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Properties

Label 18.0.33612254239...8848.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{30}\cdot 7^{9}\cdot 61^{14}$
Root discriminant $641.25$
Ramified primes $2, 3, 7, 61$
Class number $6237995652$ (GRH)
Class group $[6, 1039665942]$ (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![2489858275328, -5257925494272, 4853863902720, -2247082834368, 343019618880, 152818776720, -78281565936, 5096936940, 4500825492, -1049976215, -33488187, 34431120, -2086968, -446046, 48642, 2220, -372, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 372*x^16 + 2220*x^15 + 48642*x^14 - 446046*x^13 - 2086968*x^12 + 34431120*x^11 - 33488187*x^10 - 1049976215*x^9 + 4500825492*x^8 + 5096936940*x^7 - 78281565936*x^6 + 152818776720*x^5 + 343019618880*x^4 - 2247082834368*x^3 + 4853863902720*x^2 - 5257925494272*x + 2489858275328)
 
gp: K = bnfinit(x^18 - 3*x^17 - 372*x^16 + 2220*x^15 + 48642*x^14 - 446046*x^13 - 2086968*x^12 + 34431120*x^11 - 33488187*x^10 - 1049976215*x^9 + 4500825492*x^8 + 5096936940*x^7 - 78281565936*x^6 + 152818776720*x^5 + 343019618880*x^4 - 2247082834368*x^3 + 4853863902720*x^2 - 5257925494272*x + 2489858275328, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 372 x^{16} + 2220 x^{15} + 48642 x^{14} - 446046 x^{13} - 2086968 x^{12} + 34431120 x^{11} - 33488187 x^{10} - 1049976215 x^{9} + 4500825492 x^{8} + 5096936940 x^{7} - 78281565936 x^{6} + 152818776720 x^{5} + 343019618880 x^{4} - 2247082834368 x^{3} + 4853863902720 x^{2} - 5257925494272 x + 2489858275328 \)

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:  \(-336122542391918780239888468288923486473177906638848=-\,2^{12}\cdot 3^{30}\cdot 7^{9}\cdot 61^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $641.25$
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:  $2, 3, 7, 61$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{16} a^{6} + \frac{1}{16} a^{5} + \frac{1}{16} a^{4} - \frac{1}{16} a^{3} - \frac{1}{8} a^{2}$, $\frac{1}{16} a^{7} - \frac{1}{8} a^{4} - \frac{1}{16} a^{3} + \frac{1}{8} a^{2}$, $\frac{1}{32} a^{8} - \frac{1}{16} a^{5} - \frac{1}{32} a^{4} + \frac{1}{16} a^{3}$, $\frac{1}{128} a^{9} + \frac{1}{64} a^{7} + \frac{9}{128} a^{5} - \frac{1}{8} a^{4} + \frac{1}{32} a^{3} - \frac{1}{4} a$, $\frac{1}{768} a^{10} - \frac{1}{768} a^{9} + \frac{1}{384} a^{8} + \frac{7}{384} a^{7} + \frac{3}{256} a^{6} - \frac{25}{768} a^{5} - \frac{1}{64} a^{4} + \frac{11}{192} a^{3} - \frac{1}{6} a^{2} + \frac{7}{24} a + \frac{1}{6}$, $\frac{1}{1536} a^{11} - \frac{1}{1536} a^{10} - \frac{1}{384} a^{9} + \frac{7}{768} a^{8} - \frac{1}{512} a^{7} - \frac{25}{1536} a^{6} + \frac{21}{256} a^{5} + \frac{35}{384} a^{4} - \frac{43}{192} a^{3} - \frac{5}{48} a^{2} - \frac{1}{24} a - \frac{1}{2}$, $\frac{1}{3072} a^{12} - \frac{1}{3072} a^{10} + \frac{1}{512} a^{9} - \frac{29}{3072} a^{8} - \frac{17}{768} a^{7} + \frac{41}{3072} a^{6} + \frac{83}{1536} a^{5} + \frac{23}{256} a^{4} - \frac{29}{384} a^{3} - \frac{11}{96} a^{2} - \frac{11}{48} a + \frac{5}{12}$, $\frac{1}{12288} a^{13} - \frac{1}{6144} a^{12} + \frac{1}{4096} a^{11} - \frac{1}{3072} a^{10} - \frac{25}{12288} a^{9} + \frac{5}{2048} a^{8} + \frac{293}{12288} a^{7} - \frac{35}{1536} a^{6} + \frac{15}{128} a^{5} - \frac{17}{192} a^{4} + \frac{49}{768} a^{3} - \frac{1}{32} a^{2} - \frac{13}{96} a + \frac{5}{24}$, $\frac{1}{24576} a^{14} - \frac{1}{24576} a^{13} + \frac{1}{24576} a^{12} - \frac{1}{24576} a^{11} + \frac{1}{8192} a^{10} - \frac{9}{8192} a^{9} + \frac{1}{8192} a^{8} - \frac{307}{24576} a^{7} + \frac{85}{3072} a^{6} - \frac{7}{384} a^{5} - \frac{163}{1536} a^{4} - \frac{223}{1536} a^{3} + \frac{1}{8} a^{2} + \frac{21}{64} a - \frac{11}{48}$, $\frac{1}{49152} a^{15} - \frac{1}{49152} a^{14} + \frac{1}{49152} a^{13} - \frac{1}{49152} a^{12} + \frac{1}{16384} a^{11} - \frac{9}{16384} a^{10} - \frac{63}{16384} a^{9} - \frac{307}{49152} a^{8} - \frac{155}{6144} a^{7} + \frac{17}{768} a^{6} + \frac{209}{3072} a^{5} + \frac{257}{3072} a^{4} + \frac{11}{64} a^{3} + \frac{5}{128} a^{2} - \frac{23}{96} a - \frac{1}{2}$, $\frac{1}{98304} a^{16} - \frac{1}{49152} a^{14} + \frac{1}{49152} a^{13} - \frac{11}{49152} a^{11} - \frac{5}{16384} a^{10} + \frac{67}{49152} a^{9} + \frac{367}{98304} a^{8} - \frac{243}{16384} a^{7} - \frac{7}{512} a^{6} - \frac{115}{1024} a^{5} + \frac{487}{6144} a^{4} + \frac{235}{3072} a^{3} + \frac{67}{768} a^{2} - \frac{133}{384} a + \frac{11}{96}$, $\frac{1}{2407175201936175153280171376841411467241265418248766598414336} a^{17} + \frac{1910669217208563759188707266618434115450545417845618099}{601793800484043788320042844210352866810316354562191649603584} a^{16} + \frac{278798660335996946087203902595888659811713171902122369}{300896900242021894160021422105176433405158177281095824801792} a^{15} - \frac{1623411350969157702328514074688127904053506655610258569}{200597933494681262773347614736784288936772118187397216534528} a^{14} - \frac{18619551806546007798314826346022646821710641042421506841}{1203587600968087576640085688420705733620632709124383299207168} a^{13} - \frac{29252640412072652775896735327865120433650457755496158011}{601793800484043788320042844210352866810316354562191649603584} a^{12} - \frac{183404104614705448178044202793352831053910439352814415927}{601793800484043788320042844210352866810316354562191649603584} a^{11} + \frac{35169757603546228642147248465460515962018296035159530297}{200597933494681262773347614736784288936772118187397216534528} a^{10} - \frac{5452608585341886081857822360441555969466510543092853855239}{2407175201936175153280171376841411467241265418248766598414336} a^{9} + \frac{185633707601353852538694070620230250720084459788505025065}{25074741686835157846668451842098036117096514773424652066816} a^{8} + \frac{213102037534037516507208112032146664565702895164085609795}{16264697310379561846487644438117645048927469042221395935232} a^{7} - \frac{64620706699551980763971815052490055863113436813166590887}{4701514066281592096250334720393381771955596520017122262528} a^{6} - \frac{16775114205801377752467152967984340106281523525859380566687}{150448450121010947080010711052588216702579088640547912400896} a^{5} - \frac{682958701648918505807355411198793946958378057271570095355}{6268685421708789461667112960524509029274128693356163016704} a^{4} - \frac{2924001418272492862193097918682949447960809754210892000931}{37612112530252736770002677763147054175644772160136978100224} a^{3} + \frac{87748529035550533758476588620896406789770509241143053595}{1175378516570398024062583680098345442988899130004280565632} a^{2} - \frac{2208204392790220895122804301954700848234128533414059535851}{4701514066281592096250334720393381771955596520017122262528} a - \frac{112516451913223185340959392581455793644434206326270115847}{391792838856799341354194560032781814329633043334760188544}$

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

Class group and class number

$C_{6}\times C_{1039665942}$, which has order $6237995652$ (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:  \( 248523426321.35065 \) (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

\(\Q(\sqrt{-7}) \), 3.3.6588.1, 3.3.301401.1, 6.0.14886798192.3, 6.0.31158999040743.3, 9.9.2886075179454821059008.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
3Data not computed
$7$7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$61$61.6.4.2$x^{6} - 61 x^{3} + 7442$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
61.12.10.3$x^{12} - 61 x^{6} + 59536$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$