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Properties

Label 18.0.11881378369...6944.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{33}\cdot 7^{15}\cdot 79^{15}$
Root discriminant $687.85$
Ramified primes $2, 7, 79$
Class number $6636349440$ (GRH)
Class group $[2, 2, 2, 2, 12, 12, 2880360]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![723973068517952, -470177345986688, 300717932225024, -93030379477632, 19811453012448, -1781407473888, 487968322552, 86405002144, 7170164721, 3688040186, 324233435, 41131796, 7791772, 192596, 75140, -108, 427, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 427*x^16 - 108*x^15 + 75140*x^14 + 192596*x^13 + 7791772*x^12 + 41131796*x^11 + 324233435*x^10 + 3688040186*x^9 + 7170164721*x^8 + 86405002144*x^7 + 487968322552*x^6 - 1781407473888*x^5 + 19811453012448*x^4 - 93030379477632*x^3 + 300717932225024*x^2 - 470177345986688*x + 723973068517952)
 
gp: K = bnfinit(x^18 - 6*x^17 + 427*x^16 - 108*x^15 + 75140*x^14 + 192596*x^13 + 7791772*x^12 + 41131796*x^11 + 324233435*x^10 + 3688040186*x^9 + 7170164721*x^8 + 86405002144*x^7 + 487968322552*x^6 - 1781407473888*x^5 + 19811453012448*x^4 - 93030379477632*x^3 + 300717932225024*x^2 - 470177345986688*x + 723973068517952, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 427 x^{16} - 108 x^{15} + 75140 x^{14} + 192596 x^{13} + 7791772 x^{12} + 41131796 x^{11} + 324233435 x^{10} + 3688040186 x^{9} + 7170164721 x^{8} + 86405002144 x^{7} + 487968322552 x^{6} - 1781407473888 x^{5} + 19811453012448 x^{4} - 93030379477632 x^{3} + 300717932225024 x^{2} - 470177345986688 x + 723973068517952 \)

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:  \(-1188137836995213093753756971476827984600856939986944=-\,2^{33}\cdot 7^{15}\cdot 79^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $687.85$
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:  $2, 7, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{4} a^{3}$, $\frac{1}{16} a^{12} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{7}{16} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{12} - \frac{1}{16} a^{11} + \frac{1}{16} a^{10} - \frac{1}{16} a^{9} + \frac{3}{16} a^{8} + \frac{3}{16} a^{7} - \frac{1}{16} a^{6} - \frac{11}{32} a^{5} + \frac{15}{32} a^{4} - \frac{1}{4} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{992} a^{14} + \frac{1}{496} a^{13} - \frac{19}{992} a^{12} - \frac{7}{248} a^{11} - \frac{1}{248} a^{10} - \frac{1}{62} a^{9} + \frac{9}{62} a^{8} - \frac{27}{124} a^{7} + \frac{207}{992} a^{6} - \frac{49}{496} a^{5} + \frac{27}{992} a^{4} - \frac{95}{248} a^{3} + \frac{85}{248} a^{2} - \frac{19}{62} a + \frac{1}{4}$, $\frac{1}{1984} a^{15} - \frac{23}{1984} a^{13} - \frac{13}{496} a^{12} + \frac{13}{496} a^{11} + \frac{15}{124} a^{10} + \frac{11}{124} a^{9} - \frac{4}{31} a^{8} - \frac{105}{1984} a^{7} - \frac{1}{124} a^{6} + \frac{223}{1984} a^{5} + \frac{3}{16} a^{4} + \frac{89}{496} a^{3} - \frac{23}{62} a^{2} + \frac{107}{248} a$, $\frac{1}{21824} a^{16} + \frac{3}{21824} a^{15} - \frac{9}{21824} a^{14} - \frac{3}{704} a^{13} - \frac{123}{10912} a^{12} + \frac{63}{5456} a^{11} - \frac{87}{1364} a^{10} - \frac{335}{2728} a^{9} - \frac{3073}{21824} a^{8} + \frac{4333}{21824} a^{7} - \frac{149}{1984} a^{6} + \frac{263}{1984} a^{5} + \frac{4211}{10912} a^{4} + \frac{613}{5456} a^{3} + \frac{29}{682} a^{2} - \frac{1203}{2728} a - \frac{1}{4}$, $\frac{1}{46800087877642084973296072407297948232433037487110768729890517615485483949749708026445328235890293559693506176} a^{17} + \frac{485088957131350753365917622503776056483363209489656576810815286855483652942443719382026033495028301197529}{23400043938821042486648036203648974116216518743555384364945258807742741974874854013222664117945146779846753088} a^{16} - \frac{168117483025790426807475764230090661472449681057602479569978604908004340666717520710897560350002780768643}{1509680254117486612041808787332191878465581854422928023544855406951144643540313162143397685028719147086887296} a^{15} - \frac{888780029006610340169008138844827939708471128942015882961596346768302772327947110341728750217502735105873}{5850010984705260621662009050912243529054129685888846091236314701935685493718713503305666029486286694961688272} a^{14} + \frac{136927147740968084113362274524633110564987981616607988533346043969482007785778940704369106400769918270153095}{11700021969410521243324018101824487058108259371777692182472629403871370987437427006611332058972573389923376544} a^{13} + \frac{39418208945325419320216372203565204407302961557377893809873891884470104187828089558941942369965797834737477}{5850010984705260621662009050912243529054129685888846091236314701935685493718713503305666029486286694961688272} a^{12} - \frac{119144335345390033875876187888128895170806676479355892584011598833397112433123129350070991068928140884738035}{11700021969410521243324018101824487058108259371777692182472629403871370987437427006611332058972573389923376544} a^{11} - \frac{544487663709753923121275861426441328679524862587246212013537297413411541254501494793461279678063376559489327}{11700021969410521243324018101824487058108259371777692182472629403871370987437427006611332058972573389923376544} a^{10} + \frac{1373152669178218135965680172242156366312613526091044271598813946213162197533811546125750825012057133998637}{12476696315020550512742221382910676681533734334073785318552524024389625153225728612755352768832389645346176} a^{9} + \frac{36823578863497799538429442256247748222210125720193914684001821867641128254121995078987830344840475217070529}{23400043938821042486648036203648974116216518743555384364945258807742741974874854013222664117945146779846753088} a^{8} + \frac{4937247210634037497321718878972748071312117021989497093324652837568153658538703767526311812817349764088395289}{46800087877642084973296072407297948232433037487110768729890517615485483949749708026445328235890293559693506176} a^{7} - \frac{247092757115874780683113995911564618550639389429388797416319108871594003966975307881153199922923796716556811}{1063638360855501931211274372893135187100750851979790198406602673079215544312493364237393823542961217265761504} a^{6} - \frac{33022328708448098480854079637164522541871186963953641807186756787490640715157048012845812837832453827539617}{365625686544078788853875565682015220565883105368052880702269668870980343357419593956604126842892918435105517} a^{5} - \frac{2233584769188208619057056789807303564731943001787172192056458528732720406767263250854194811636188657810220517}{11700021969410521243324018101824487058108259371777692182472629403871370987437427006611332058972573389923376544} a^{4} - \frac{1178886193086808664833922506084426191314114509943241801715233822253215475911955643553486693387492716077512091}{5850010984705260621662009050912243529054129685888846091236314701935685493718713503305666029486286694961688272} a^{3} - \frac{239161928968579212847812612347423076917699324141233725294447533308114485078333629245784504798622101453910253}{2925005492352630310831004525456121764527064842944423045618157350967842746859356751652833014743143347480844136} a^{2} - \frac{529725310310424740997911075816153501569153126545329947988106568955086555235771792168487484887674017093129591}{1462502746176315155415502262728060882263532421472211522809078675483921373429678375826416507371571673740422068} a + \frac{45903090430554649855299561481237656950064544263184119889253176467413367180116679182059402381827339211215}{4288864358288314238755138600375545109277221177337863703252430133383933646421344210634652514286133940587748}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{12}\times C_{12}\times C_{2880360}$, which has order $6636349440$ (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:  \( 20721910881.979282 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-1106}) \), 3.3.305809.1, 3.3.49928.1, 6.0.26478636491772416.1, Deg 6, 9.9.14642685979950146048.3

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
7Data not computed
79Data not computed