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Properties

Label 18.0.70915645391...8432.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{6}\cdot 7^{15}\cdot 29^{8}$
Root discriminant $51.75$
Ramified primes $2, 3, 7, 29$
Class number $81$ (GRH)
Class group $[3, 27]$ (GRH)
Galois group $C_2\times He_3:C_2$ (as 18T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![89383, 0, 831922, 0, 2697233, 0, 3626295, 0, 1886613, 0, 420932, 0, 47384, 0, 2829, 0, 85, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 85*x^16 + 2829*x^14 + 47384*x^12 + 420932*x^10 + 1886613*x^8 + 3626295*x^6 + 2697233*x^4 + 831922*x^2 + 89383)
 
gp: K = bnfinit(x^18 + 85*x^16 + 2829*x^14 + 47384*x^12 + 420932*x^10 + 1886613*x^8 + 3626295*x^6 + 2697233*x^4 + 831922*x^2 + 89383, 1)
 

Normalized defining polynomial

\( x^{18} + 85 x^{16} + 2829 x^{14} + 47384 x^{12} + 420932 x^{10} + 1886613 x^{8} + 3626295 x^{6} + 2697233 x^{4} + 831922 x^{2} + 89383 \)

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:  \(-7091564539152983760000702738432=-\,2^{12}\cdot 3^{6}\cdot 7^{15}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.75$
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:  $2, 3, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{728} a^{12} - \frac{5}{364} a^{10} - \frac{95}{728} a^{8} + \frac{31}{364} a^{6} - \frac{1}{2} a^{5} + \frac{15}{52} a^{4} - \frac{1}{2} a^{2} - \frac{41}{104}$, $\frac{1}{728} a^{13} - \frac{5}{364} a^{11} + \frac{87}{728} a^{9} - \frac{1}{4} a^{8} - \frac{15}{91} a^{7} - \frac{11}{52} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{41}{104} a - \frac{1}{4}$, $\frac{1}{728} a^{14} - \frac{1}{56} a^{10} - \frac{20}{91} a^{8} - \frac{1}{4} a^{7} + \frac{51}{364} a^{6} - \frac{1}{4} a^{5} - \frac{3}{26} a^{4} - \frac{1}{2} a^{3} - \frac{15}{104} a^{2} - \frac{1}{4} a + \frac{4}{13}$, $\frac{1}{728} a^{15} - \frac{1}{56} a^{11} + \frac{11}{364} a^{9} - \frac{10}{91} a^{7} - \frac{1}{4} a^{6} + \frac{5}{13} a^{5} - \frac{1}{4} a^{4} - \frac{41}{104} a^{3} + \frac{4}{13} a - \frac{1}{4}$, $\frac{1}{5235530983776184} a^{16} + \frac{2320564150055}{5235530983776184} a^{14} + \frac{431763349393}{2617765491888092} a^{12} + \frac{288255826048973}{5235530983776184} a^{10} + \frac{9415612916685}{57533307514024} a^{8} + \frac{16906139355491}{93491624710289} a^{6} - \frac{1}{2} a^{5} - \frac{347201210713879}{747932997682312} a^{4} - \frac{1}{2} a^{3} + \frac{163879321895435}{747932997682312} a^{2} - \frac{1253739376231}{57533307514024}$, $\frac{1}{591615001166708792} a^{17} - \frac{386029261569607}{591615001166708792} a^{15} + \frac{90111674665359}{147903750291677198} a^{13} + \frac{752681229493675}{6501263749084712} a^{11} - \frac{62789400661970715}{591615001166708792} a^{9} - \frac{1}{4} a^{8} + \frac{5118318330262853}{21129107184525314} a^{7} - \frac{1}{4} a^{6} - \frac{33889119491389871}{84516428738101256} a^{5} + \frac{16647171924663311}{84516428738101256} a^{3} + \frac{1}{4} a^{2} + \frac{18250526523811617}{84516428738101256} a$

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

Class group and class number

$C_{3}\times C_{27}$, which has order $81$ (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:  \( -\frac{79618166364557}{591615001166708792} a^{17} + \frac{3427963479}{1308882745944046} a^{16} - \frac{6737486028277981}{591615001166708792} a^{15} + \frac{20138507901}{93491624710289} a^{14} - \frac{31815024878658785}{84516428738101256} a^{13} + \frac{4469383658963}{654441372972023} a^{12} - \frac{3689209519489847477}{591615001166708792} a^{11} + \frac{69411793591149}{654441372972023} a^{10} - \frac{4017360259230191103}{73951875145838599} a^{9} + \frac{1077002117378037}{1308882745944046} a^{8} - \frac{69148299274913147683}{295807500583354396} a^{7} + \frac{134390593119602}{50341644074771} a^{6} - \frac{33965195666038953815}{84516428738101256} a^{5} + \frac{39761001045336}{93491624710289} a^{4} - \frac{18236314724571599053}{84516428738101256} a^{3} - \frac{652360369688901}{93491624710289} a^{2} - \frac{360580447844055683}{10564553592262657} a - \frac{447820411359325}{186983249420578} \) (order $14$)
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:  \( 11573439.6965 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times He_3:C_2$ (as 18T42):

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 $C_2\times He_3:C_2$
Character table for $C_2\times He_3:C_2$

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{7})\), 9.9.1006519075055424.1

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

Sibling fields

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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$