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Properties

Label 18.0.28065512141...4832.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{24}\cdot 3^{21}\cdot 7^{14}\cdot 11^{9}$
Root discriminant $136.78$
Ramified primes $2, 3, 7, 11$
Class number $2991816$ (GRH)
Class group $[3, 3, 18, 18, 1026]$ (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![37617238057, 5908781970, -10099126569, 10833272058, 5662902273, -2285780196, 2860912354, -698258490, 417136110, -64909088, 29051985, -2735124, 1082299, -56064, 22707, -460, 246, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 246*x^16 - 460*x^15 + 22707*x^14 - 56064*x^13 + 1082299*x^12 - 2735124*x^11 + 29051985*x^10 - 64909088*x^9 + 417136110*x^8 - 698258490*x^7 + 2860912354*x^6 - 2285780196*x^5 + 5662902273*x^4 + 10833272058*x^3 - 10099126569*x^2 + 5908781970*x + 37617238057)
 
gp: K = bnfinit(x^18 + 246*x^16 - 460*x^15 + 22707*x^14 - 56064*x^13 + 1082299*x^12 - 2735124*x^11 + 29051985*x^10 - 64909088*x^9 + 417136110*x^8 - 698258490*x^7 + 2860912354*x^6 - 2285780196*x^5 + 5662902273*x^4 + 10833272058*x^3 - 10099126569*x^2 + 5908781970*x + 37617238057, 1)
 

Normalized defining polynomial

\( x^{18} + 246 x^{16} - 460 x^{15} + 22707 x^{14} - 56064 x^{13} + 1082299 x^{12} - 2735124 x^{11} + 29051985 x^{10} - 64909088 x^{9} + 417136110 x^{8} - 698258490 x^{7} + 2860912354 x^{6} - 2285780196 x^{5} + 5662902273 x^{4} + 10833272058 x^{3} - 10099126569 x^{2} + 5908781970 x + 37617238057 \)

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:  \(-280655121419454569082411120580815224832=-\,2^{24}\cdot 3^{21}\cdot 7^{14}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $136.78$
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:  $2, 3, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} + \frac{1}{7} a^{14} - \frac{3}{7} a^{13} - \frac{1}{7} a^{11} + \frac{1}{7} a^{10} + \frac{3}{7} a^{8} - \frac{1}{7} a^{7} - \frac{1}{7} a^{6}$, $\frac{1}{42833} a^{16} + \frac{209}{6119} a^{15} + \frac{12757}{42833} a^{14} - \frac{2685}{42833} a^{13} - \frac{14946}{42833} a^{12} + \frac{9102}{42833} a^{11} - \frac{16199}{42833} a^{10} - \frac{21158}{42833} a^{9} - \frac{382}{1477} a^{8} + \frac{2799}{6119} a^{7} + \frac{10697}{42833} a^{6} + \frac{1126}{6119} a^{5} - \frac{1839}{6119} a^{4} + \frac{2080}{6119} a^{3} + \frac{832}{6119} a^{2} - \frac{44}{6119} a - \frac{2029}{6119}$, $\frac{1}{17526078771953215728181161473848750587050686728722009005088769924981914715067495407924359957} a^{17} + \frac{171702551792747245324215220532205782774373286508098125248216460910946579505487745624004}{17526078771953215728181161473848750587050686728722009005088769924981914715067495407924359957} a^{16} - \frac{709874751931063687322382827123845572396965676247329774313664196127862910368716401135282184}{17526078771953215728181161473848750587050686728722009005088769924981914715067495407924359957} a^{15} - \frac{3765615051280586353227419182053022465119283820923828337018456317596050881439036084414289449}{17526078771953215728181161473848750587050686728722009005088769924981914715067495407924359957} a^{14} - \frac{8700517040592626426568339508592770553305846172721274455534306413519173249440856733963784776}{17526078771953215728181161473848750587050686728722009005088769924981914715067495407924359957} a^{13} + \frac{6257101254619593939611222963394263816834001040646469142607146969114742821503860391412291760}{17526078771953215728181161473848750587050686728722009005088769924981914715067495407924359957} a^{12} + \frac{7367393430264722454646683380972149719534531627579430179914690331592686607905326840913444180}{17526078771953215728181161473848750587050686728722009005088769924981914715067495407924359957} a^{11} - \frac{2147484418584130105534652867958605138412217905799145573509065674045845700851452033633503231}{17526078771953215728181161473848750587050686728722009005088769924981914715067495407924359957} a^{10} - \frac{76636303402540039631338637418555746133295778798514863178122135615769345512247878921147975}{2503725538850459389740165924835535798150098104103144143584109989283130673581070772560622851} a^{9} - \frac{6675574176925003813898862223910079666610438720237633914708224558153084682180238569985775989}{17526078771953215728181161473848750587050686728722009005088769924981914715067495407924359957} a^{8} + \frac{5115772573701153742366771247338315109185436281094926548116810732296108055344722685611243447}{17526078771953215728181161473848750587050686728722009005088769924981914715067495407924359957} a^{7} + \frac{8429953433676140962294269236936869224352519858887422174552206019147625036379555845146759956}{17526078771953215728181161473848750587050686728722009005088769924981914715067495407924359957} a^{6} + \frac{18340297517831125696024352205815123951366016266018599961907072097559382229113931284279913}{86335363408636530680695376718466751660348210486315315296003792733901057709692095605538719} a^{5} + \frac{336988854579790582049532553945517409379787067604087285699657782818159604005709547195767885}{2503725538850459389740165924835535798150098104103144143584109989283130673581070772560622851} a^{4} + \frac{710312768731188053400348308540739338728673333872018928727564514261584018314316212270621691}{2503725538850459389740165924835535798150098104103144143584109989283130673581070772560622851} a^{3} - \frac{950156892221169159987887065427242774049816657591570003374683782734590795275295176692450058}{2503725538850459389740165924835535798150098104103144143584109989283130673581070772560622851} a^{2} + \frac{1040669859012607034229757296659073915011214759549375285673741128347094485048840183449240777}{2503725538850459389740165924835535798150098104103144143584109989283130673581070772560622851} a - \frac{973135117551101967432063768216457406181956684684388670448298897927669075578274988794673916}{2503725538850459389740165924835535798150098104103144143584109989283130673581070772560622851}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{18}\times C_{18}\times C_{1026}$, which has order $2991816$ (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:  \( 296124.35954857944 \) (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

\(\Q(\sqrt{-33}) \), 3.3.756.1, \(\Q(\zeta_{7})^+\), 6.0.36514291968.4, 6.0.5522223168.10, 9.9.1037426999616.1

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 R ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$11$11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.12.6.1$x^{12} + 242 x^{8} + 21296 x^{6} + 14641 x^{4} + 1932612 x^{2} + 113379904$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$