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Properties

Label 18.0.13126401193...0608.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{33}\cdot 7^{8}$
Root discriminant $28.25$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3^2:S_3$ (as 18T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, -486, 2187, -4104, 5994, -3969, 3348, -2430, 1458, -933, 657, -324, 169, -75, 33, -13, 6, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 6*x^16 - 13*x^15 + 33*x^14 - 75*x^13 + 169*x^12 - 324*x^11 + 657*x^10 - 933*x^9 + 1458*x^8 - 2430*x^7 + 3348*x^6 - 3969*x^5 + 5994*x^4 - 4104*x^3 + 2187*x^2 - 486*x + 81)
 
gp: K = bnfinit(x^18 - 3*x^17 + 6*x^16 - 13*x^15 + 33*x^14 - 75*x^13 + 169*x^12 - 324*x^11 + 657*x^10 - 933*x^9 + 1458*x^8 - 2430*x^7 + 3348*x^6 - 3969*x^5 + 5994*x^4 - 4104*x^3 + 2187*x^2 - 486*x + 81, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 6 x^{16} - 13 x^{15} + 33 x^{14} - 75 x^{13} + 169 x^{12} - 324 x^{11} + 657 x^{10} - 933 x^{9} + 1458 x^{8} - 2430 x^{7} + 3348 x^{6} - 3969 x^{5} + 5994 x^{4} - 4104 x^{3} + 2187 x^{2} - 486 x + 81 \)

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:  \(-131264011932220807356100608=-\,2^{12}\cdot 3^{33}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.25$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3}$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{6} + \frac{4}{9} a^{4} + \frac{4}{9} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{6} + \frac{1}{9} a^{5} + \frac{1}{3} a^{4} + \frac{2}{9} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{9} a^{9} - \frac{1}{9} a^{6} + \frac{1}{9} a^{3} + \frac{1}{3}$, $\frac{1}{9} a^{10} + \frac{1}{9} a^{6} - \frac{4}{9} a^{4} + \frac{4}{9} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{6} - \frac{1}{9} a^{5} - \frac{1}{3} a^{4} - \frac{1}{9} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{81} a^{12} - \frac{1}{27} a^{11} - \frac{1}{27} a^{10} - \frac{1}{81} a^{9} - \frac{1}{27} a^{8} - \frac{1}{27} a^{7} + \frac{4}{81} a^{6} - \frac{1}{9} a^{5} + \frac{4}{9} a^{4} + \frac{2}{27} a^{3} - \frac{1}{3} a + \frac{4}{9}$, $\frac{1}{243} a^{13} - \frac{4}{81} a^{11} - \frac{1}{243} a^{10} - \frac{2}{81} a^{9} - \frac{1}{81} a^{8} - \frac{5}{243} a^{7} + \frac{10}{81} a^{6} - \frac{4}{27} a^{5} - \frac{1}{81} a^{4} + \frac{5}{27} a^{3} + \frac{10}{27} a - \frac{2}{9}$, $\frac{1}{243} a^{14} - \frac{10}{243} a^{11} + \frac{4}{81} a^{10} + \frac{4}{81} a^{9} + \frac{13}{243} a^{8} - \frac{2}{81} a^{7} + \frac{13}{81} a^{6} - \frac{1}{81} a^{5} + \frac{2}{27} a^{4} + \frac{8}{27} a^{3} - \frac{8}{27} a^{2} - \frac{2}{9} a - \frac{2}{9}$, $\frac{1}{2187} a^{15} - \frac{1}{729} a^{14} - \frac{1}{729} a^{13} - \frac{7}{2187} a^{12} + \frac{14}{729} a^{11} - \frac{10}{729} a^{10} - \frac{62}{2187} a^{9} + \frac{13}{243} a^{8} - \frac{2}{243} a^{7} + \frac{5}{243} a^{6} - \frac{7}{81} a^{5} + \frac{14}{81} a^{4} + \frac{17}{81} a^{3} - \frac{1}{9} a^{2} - \frac{23}{81}$, $\frac{1}{37179} a^{16} + \frac{5}{37179} a^{15} - \frac{2}{4131} a^{14} + \frac{50}{37179} a^{13} - \frac{149}{37179} a^{12} + \frac{14}{1377} a^{11} - \frac{599}{37179} a^{10} - \frac{8}{2187} a^{9} + \frac{160}{4131} a^{8} + \frac{145}{4131} a^{7} - \frac{299}{4131} a^{6} + \frac{173}{1377} a^{5} + \frac{6}{17} a^{4} - \frac{140}{1377} a^{3} + \frac{139}{459} a^{2} + \frac{40}{1377} a + \frac{131}{1377}$, $\frac{1}{711197091} a^{17} - \frac{4817}{711197091} a^{16} + \frac{14318}{237065697} a^{15} - \frac{244756}{711197091} a^{14} - \frac{31411}{19221543} a^{13} + \frac{286421}{79021899} a^{12} - \frac{4243061}{711197091} a^{11} - \frac{25196795}{711197091} a^{10} - \frac{6408146}{237065697} a^{9} + \frac{13360}{4648347} a^{8} + \frac{2793569}{79021899} a^{7} - \frac{43007}{516483} a^{6} - \frac{655084}{26340633} a^{5} + \frac{4233056}{26340633} a^{4} - \frac{312091}{8780211} a^{3} + \frac{11979586}{26340633} a^{2} + \frac{3060337}{26340633} a + \frac{887917}{8780211}$

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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{41174}{13945041} a^{17} + \frac{342251}{41835123} a^{16} - \frac{226720}{13945041} a^{15} + \frac{167620}{4648347} a^{14} - \frac{104438}{1130679} a^{13} + \frac{957607}{4648347} a^{12} - \frac{6489958}{13945041} a^{11} + \frac{36927530}{41835123} a^{10} - \frac{25318280}{13945041} a^{9} + \frac{3847247}{1549449} a^{8} - \frac{18634082}{4648347} a^{7} + \frac{10281956}{1549449} a^{6} - \frac{4626214}{516483} a^{5} + \frac{16341283}{1549449} a^{4} - \frac{951680}{57387} a^{3} + \frac{5000630}{516483} a^{2} - \frac{9591916}{1549449} a + \frac{712468}{516483} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4215592.441455651 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:S_3$ (as 18T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $C_3^2:S_3$
Character table for $C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.972.2 x3, 6.0.2834352.2, 9.3.2204910445248.2

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

Sibling fields

Degree 9 siblings: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.6.11.7$x^{6} + 21$$6$$1$$11$$S_3$$[5/2]_{2}$
3.6.11.7$x^{6} + 21$$6$$1$$11$$S_3$$[5/2]_{2}$
3.6.11.7$x^{6} + 21$$6$$1$$11$$S_3$$[5/2]_{2}$
$7$7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$