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Properties

Label 18.0.93735171709...7664.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{30}\cdot 3^{18}\cdot 41^{3}\cdot 83^{6}$
Root discriminant $77.15$
Ramified primes $2, 3, 41, 83$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 18T903

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![20992, -94464, 212544, -286464, 239880, -108036, 3393, 24432, -11268, 728, 870, -828, 318, 72, -18, -8, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^16 - 8*x^15 - 18*x^14 + 72*x^13 + 318*x^12 - 828*x^11 + 870*x^10 + 728*x^9 - 11268*x^8 + 24432*x^7 + 3393*x^6 - 108036*x^5 + 239880*x^4 - 286464*x^3 + 212544*x^2 - 94464*x + 20992)
 
gp: K = bnfinit(x^18 - 6*x^16 - 8*x^15 - 18*x^14 + 72*x^13 + 318*x^12 - 828*x^11 + 870*x^10 + 728*x^9 - 11268*x^8 + 24432*x^7 + 3393*x^6 - 108036*x^5 + 239880*x^4 - 286464*x^3 + 212544*x^2 - 94464*x + 20992, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{16} - 8 x^{15} - 18 x^{14} + 72 x^{13} + 318 x^{12} - 828 x^{11} + 870 x^{10} + 728 x^{9} - 11268 x^{8} + 24432 x^{7} + 3393 x^{6} - 108036 x^{5} + 239880 x^{4} - 286464 x^{3} + 212544 x^{2} - 94464 x + 20992 \)

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:  \(-9373517170938828381535914021617664=-\,2^{30}\cdot 3^{18}\cdot 41^{3}\cdot 83^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $77.15$
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:  $2, 3, 41, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{64} a^{14} - \frac{1}{8} a^{13} + \frac{13}{32} a^{12} - \frac{3}{8} a^{11} - \frac{9}{32} a^{10} + \frac{3}{8} a^{9} - \frac{1}{32} a^{8} + \frac{5}{16} a^{7} + \frac{3}{32} a^{6} - \frac{3}{8} a^{5} - \frac{1}{16} a^{4} + \frac{1}{4} a^{3} + \frac{1}{64} a^{2} - \frac{3}{16} a + \frac{1}{8}$, $\frac{1}{256} a^{15} - \frac{19}{128} a^{13} - \frac{9}{32} a^{12} + \frac{55}{128} a^{11} - \frac{7}{32} a^{10} - \frac{1}{128} a^{9} + \frac{17}{64} a^{8} + \frac{19}{128} a^{7} - \frac{5}{32} a^{6} - \frac{17}{64} a^{5} - \frac{1}{16} a^{4} + \frac{65}{256} a^{3} - \frac{1}{64} a^{2} - \frac{3}{32} a + \frac{1}{4}$, $\frac{1}{8192} a^{16} + \frac{3}{2048} a^{15} - \frac{31}{4096} a^{14} + \frac{59}{512} a^{13} - \frac{433}{4096} a^{12} - \frac{157}{512} a^{11} - \frac{2041}{4096} a^{10} + \frac{507}{2048} a^{9} - \frac{1085}{4096} a^{8} + \frac{51}{128} a^{7} + \frac{787}{2048} a^{6} - \frac{3}{32} a^{5} + \frac{3553}{8192} a^{4} - \frac{239}{1024} a^{3} - \frac{67}{256} a^{2} + \frac{9}{128} a + \frac{17}{128}$, $\frac{1}{97174846048156062278434816} a^{17} + \frac{1475037367229912590709}{24293711512039015569608704} a^{16} - \frac{812305558647435721877}{2557232790740949007327232} a^{15} - \frac{8418988700714361268399}{1518356969502438473100544} a^{14} + \frac{9599826444088301317949647}{48587423024078031139217408} a^{13} - \frac{26978391694246798466733}{64610934872444190344704} a^{12} + \frac{10981594585782783682122759}{48587423024078031139217408} a^{11} + \frac{4029795048363902295911479}{24293711512039015569608704} a^{10} - \frac{10334042890439532120207}{214041511119286480789504} a^{9} - \frac{2662652114643440123545593}{6073427878009753892402176} a^{8} - \frac{9527390801472651495406413}{24293711512039015569608704} a^{7} - \frac{514670039436687739607037}{3036713939004876946201088} a^{6} - \frac{1797400080773638094663749}{5114465581481898014654464} a^{5} + \frac{612256084852531103332565}{6073427878009753892402176} a^{4} - \frac{573211419901181777877707}{3036713939004876946201088} a^{3} + \frac{710433174231870520203245}{1518356969502438473100544} a^{2} - \frac{711877863451746338198271}{1518356969502438473100544} a + \frac{19321242053576539170493}{189794621187804809137568}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( \frac{31768113209358041829}{1518356969502438473100544} a^{17} - \frac{120184799374643726697}{759178484751219236550272} a^{16} - \frac{23357103403040031437}{39956762355327328239488} a^{15} - \frac{97088816046955916115}{379589242375609618275136} a^{14} + \frac{1128688684431995239179}{759178484751219236550272} a^{13} + \frac{80395521499261285581}{8076366859055523793088} a^{12} + \frac{12417395581733548864707}{759178484751219236550272} a^{11} - \frac{10446687814831583539581}{189794621187804809137568} a^{10} + \frac{72938094597316799609}{3344398611238851262336} a^{9} - \frac{16569688792158666855333}{379589242375609618275136} a^{8} - \frac{200871555629538248309241}{379589242375609618275136} a^{7} + \frac{294862252750319166599315}{189794621187804809137568} a^{6} - \frac{19008848312510544265977}{79913524710654656478976} a^{5} - \frac{4230729911276381301680979}{759178484751219236550272} a^{4} + \frac{618060060759703487370659}{47448655296951202284392} a^{3} - \frac{690118604223860173127499}{47448655296951202284392} a^{2} + \frac{246824924262841572033801}{23724327648475601142196} a - \frac{41620373157065186208763}{11862163824237800571098} \) (order $4$)
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:  \( 23256887564.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T903:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 559872
The 174 conjugacy class representatives for t18n903 are not computed
Character table for t18n903 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 3.1.83.1, 6.0.440896.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ $18$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ $18$

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$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.8.16.21$x^{8} + 12 x^{7} + 20 x^{5} + 16 x^{4} + 40 x + 20$$4$$2$$16$$Q_8:C_2$$[2, 3, 3]^{2}$
3Data not computed
41Data not computed
$83$83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.12.6.1$x^{12} + 38881516 x^{6} - 3939040643 x^{2} + 377943071614564$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$