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Properties

Label 18.0.49074206261...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{31}\cdot 5^{9}\cdot 7^{9}\cdot 67^{15}$
Root discriminant $2070.90$
Ramified primes $2, 3, 5, 7, 67$
Class number $11913096384$ (GRH)
Class group $[6, 6, 6, 378, 145908]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![463589489758381849, -151213514911900860, 17012485372731978, 2951327570583606, -281050838668113, -93984714115572, 18917369006865, 1253829209316, -302274108753, -17494139554, 4227681804, 153139332, -45303006, -1075578, 272031, 5406, -687, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 687*x^16 + 5406*x^15 + 272031*x^14 - 1075578*x^13 - 45303006*x^12 + 153139332*x^11 + 4227681804*x^10 - 17494139554*x^9 - 302274108753*x^8 + 1253829209316*x^7 + 18917369006865*x^6 - 93984714115572*x^5 - 281050838668113*x^4 + 2951327570583606*x^3 + 17012485372731978*x^2 - 151213514911900860*x + 463589489758381849)
 
gp: K = bnfinit(x^18 - 6*x^17 - 687*x^16 + 5406*x^15 + 272031*x^14 - 1075578*x^13 - 45303006*x^12 + 153139332*x^11 + 4227681804*x^10 - 17494139554*x^9 - 302274108753*x^8 + 1253829209316*x^7 + 18917369006865*x^6 - 93984714115572*x^5 - 281050838668113*x^4 + 2951327570583606*x^3 + 17012485372731978*x^2 - 151213514911900860*x + 463589489758381849, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 687 x^{16} + 5406 x^{15} + 272031 x^{14} - 1075578 x^{13} - 45303006 x^{12} + 153139332 x^{11} + 4227681804 x^{10} - 17494139554 x^{9} - 302274108753 x^{8} + 1253829209316 x^{7} + 18917369006865 x^{6} - 93984714115572 x^{5} - 281050838668113 x^{4} + 2951327570583606 x^{3} + 17012485372731978 x^{2} - 151213514911900860 x + 463589489758381849 \)

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:  \(-490742062616166373662983522813549807415554217709176000000000=-\,2^{12}\cdot 3^{31}\cdot 5^{9}\cdot 7^{9}\cdot 67^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2070.90$
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:  $2, 3, 5, 7, 67$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{7} a^{4} + \frac{1}{7} a^{3} - \frac{1}{7} a^{2} + \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{147} a^{5} - \frac{1}{147} a^{4} - \frac{38}{147} a^{3} - \frac{38}{147} a^{2} - \frac{10}{21} a - \frac{8}{147}$, $\frac{1}{147} a^{6} + \frac{1}{49} a^{4} - \frac{34}{147} a^{3} - \frac{1}{49} a^{2} + \frac{2}{49} a + \frac{13}{147}$, $\frac{1}{1029} a^{7} + \frac{2}{1029} a^{6} - \frac{2}{1029} a^{5} - \frac{44}{1029} a^{4} - \frac{1}{21} a^{3} - \frac{377}{1029} a^{2} - \frac{85}{343} a - \frac{55}{343}$, $\frac{1}{7203} a^{8} + \frac{1}{7203} a^{6} - \frac{4}{2401} a^{5} - \frac{262}{7203} a^{4} - \frac{625}{2401} a^{3} + \frac{1766}{7203} a^{2} - \frac{3190}{7203} a - \frac{3037}{7203}$, $\frac{1}{907578} a^{9} + \frac{11}{302526} a^{8} - \frac{65}{302526} a^{7} + \frac{11}{21609} a^{6} - \frac{25}{21609} a^{5} + \frac{1487}{43218} a^{4} - \frac{16349}{43218} a^{3} - \frac{46693}{302526} a^{2} + \frac{16208}{151263} a - \frac{241145}{907578}$, $\frac{1}{907578} a^{10} - \frac{4}{151263} a^{8} - \frac{53}{302526} a^{7} + \frac{41}{21609} a^{6} - \frac{145}{43218} a^{5} + \frac{1514}{21609} a^{4} + \frac{33700}{151263} a^{3} - \frac{641}{6174} a^{2} + \frac{46675}{907578} a - \frac{124109}{302526}$, $\frac{1}{6353046} a^{11} + \frac{1}{6353046} a^{10} - \frac{1}{6353046} a^{9} + \frac{6}{117649} a^{8} - \frac{46}{1058841} a^{7} - \frac{493}{302526} a^{6} - \frac{433}{302526} a^{5} - \frac{131267}{2117682} a^{4} + \frac{166714}{1058841} a^{3} + \frac{116125}{6353046} a^{2} - \frac{653053}{3176523} a - \frac{384017}{3176523}$, $\frac{1}{222356610} a^{12} - \frac{1}{222356610} a^{11} - \frac{2}{4117715} a^{10} - \frac{29}{74118870} a^{9} + \frac{81}{1176490} a^{8} - \frac{30539}{74118870} a^{7} + \frac{1769}{1512630} a^{6} + \frac{75161}{37059435} a^{5} + \frac{691463}{74118870} a^{4} - \frac{43783282}{111178305} a^{3} - \frac{2307499}{22235661} a^{2} - \frac{4169063}{10588410} a + \frac{13583141}{37059435}$, $\frac{1}{222356610} a^{13} - \frac{2}{111178305} a^{11} - \frac{1}{2470629} a^{10} - \frac{73}{222356610} a^{9} + \frac{2017}{37059435} a^{8} + \frac{12706}{37059435} a^{7} + \frac{120829}{37059435} a^{6} + \frac{1256}{1058841} a^{5} + \frac{1560319}{22235661} a^{4} - \frac{9513719}{37059435} a^{3} - \frac{26962499}{111178305} a^{2} - \frac{1021313}{24706290} a - \frac{20770007}{111178305}$, $\frac{1}{32686421670} a^{14} + \frac{1}{16343210835} a^{13} - \frac{13}{6537284334} a^{12} + \frac{523}{32686421670} a^{11} + \frac{158}{466948881} a^{10} - \frac{4756}{16343210835} a^{9} - \frac{37979}{5447736945} a^{8} - \frac{3255779}{10895473890} a^{7} - \frac{2589269}{2179094778} a^{6} - \frac{63459581}{32686421670} a^{5} - \frac{445560449}{16343210835} a^{4} - \frac{2202504791}{4669488810} a^{3} + \frac{8688015187}{32686421670} a^{2} + \frac{785320580}{3268642167} a - \frac{371839177}{16343210835}$, $\frac{1}{2406799286827110} a^{15} - \frac{1}{366554871585} a^{14} + \frac{2620657}{1203399643413555} a^{13} - \frac{339631}{160453285788474} a^{12} - \frac{104754191}{2406799286827110} a^{11} - \frac{146068499}{1203399643413555} a^{10} + \frac{199597819}{1203399643413555} a^{9} - \frac{5771655877}{114609489848910} a^{8} + \frac{131348873417}{401133214471185} a^{7} - \frac{289796522249}{240679928682711} a^{6} - \frac{4068524529694}{1203399643413555} a^{5} - \frac{4793965282972}{240679928682711} a^{4} - \frac{52778797198619}{160453285788474} a^{3} + \frac{353163015661937}{1203399643413555} a^{2} + \frac{93942495595754}{240679928682711} a + \frac{826240901376119}{2406799286827110}$, $\frac{1}{2406799286827110} a^{16} - \frac{2368}{240679928682711} a^{14} - \frac{119501}{802266428942370} a^{13} - \frac{1736117}{1203399643413555} a^{12} - \frac{7713968}{1203399643413555} a^{11} + \frac{47970449}{2406799286827110} a^{10} + \frac{45243904}{171914234773365} a^{9} - \frac{4144082692}{401133214471185} a^{8} - \frac{411789821311}{1203399643413555} a^{7} + \frac{1077140516624}{401133214471185} a^{6} - \frac{1488931246993}{2406799286827110} a^{5} - \frac{6361212611108}{401133214471185} a^{4} - \frac{291716531436439}{2406799286827110} a^{3} + \frac{127471966253308}{1203399643413555} a^{2} - \frac{99352340248423}{240679928682711} a - \frac{1475349347407}{5131768202190}$, $\frac{1}{15680679008467628450652106586493922686121004022419190} a^{17} + \frac{485345204888333411990557571249517038}{2613446501411271408442017764415653781020167337069865} a^{16} - \frac{635339365620837934100023249964056799}{15680679008467628450652106586493922686121004022419190} a^{15} - \frac{88145928776570739297319446507031451739562}{7840339504233814225326053293246961343060502011209595} a^{14} - \frac{365217917870731065409994148016515860693908}{1120048500604830603618007613320994477580071715887085} a^{13} - \frac{132676129367728260828077941494577975429273}{448019400241932241447203045328397791032028686354834} a^{12} - \frac{4113916219144064367507866871217180212374623}{106671285771888628916000725078189950245721115798770} a^{11} + \frac{1038311120058991866355972174005884532520004461}{15680679008467628450652106586493922686121004022419190} a^{10} - \frac{2478960122200512859254266728438399920274375921}{5226893002822542816884035528831307562040334674139730} a^{9} + \frac{221224298817721714478835446132158633845605539321}{15680679008467628450652106586493922686121004022419190} a^{8} + \frac{489155570799870787833244035744001821844404509789}{1742297667607514272294678509610435854013444891379910} a^{7} + \frac{2355781942613626350931093123701371334469451289749}{1120048500604830603618007613320994477580071715887085} a^{6} - \frac{1673529682044403098880625943478805531225889572578}{1120048500604830603618007613320994477580071715887085} a^{5} - \frac{1235732578354687931633694657824573424351631098472}{160006928657832943374001087617284925368581673698155} a^{4} - \frac{4559586045265391828367341338335050913203465476625501}{15680679008467628450652106586493922686121004022419190} a^{3} + \frac{252048112068586392643793459281769557078999583228663}{580765889202504757431559503203478618004481630459970} a^{2} - \frac{172325242366752166072133128514006061287928634893208}{7840339504233814225326053293246961343060502011209595} a + \frac{56781081083299004365514134326203085766162418120}{125435397235962150633166199395999701512847004419}$

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

Class group and class number

$C_{6}\times C_{6}\times C_{6}\times C_{378}\times C_{145908}$, which has order $11913096384$ (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:  \( 7232933732765.735 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-7035}) \), 3.1.253260.2 x3, 3.3.363609.1, 6.0.451229315166000.1, Deg 6, Deg 6 x2, 9.3.238630827218079585300984000.1 x3

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

Sibling fields

Degree 6 sibling: data not computed
Degree 9 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 R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
$67$67.6.5.3$x^{6} - 17152$$6$$1$$5$$C_6$$[\ ]_{6}$
67.6.5.3$x^{6} - 17152$$6$$1$$5$$C_6$$[\ ]_{6}$
67.6.5.3$x^{6} - 17152$$6$$1$$5$$C_6$$[\ ]_{6}$