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Properties

Label 18.0.13991415038...6391.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 7^{12}\cdot 71^{3}$
Root discriminant $32.22$
Ramified primes $3, 7, 71$
Class number $20$
Class group $[2, 10]$
Galois group $C_6\times A_4$ (as 18T25)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3079, -21321, 71619, -147022, 203559, -201378, 152588, -96642, 54783, -27983, 12642, -5094, 1867, -648, 207, -53, 15, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 15*x^16 - 53*x^15 + 207*x^14 - 648*x^13 + 1867*x^12 - 5094*x^11 + 12642*x^10 - 27983*x^9 + 54783*x^8 - 96642*x^7 + 152588*x^6 - 201378*x^5 + 203559*x^4 - 147022*x^3 + 71619*x^2 - 21321*x + 3079)
 
gp: K = bnfinit(x^18 - 3*x^17 + 15*x^16 - 53*x^15 + 207*x^14 - 648*x^13 + 1867*x^12 - 5094*x^11 + 12642*x^10 - 27983*x^9 + 54783*x^8 - 96642*x^7 + 152588*x^6 - 201378*x^5 + 203559*x^4 - 147022*x^3 + 71619*x^2 - 21321*x + 3079, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 15 x^{16} - 53 x^{15} + 207 x^{14} - 648 x^{13} + 1867 x^{12} - 5094 x^{11} + 12642 x^{10} - 27983 x^{9} + 54783 x^{8} - 96642 x^{7} + 152588 x^{6} - 201378 x^{5} + 203559 x^{4} - 147022 x^{3} + 71619 x^{2} - 21321 x + 3079 \)

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:  \(-1399141503834185765244506391=-\,3^{24}\cdot 7^{12}\cdot 71^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.22$
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:  $3, 7, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{4}{9} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{4}{9}$, $\frac{1}{90} a^{16} + \frac{1}{30} a^{15} - \frac{1}{15} a^{14} + \frac{7}{45} a^{13} - \frac{1}{6} a^{12} + \frac{1}{15} a^{11} - \frac{2}{15} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{23}{90} a^{7} - \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{3}{10} a^{4} + \frac{1}{6} a^{3} + \frac{7}{15} a^{2} - \frac{5}{18} a - \frac{13}{30}$, $\frac{1}{639939212659185367209536700} a^{17} - \frac{21967294195567469278742}{10665653544319756120158945} a^{16} + \frac{5107514655953985601535471}{127987842531837073441907340} a^{15} + \frac{19074462002513537620189553}{159984803164796341802384175} a^{14} - \frac{8780576039877437796433199}{213313070886395122403178900} a^{13} + \frac{80001190986576052104134201}{639939212659185367209536700} a^{12} + \frac{795530300026956215365103}{4266261417727902448063578} a^{11} + \frac{15398577276565237237792363}{53328267721598780600794725} a^{10} - \frac{265508301113543506190609}{7110435696213170746772630} a^{9} + \frac{278307098898915871424538487}{639939212659185367209536700} a^{8} + \frac{5217255433060494027274914}{17776089240532926866931575} a^{7} - \frac{5693115202730221986569681}{63993921265918536720953670} a^{6} - \frac{8091382689431073474081319}{35552178481065853733863150} a^{5} + \frac{1780890687554732302064753}{53328267721598780600794725} a^{4} + \frac{99969388300806217145795449}{213313070886395122403178900} a^{3} - \frac{166004248926398179900849921}{639939212659185367209536700} a^{2} - \frac{5588298379108329075587059}{17776089240532926866931575} a - \frac{259328890004884614626871913}{639939212659185367209536700}$

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

Class group and class number

$C_{2}\times C_{10}$, which has order $20$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 54408.4888887 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times A_4$ (as 18T25):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 24 conjugacy class representatives for $C_6\times A_4$
Character table for $C_6\times A_4$ is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.3.3969.2, 3.3.3969.1, \(\Q(\zeta_{7})^+\), 6.0.1118460231.1, 9.9.62523502209.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
7Data not computed
$71$$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$