frac{1}{3} a^{5} + \frac{5}{63} a^{4} - \frac{2}{63} a^{3} - \frac{1}{7} a^{2} + \frac{29}{63} a + \frac{2}{63}$, $\frac{1}{63372141} a^{17} - \frac{224299}{63372141} a^{16} - \frac{1070086}{63372141} a^{15} - \frac{817975}{21124047} a^{14} - \frac{868933}{21124047} a^{13} + \frac{697190}{21124047} a^{12} - \frac{907520}{63372141} a^{11} + \frac{173146}{1242591} a^{10} - \frac{91230}{7041349} a^{9} + \frac{4491512}{9053163} a^{8} - \frac{29874263}{63372141} a^{7} - \frac{3799676}{63372141} a^{6} + \frac{3164599}{21124047} a^{5} - \frac{3711178}{21124047} a^{4} + \frac{9308780}{21124047} a^{3} - \frac{28733260}{63372141} a^{2} - \frac{3278182}{7041349} a - \frac{1112959}{21124047}$

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:  \( -\frac{10466}{16371} a^{17} + \frac{37445}{16371} a^{16} - \frac{182978}{16371} a^{15} + \frac{310774}{16371} a^{14} - \frac{600928}{16371} a^{13} + \frac{696748}{16371} a^{12} - \frac{1386710}{16371} a^{11} + \frac{14896}{963} a^{10} - \frac{1885664}{16371} a^{9} - \frac{1225399}{16371} a^{8} - \frac{2854802}{16371} a^{7} - \frac{4171054}{16371} a^{6} - \frac{4304878}{16371} a^{5} - \frac{4336217}{16371} a^{4} - \frac{2475484}{16371} a^{3} - \frac{1347430}{16371} a^{2} - \frac{364730}{16371} a - \frac{27730}{16371} \) (order $6$)
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:  \( 119253.839734 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3^2$ (as 18T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 27 conjugacy class representatives for $C_3\times S_3^2$
Character table for $C_3\times S_3^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.3.1620.1, 6.0.314928.1, 6.0.7873200.1

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

Sibling fields

Degree 12 sibling: data not computed
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 R R ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
3Data not computed
$5$5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$