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Properties

Label 18.0.21028741084...5408.5
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 7^{12}\cdot 157^{9}$
Root discriminant $91.70$
Ramified primes $2, 7, 157$
Class number $13608$ (GRH)
Class group $[6, 18, 126]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1769766817, 55016416, 64290142, -50646490, -69861, -1431434, -196809, -540496, 40103, -17244, 45764, 3902, 3336, 826, 81, 38, 19, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 19*x^16 + 38*x^15 + 81*x^14 + 826*x^13 + 3336*x^12 + 3902*x^11 + 45764*x^10 - 17244*x^9 + 40103*x^8 - 540496*x^7 - 196809*x^6 - 1431434*x^5 - 69861*x^4 - 50646490*x^3 + 64290142*x^2 + 55016416*x + 1769766817)
 
gp: K = bnfinit(x^18 - 2*x^17 + 19*x^16 + 38*x^15 + 81*x^14 + 826*x^13 + 3336*x^12 + 3902*x^11 + 45764*x^10 - 17244*x^9 + 40103*x^8 - 540496*x^7 - 196809*x^6 - 1431434*x^5 - 69861*x^4 - 50646490*x^3 + 64290142*x^2 + 55016416*x + 1769766817, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 19 x^{16} + 38 x^{15} + 81 x^{14} + 826 x^{13} + 3336 x^{12} + 3902 x^{11} + 45764 x^{10} - 17244 x^{9} + 40103 x^{8} - 540496 x^{7} - 196809 x^{6} - 1431434 x^{5} - 69861 x^{4} - 50646490 x^{3} + 64290142 x^{2} + 55016416 x + 1769766817 \)

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:  \(-210287410844167058761982755564945408=-\,2^{18}\cdot 7^{12}\cdot 157^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.70$
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:  $2, 7, 157$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{26} a^{9} + \frac{1}{26} a^{8} - \frac{5}{26} a^{7} + \frac{1}{26} a^{6} - \frac{1}{13} a^{5} + \frac{2}{13} a^{4} - \frac{7}{26} a^{3} - \frac{9}{26} a^{2} + \frac{1}{13} a$, $\frac{1}{26} a^{10} - \frac{3}{13} a^{8} + \frac{3}{13} a^{7} - \frac{3}{26} a^{6} + \frac{3}{13} a^{5} - \frac{11}{26} a^{4} - \frac{1}{13} a^{3} + \frac{11}{26} a^{2} - \frac{1}{13} a$, $\frac{1}{26} a^{11} - \frac{1}{26} a^{8} + \frac{3}{13} a^{7} - \frac{1}{26} a^{6} - \frac{5}{13} a^{5} - \frac{2}{13} a^{4} - \frac{5}{26} a^{3} + \frac{9}{26} a^{2} - \frac{1}{26} a$, $\frac{1}{52} a^{12} + \frac{7}{52} a^{8} - \frac{3}{26} a^{7} + \frac{1}{13} a^{6} - \frac{3}{26} a^{5} - \frac{7}{26} a^{4} - \frac{6}{13} a^{3} + \frac{4}{13} a^{2} + \frac{1}{26} a - \frac{1}{4}$, $\frac{1}{52} a^{13} - \frac{1}{52} a^{9} + \frac{3}{13} a^{8} - \frac{2}{13} a^{7} + \frac{3}{13} a^{6} + \frac{1}{26} a^{5} - \frac{1}{13} a^{4} + \frac{5}{13} a^{3} - \frac{1}{13} a^{2} + \frac{23}{52} a$, $\frac{1}{52} a^{14} - \frac{1}{52} a^{10} + \frac{3}{26} a^{8} - \frac{3}{26} a^{7} - \frac{5}{26} a^{6} - \frac{3}{26} a^{5} - \frac{1}{26} a^{4} - \frac{6}{13} a^{3} + \frac{1}{52} a^{2} + \frac{1}{26} a - \frac{1}{2}$, $\frac{1}{676} a^{15} - \frac{3}{338} a^{14} - \frac{1}{676} a^{13} - \frac{1}{169} a^{12} - \frac{1}{676} a^{11} - \frac{5}{338} a^{10} - \frac{7}{676} a^{9} + \frac{3}{13} a^{8} - \frac{23}{338} a^{7} + \frac{2}{169} a^{6} + \frac{33}{338} a^{5} - \frac{62}{169} a^{4} + \frac{15}{52} a^{3} + \frac{125}{338} a^{2} - \frac{19}{52} a$, $\frac{1}{229036398371139268} a^{16} + \frac{3039848545121}{4404546122521909} a^{15} - \frac{1782022377230821}{229036398371139268} a^{14} + \frac{751090469080591}{229036398371139268} a^{13} - \frac{529273907442325}{114518199185569634} a^{12} - \frac{498946465630461}{57259099592784817} a^{11} + \frac{3166669283035919}{229036398371139268} a^{10} - \frac{1478684340723333}{229036398371139268} a^{9} + \frac{44826626996735317}{229036398371139268} a^{8} - \frac{27975383783684991}{114518199185569634} a^{7} + \frac{12276043081933822}{57259099592784817} a^{6} - \frac{39214608057428691}{114518199185569634} a^{5} + \frac{6553396290071879}{32719485481591324} a^{4} + \frac{42517306323782935}{114518199185569634} a^{3} + \frac{6768624122828903}{32719485481591324} a^{2} - \frac{42906599460717}{193606422967996} a + \frac{94205324712293}{193606422967996}$, $\frac{1}{41805414025863443198227588295994082809494284} a^{17} + \frac{648932766360655686290583}{853171714813539657106685475428450669581516} a^{16} + \frac{15198274346630892607897578794038553398097}{41805414025863443198227588295994082809494284} a^{15} + \frac{2484124606802735640982004091160556478493}{3215801078912572553709814484307237139191868} a^{14} - \frac{319964458865543823543025459829965119303859}{41805414025863443198227588295994082809494284} a^{13} - \frac{386790342953017572378974537910714762114645}{41805414025863443198227588295994082809494284} a^{12} + \frac{726176069580190006840310151658665658259209}{41805414025863443198227588295994082809494284} a^{11} + \frac{56602505101132546768476921914436632687235}{3215801078912572553709814484307237139191868} a^{10} + \frac{342423181343884713320687692502652195709971}{20902707012931721599113794147997041404747142} a^{9} + \frac{659222268384113849250661616999662654643169}{10451353506465860799556897073998520702373571} a^{8} - \frac{265634485176408431506881538120508713551294}{10451353506465860799556897073998520702373571} a^{7} + \frac{181249040505496629965935322153592513474458}{1493050500923694399936699581999788671767653} a^{6} + \frac{18823028569925167458940842594109534526096407}{41805414025863443198227588295994082809494284} a^{5} + \frac{16435316263597835411519755642601241114650007}{41805414025863443198227588295994082809494284} a^{4} - \frac{8604155647067466312306964295818455106442089}{41805414025863443198227588295994082809494284} a^{3} + \frac{106004549956479032387959844925869907692919}{853171714813539657106685475428450669581516} a^{2} - \frac{23556419488777799377913704436772162131797}{114850038532591876918207660153829897828281} a - \frac{2939517810191325506848661627440856860292}{8834618348660913609092896934909992140637}$

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

Class group and class number

$C_{6}\times C_{18}\times C_{126}$, which has order $13608$ (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:  \( 7779732.79418 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-157}) \), 3.1.30772.1 x3, \(\Q(\zeta_{7})^+\), 6.0.594663237952.1, 6.0.12135984448.5 x2, 6.0.594663237952.6, 9.3.29138498659648.8 x3

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

Sibling fields

Degree 6 sibling: data not computed
Degree 9 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$2$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
$157$157.6.3.1$x^{6} - 314 x^{4} + 24649 x^{2} - 870725925$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
157.6.3.1$x^{6} - 314 x^{4} + 24649 x^{2} - 870725925$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
157.6.3.1$x^{6} - 314 x^{4} + 24649 x^{2} - 870725925$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$