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Properties

Label 18.0.43203768346...6016.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{16}\cdot 3^{33}\cdot 17^{9}$
Root discriminant $57.22$
Ramified primes $2, 3, 17$
Class number $12$ (GRH)
Class group $[2, 6]$ (GRH)
Galois group $C_2\times D_9:C_3$ (as 18T45)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1939407, 3181869, 4963797, 2069244, 2786436, -723762, 662610, -311886, 344619, -219663, 106011, -25632, 3096, -396, 432, -180, 51, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 51*x^16 - 180*x^15 + 432*x^14 - 396*x^13 + 3096*x^12 - 25632*x^11 + 106011*x^10 - 219663*x^9 + 344619*x^8 - 311886*x^7 + 662610*x^6 - 723762*x^5 + 2786436*x^4 + 2069244*x^3 + 4963797*x^2 + 3181869*x + 1939407)
 
gp: K = bnfinit(x^18 - 9*x^17 + 51*x^16 - 180*x^15 + 432*x^14 - 396*x^13 + 3096*x^12 - 25632*x^11 + 106011*x^10 - 219663*x^9 + 344619*x^8 - 311886*x^7 + 662610*x^6 - 723762*x^5 + 2786436*x^4 + 2069244*x^3 + 4963797*x^2 + 3181869*x + 1939407, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 51 x^{16} - 180 x^{15} + 432 x^{14} - 396 x^{13} + 3096 x^{12} - 25632 x^{11} + 106011 x^{10} - 219663 x^{9} + 344619 x^{8} - 311886 x^{7} + 662610 x^{6} - 723762 x^{5} + 2786436 x^{4} + 2069244 x^{3} + 4963797 x^{2} + 3181869 x + 1939407 \)

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:  \(-43203768346609530320797232726016=-\,2^{16}\cdot 3^{33}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.22$
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:  $2, 3, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{11} - \frac{2}{5} a^{10} + \frac{2}{5} a^{9} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{1}{5} a^{4} + \frac{1}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{1}{5} a^{5} + \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{25} a^{15} - \frac{1}{25} a^{14} + \frac{1}{25} a^{12} + \frac{2}{25} a^{11} + \frac{2}{25} a^{10} - \frac{11}{25} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{6}{25} a^{6} - \frac{4}{25} a^{5} + \frac{4}{25} a^{3} - \frac{7}{25} a^{2} + \frac{8}{25} a + \frac{11}{25}$, $\frac{1}{25} a^{16} - \frac{1}{25} a^{14} + \frac{1}{25} a^{13} - \frac{2}{25} a^{12} - \frac{6}{25} a^{11} + \frac{1}{25} a^{10} - \frac{1}{25} a^{9} - \frac{2}{5} a^{8} - \frac{11}{25} a^{7} - \frac{2}{5} a^{6} - \frac{4}{25} a^{5} + \frac{4}{25} a^{4} + \frac{2}{25} a^{3} + \frac{11}{25} a^{2} + \frac{9}{25} a - \frac{4}{25}$, $\frac{1}{39966893359969266887315481764524404459348473155770084483475} a^{17} + \frac{256264577638087617908765847985316166137455913536419722232}{39966893359969266887315481764524404459348473155770084483475} a^{16} + \frac{38944985351974779361994171873045139325750787509372482448}{39966893359969266887315481764524404459348473155770084483475} a^{15} - \frac{43846694322466477770981614093882823571643082904446548757}{1598675734398770675492619270580976178373938926230803379339} a^{14} - \frac{503889541542100823521707069486002298779804473928649781457}{7993378671993853377463096352904880891869694631154016896695} a^{13} + \frac{2156938979604797504773654008755112290591679581087566295324}{39966893359969266887315481764524404459348473155770084483475} a^{12} + \frac{4609852933764596659212125219438253711966904045581158972902}{39966893359969266887315481764524404459348473155770084483475} a^{11} - \frac{3069276248755652593773700152582978954183861286205443361531}{39966893359969266887315481764524404459348473155770084483475} a^{10} - \frac{7237286308224322906702773157463970189737454695097025237021}{39966893359969266887315481764524404459348473155770084483475} a^{9} - \frac{17629381965746738239523698289990915566574603554739569006476}{39966893359969266887315481764524404459348473155770084483475} a^{8} - \frac{4206808164642647836011476620616614753953459218837070771232}{39966893359969266887315481764524404459348473155770084483475} a^{7} - \frac{481443613943981015579429199772857534611497917071278351318}{39966893359969266887315481764524404459348473155770084483475} a^{6} - \frac{579920805255501556576424957150257731928370286026501704369}{1598675734398770675492619270580976178373938926230803379339} a^{5} + \frac{2733326583191505294920225357927850330221306039114570919669}{7993378671993853377463096352904880891869694631154016896695} a^{4} + \frac{13792771562436006061258589778268243039072632909718787417251}{39966893359969266887315481764524404459348473155770084483475} a^{3} - \frac{13125628712326361785992783981484118320969801463764411553027}{39966893359969266887315481764524404459348473155770084483475} a^{2} - \frac{18067374678437851629395752019592361950524102298783774446939}{39966893359969266887315481764524404459348473155770084483475} a + \frac{18626045964253860841891564378930495935246111915475843761226}{39966893359969266887315481764524404459348473155770084483475}$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$ (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:  \( -1 \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 54838383.634739265 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_9:C_3$ (as 18T45):

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 $C_2\times D_9:C_3$
Character table for $C_2\times D_9:C_3$

Intermediate fields

\(\Q(\sqrt{-51}) \), 3.1.108.1, 6.0.171915696.7, 9.1.11019960576.1

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

Sibling fields

Degree 18 sibling: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$17$17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$