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Properties

Label 18.0.51620863001...8544.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{6}\cdot 11^{9}\cdot 523^{14}$
Root discriminant $1244.96$
Ramified primes $2, 3, 11, 523$
Class number $15863054324$ (GRH)
Class group $[2, 7931527162]$ (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![10883991773043, 6621321395790, 187960310919, -1043363367495, -27367760547, 195609354075, 32763269619, -13326944589, -3261383845, 437794222, 144682296, -7365053, -3490681, 56007, 47557, -46, -339, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 339*x^16 - 46*x^15 + 47557*x^14 + 56007*x^13 - 3490681*x^12 - 7365053*x^11 + 144682296*x^10 + 437794222*x^9 - 3261383845*x^8 - 13326944589*x^7 + 32763269619*x^6 + 195609354075*x^5 - 27367760547*x^4 - 1043363367495*x^3 + 187960310919*x^2 + 6621321395790*x + 10883991773043)
 
gp: K = bnfinit(x^18 - x^17 - 339*x^16 - 46*x^15 + 47557*x^14 + 56007*x^13 - 3490681*x^12 - 7365053*x^11 + 144682296*x^10 + 437794222*x^9 - 3261383845*x^8 - 13326944589*x^7 + 32763269619*x^6 + 195609354075*x^5 - 27367760547*x^4 - 1043363367495*x^3 + 187960310919*x^2 + 6621321395790*x + 10883991773043, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 339 x^{16} - 46 x^{15} + 47557 x^{14} + 56007 x^{13} - 3490681 x^{12} - 7365053 x^{11} + 144682296 x^{10} + 437794222 x^{9} - 3261383845 x^{8} - 13326944589 x^{7} + 32763269619 x^{6} + 195609354075 x^{5} - 27367760547 x^{4} - 1043363367495 x^{3} + 187960310919 x^{2} + 6621321395790 x + 10883991773043 \)

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:  \(-51620863001368675404465326891529689310638927567087468544=-\,2^{18}\cdot 3^{6}\cdot 11^{9}\cdot 523^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1244.96$
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:  $2, 3, 11, 523$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{9} a^{5} + \frac{1}{9} a^{4} - \frac{4}{9} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{4} + \frac{2}{9} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{4}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{5}$, $\frac{1}{18} a^{12} - \frac{1}{18} a^{10} - \frac{1}{18} a^{6} + \frac{1}{18} a^{4} - \frac{1}{2}$, $\frac{1}{54} a^{13} + \frac{1}{54} a^{12} + \frac{1}{54} a^{11} - \frac{1}{18} a^{10} + \frac{5}{54} a^{7} + \frac{5}{54} a^{6} - \frac{1}{54} a^{5} + \frac{5}{18} a^{4} - \frac{4}{9} a^{3} - \frac{1}{3} a^{2} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{162} a^{14} + \frac{1}{162} a^{13} + \frac{2}{81} a^{12} - \frac{1}{54} a^{11} - \frac{1}{54} a^{10} - \frac{1}{27} a^{9} - \frac{7}{162} a^{8} + \frac{17}{162} a^{7} + \frac{10}{81} a^{6} - \frac{1}{6} a^{5} + \frac{7}{54} a^{4} - \frac{2}{9} a^{3} - \frac{7}{18} a^{2} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{162} a^{15} - \frac{1}{162} a^{12} - \frac{1}{54} a^{11} - \frac{1}{54} a^{10} - \frac{1}{162} a^{9} + \frac{1}{27} a^{8} + \frac{1}{27} a^{7} + \frac{19}{162} a^{6} + \frac{5}{54} a^{5} + \frac{17}{54} a^{4} + \frac{1}{18} a^{3} - \frac{4}{9} a^{2} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{4266659241924822} a^{16} + \frac{434971609613}{4266659241924822} a^{15} + \frac{14670056923}{11562762173238} a^{14} + \frac{1512107501}{581367930498} a^{13} + \frac{109192139983873}{4266659241924822} a^{12} - \frac{28301942004088}{711109873654137} a^{11} + \frac{65388158784001}{2133329620962411} a^{10} - \frac{180919181323199}{4266659241924822} a^{9} - \frac{19720664969735}{474073249102758} a^{8} + \frac{561130088114719}{4266659241924822} a^{7} - \frac{462028725900817}{4266659241924822} a^{6} + \frac{40444626669094}{711109873654137} a^{5} - \frac{250916977642876}{711109873654137} a^{4} - \frac{47529074849}{882817968534} a^{3} + \frac{41958673563748}{237036624551379} a^{2} - \frac{12868297859927}{79012208183793} a - \frac{23563679575529}{52674805455862}$, $\frac{1}{195662776036544418950895976821024306142406822982899959613967226473348114} a^{17} + \frac{3912997598525681325184430136665109199592377295682036031}{97831388018272209475447988410512153071203411491449979806983613236674057} a^{16} + \frac{18934928128546215648782690819853937753763584130483250642338788711279}{21740308448504935438988441869002700682489646998099995512663025163705346} a^{15} - \frac{4471136480275507570476053719473183540594044210386096385415188612428}{2386131415079809987206048497817369587102522231498779995292283249674977} a^{14} - \frac{797306375194409627050174020979659938476597095412193068551977317454230}{97831388018272209475447988410512153071203411491449979806983613236674057} a^{13} + \frac{59966249470293962372208340995398212718521954426584436061222848966493}{7246769482834978479662813956334233560829882332699998504221008387901782} a^{12} - \frac{10210865842193567725900076077908256721202307249008820900072024932467449}{195662776036544418950895976821024306142406822982899959613967226473348114} a^{11} + \frac{5908348020229725668026113534369454643570050076405188475063480957839039}{195662776036544418950895976821024306142406822982899959613967226473348114} a^{10} + \frac{1031659625468029931571992650896014049366960437895080005431319576190903}{21740308448504935438988441869002700682489646998099995512663025163705346} a^{9} - \frac{4696541952421655761416979116952762480699210066476173256945738077870470}{97831388018272209475447988410512153071203411491449979806983613236674057} a^{8} - \frac{15805982536664637048694647166991064152340741998586017443779182074458857}{97831388018272209475447988410512153071203411491449979806983613236674057} a^{7} - \frac{74433396422060202111621545836547416426822272079025344261920769274227}{1278841673441466790528731874647217687205273352829411500744883833159138} a^{6} - \frac{7878842683359826091205223651543392401616360690114608061981944140795709}{65220925345514806316965325607008102047468940994299986537989075491116038} a^{5} - \frac{334612575598709081268838773809472202939475911673730714588290122327009}{21740308448504935438988441869002700682489646998099995512663025163705346} a^{4} - \frac{4563132355467163746705573506285882966708754020057641560806646830212067}{10870154224252467719494220934501350341244823499049997756331512581852673} a^{3} + \frac{34362982630590232902650279626116716272944344104711816455718071069343}{71046759635637043918262881924845427066959630712745083374715768508841} a^{2} + \frac{47507884282062650384952050640857520359159115392926006493927770507943}{402598304601943248870156330907457420046104574038888805790056021550099} a - \frac{7275843033889886777638988913580260154877411246376085929717555857403}{47364506423758029278841921283230284711306420475163388916477179005894}$

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

Class group and class number

$C_{2}\times C_{7931527162}$, which has order $15863054324$ (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:  \( 1120251490200.2412 \) (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

\(\Q(\sqrt{-11}) \), 3.3.273529.1, 3.3.12552.1, 6.0.99582909522371.1, 6.0.209702649024.1, 9.9.147960417197346328355328.1

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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
$2$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.18.23$x^{12} + 52 x^{10} - 28 x^{8} + 8 x^{6} + 64 x^{4} - 32 x^{2} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$11$11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
523Data not computed