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Properties

Label 18.0.13991415038...6391.2
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 $28$
Class group $[2, 14]$
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![27, -162, 567, -963, 1647, -1296, 2454, -2052, 3594, -2630, 2610, -1320, 1144, -378, 249, -52, 27, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 27*x^16 - 52*x^15 + 249*x^14 - 378*x^13 + 1144*x^12 - 1320*x^11 + 2610*x^10 - 2630*x^9 + 3594*x^8 - 2052*x^7 + 2454*x^6 - 1296*x^5 + 1647*x^4 - 963*x^3 + 567*x^2 - 162*x + 27)
 
gp: K = bnfinit(x^18 - 3*x^17 + 27*x^16 - 52*x^15 + 249*x^14 - 378*x^13 + 1144*x^12 - 1320*x^11 + 2610*x^10 - 2630*x^9 + 3594*x^8 - 2052*x^7 + 2454*x^6 - 1296*x^5 + 1647*x^4 - 963*x^3 + 567*x^2 - 162*x + 27, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 27 x^{16} - 52 x^{15} + 249 x^{14} - 378 x^{13} + 1144 x^{12} - 1320 x^{11} + 2610 x^{10} - 2630 x^{9} + 3594 x^{8} - 2052 x^{7} + 2454 x^{6} - 1296 x^{5} + 1647 x^{4} - 963 x^{3} + 567 x^{2} - 162 x + 27 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{9} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{10} + \frac{1}{6} a^{7} - \frac{1}{2} a^{6} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{12} a^{14} + \frac{1}{6} a^{11} - \frac{1}{6} a^{8} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{288} a^{15} - \frac{1}{32} a^{14} - \frac{1}{48} a^{13} - \frac{11}{144} a^{12} - \frac{5}{24} a^{11} + \frac{1}{48} a^{10} + \frac{35}{144} a^{9} + \frac{1}{48} a^{8} - \frac{7}{48} a^{7} - \frac{49}{144} a^{6} - \frac{11}{48} a^{5} + \frac{5}{12} a^{4} + \frac{7}{16} a^{3} + \frac{7}{16} a^{2} - \frac{7}{32} a + \frac{7}{32}$, $\frac{1}{288} a^{16} + \frac{1}{32} a^{14} + \frac{5}{72} a^{13} - \frac{1}{16} a^{12} - \frac{3}{16} a^{11} + \frac{7}{72} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} + \frac{13}{72} a^{7} + \frac{1}{24} a^{6} + \frac{3}{16} a^{5} - \frac{23}{48} a^{4} - \frac{7}{24} a^{3} - \frac{9}{32} a^{2} - \frac{1}{4} a + \frac{15}{32}$, $\frac{1}{259799469467366364768} a^{17} + \frac{191545389012242411}{129899734733683182384} a^{16} - \frac{37085662695292315}{32474933683420795596} a^{15} + \frac{7203144186943755443}{259799469467366364768} a^{14} - \frac{1820188985336922637}{64949867366841591192} a^{13} + \frac{2371394709029162945}{64949867366841591192} a^{12} - \frac{8443073237142724403}{64949867366841591192} a^{11} - \frac{13211725269706315633}{129899734733683182384} a^{10} + \frac{21732998272873445927}{129899734733683182384} a^{9} - \frac{17458275594149286997}{129899734733683182384} a^{8} + \frac{6911690384491709255}{129899734733683182384} a^{7} + \frac{2424558289288340936}{8118733420855198899} a^{6} + \frac{946548512802315067}{7216651929649065688} a^{5} + \frac{3046660919461861493}{10824977894473598532} a^{4} - \frac{17205447113794914829}{86599823155788788256} a^{3} - \frac{1056701616746440251}{7216651929649065688} a^{2} - \frac{5442494542566918445}{14433303859298131376} a + \frac{10896394388098387827}{28866607718596262752}$

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

Class group and class number

$C_{2}\times C_{14}$, which has order $28$

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

\(\Q(\zeta_{7})^+\), 3.3.3969.1, 3.3.3969.2, \(\Q(\zeta_{9})^+\), 6.0.465831.1, 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$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$$0$Trivial$[\ ]$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$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}$