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Properties

Label 18.0.10537754066...5744.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{30}\cdot 3^{31}\cdot 7^{9}\cdot 13^{14}$
Root discriminant $409.61$
Ramified primes $2, 3, 7, 13$
Class number $184620384$ (GRH)
Class group $[2, 2, 2, 6, 3846258]$ (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![8052186349, -6212683488, 2920222299, 228252030, 266849940, -114235734, 34894203, 4485624, 12321687, -2492512, 1383927, 142518, 162195, -22110, 4776, -66, 60, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 60*x^16 - 66*x^15 + 4776*x^14 - 22110*x^13 + 162195*x^12 + 142518*x^11 + 1383927*x^10 - 2492512*x^9 + 12321687*x^8 + 4485624*x^7 + 34894203*x^6 - 114235734*x^5 + 266849940*x^4 + 228252030*x^3 + 2920222299*x^2 - 6212683488*x + 8052186349)
 
gp: K = bnfinit(x^18 - 6*x^17 + 60*x^16 - 66*x^15 + 4776*x^14 - 22110*x^13 + 162195*x^12 + 142518*x^11 + 1383927*x^10 - 2492512*x^9 + 12321687*x^8 + 4485624*x^7 + 34894203*x^6 - 114235734*x^5 + 266849940*x^4 + 228252030*x^3 + 2920222299*x^2 - 6212683488*x + 8052186349, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 60 x^{16} - 66 x^{15} + 4776 x^{14} - 22110 x^{13} + 162195 x^{12} + 142518 x^{11} + 1383927 x^{10} - 2492512 x^{9} + 12321687 x^{8} + 4485624 x^{7} + 34894203 x^{6} - 114235734 x^{5} + 266849940 x^{4} + 228252030 x^{3} + 2920222299 x^{2} - 6212683488 x + 8052186349 \)

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:  \(-105377540665610698021122402053144975812532895744=-\,2^{30}\cdot 3^{31}\cdot 7^{9}\cdot 13^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $409.61$
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:  $2, 3, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{2} - \frac{1}{9} a + \frac{1}{9}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{9} + \frac{1}{9} a^{3} + \frac{1}{9}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{10} + \frac{1}{9} a^{4} + \frac{1}{9} a$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{1}{9} a^{5} + \frac{1}{9} a - \frac{1}{9}$, $\frac{1}{981} a^{15} - \frac{49}{981} a^{14} + \frac{16}{327} a^{13} + \frac{7}{981} a^{12} - \frac{47}{981} a^{11} - \frac{116}{981} a^{10} + \frac{74}{981} a^{9} + \frac{40}{327} a^{8} + \frac{53}{327} a^{7} + \frac{151}{981} a^{6} - \frac{358}{981} a^{5} + \frac{59}{327} a^{4} - \frac{296}{981} a^{3} - \frac{467}{981} a^{2} + \frac{331}{981} a - \frac{82}{981}$, $\frac{1}{120663} a^{16} + \frac{55}{120663} a^{15} + \frac{626}{13407} a^{14} - \frac{5683}{120663} a^{13} - \frac{4987}{120663} a^{12} - \frac{46}{981} a^{11} - \frac{140}{1107} a^{10} + \frac{949}{120663} a^{9} - \frac{525}{4469} a^{8} - \frac{11762}{120663} a^{7} + \frac{9460}{120663} a^{6} + \frac{3101}{13407} a^{5} + \frac{51575}{120663} a^{4} + \frac{55295}{120663} a^{3} - \frac{18041}{40221} a^{2} - \frac{17978}{120663} a - \frac{1189}{2943}$, $\frac{1}{4273225069772591538748562422312528379002503535480159978237207656187620597} a^{17} + \frac{4280948287361142889347427972204622247576694711173023456106915842686}{1424408356590863846249520807437509459667501178493386659412402552062540199} a^{16} - \frac{898729556734230557364133865599730988759403151324682615964685147971330}{4273225069772591538748562422312528379002503535480159978237207656187620597} a^{15} - \frac{76399320907928888438481834912996288760252664517076972916609714355926515}{4273225069772591538748562422312528379002503535480159978237207656187620597} a^{14} - \frac{41902224738015676302263113263163871416984646430076631698785945334895788}{1424408356590863846249520807437509459667501178493386659412402552062540199} a^{13} - \frac{20414344957445868761716986498828789467739872849099537627059714342545065}{4273225069772591538748562422312528379002503535480159978237207656187620597} a^{12} - \frac{11744385721531157852042143378889918967845179797212264636522717260514045}{4273225069772591538748562422312528379002503535480159978237207656187620597} a^{11} - \frac{217292811378360543738971268576173918826057544491843001032480877489871872}{1424408356590863846249520807437509459667501178493386659412402552062540199} a^{10} + \frac{122317409444945123452188497191340573509487274942222606604380408372343703}{4273225069772591538748562422312528379002503535480159978237207656187620597} a^{9} - \frac{689666642999184523594844121535960975427891336799156163121993530974730691}{4273225069772591538748562422312528379002503535480159978237207656187620597} a^{8} - \frac{210154916581900902553865946573760797944655623552423987403427331609015417}{1424408356590863846249520807437509459667501178493386659412402552062540199} a^{7} - \frac{3644547916242747439969328994675166078350124297648331492114414940765197}{104225001701770525335330790788110448268353744767808779957005064785063917} a^{6} + \frac{948573420770338010557425292149752547338070078862201065942100779440736344}{4273225069772591538748562422312528379002503535480159978237207656187620597} a^{5} - \frac{487349391817763675230768847829072530105255220933307250619031483808220442}{1424408356590863846249520807437509459667501178493386659412402552062540199} a^{4} - \frac{1509391567365676918936921004944124427774311432982498125678484708873337970}{4273225069772591538748562422312528379002503535480159978237207656187620597} a^{3} - \frac{1055215380463234223484394901686154274982536691022380892992124909979529247}{4273225069772591538748562422312528379002503535480159978237207656187620597} a^{2} + \frac{54205044896326286874540370049477970359215759165496879107122249842165726}{1424408356590863846249520807437509459667501178493386659412402552062540199} a - \frac{34993873679573881971437709042280091459432828351959822721879462082449012}{104225001701770525335330790788110448268353744767808779957005064785063917}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{6}\times C_{3846258}$, which has order $184620384$ (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:  \( 96135527.08384958 \) (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

\(\Q(\sqrt{-21}) \), 3.3.13689.2, 3.3.2808.1, 6.0.12340671610176.4, 6.0.129816400896.2, 9.9.460990789028310528.2

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.12.24.307$x^{12} + 28 x^{11} - 2 x^{10} + 16 x^{9} + 26 x^{8} + 8 x^{7} + 20 x^{6} - 24 x^{5} - 8 x^{4} + 32 x^{3} + 32 x^{2} + 32 x + 24$$4$$3$$24$$C_6\times C_2$$[2, 3]^{3}$
3Data not computed
$7$7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$13$13.6.4.2$x^{6} - 13 x^{3} + 338$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.12.10.3$x^{12} - 13 x^{6} + 2704$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$