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Properties

Label 18.0.16033716621...8352.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{24}\cdot 13^{9}\cdot 37^{14}$
Root discriminant $150.68$
Ramified primes $2, 13, 37$
Class number $1573312$ (GRH)
Class group $[2, 2, 2, 196664]$ (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![54985356817, 5368141552, 24679661181, 1294777734, 5024127315, 75541038, 622617494, -9253226, 52668918, -1924606, 3197563, -156556, 141003, -7220, 4421, -186, 92, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 92*x^16 - 186*x^15 + 4421*x^14 - 7220*x^13 + 141003*x^12 - 156556*x^11 + 3197563*x^10 - 1924606*x^9 + 52668918*x^8 - 9253226*x^7 + 622617494*x^6 + 75541038*x^5 + 5024127315*x^4 + 1294777734*x^3 + 24679661181*x^2 + 5368141552*x + 54985356817)
 
gp: K = bnfinit(x^18 - 2*x^17 + 92*x^16 - 186*x^15 + 4421*x^14 - 7220*x^13 + 141003*x^12 - 156556*x^11 + 3197563*x^10 - 1924606*x^9 + 52668918*x^8 - 9253226*x^7 + 622617494*x^6 + 75541038*x^5 + 5024127315*x^4 + 1294777734*x^3 + 24679661181*x^2 + 5368141552*x + 54985356817, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 92 x^{16} - 186 x^{15} + 4421 x^{14} - 7220 x^{13} + 141003 x^{12} - 156556 x^{11} + 3197563 x^{10} - 1924606 x^{9} + 52668918 x^{8} - 9253226 x^{7} + 622617494 x^{6} + 75541038 x^{5} + 5024127315 x^{4} + 1294777734 x^{3} + 24679661181 x^{2} + 5368141552 x + 54985356817 \)

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:  \(-1603371662107021018978486840742329188352=-\,2^{24}\cdot 13^{9}\cdot 37^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $150.68$
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:  $2, 13, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{46} a^{16} + \frac{1}{46} a^{15} + \frac{5}{46} a^{14} - \frac{4}{23} a^{13} - \frac{9}{46} a^{12} + \frac{5}{46} a^{11} + \frac{5}{23} a^{10} - \frac{1}{46} a^{9} - \frac{9}{23} a^{8} - \frac{11}{23} a^{7} - \frac{11}{46} a^{6} - \frac{6}{23} a^{5} + \frac{7}{46} a^{4} - \frac{2}{23} a^{3} - \frac{10}{23} a^{2} - \frac{10}{23} a + \frac{15}{46}$, $\frac{1}{4740646267467658011062617208121341612208543712126035731802886} a^{17} - \frac{15532932986738359793215913437594970970606669377432723486124}{2370323133733829005531308604060670806104271856063017865901443} a^{16} + \frac{100938350212584163332838183613426775095912625990621099390599}{2370323133733829005531308604060670806104271856063017865901443} a^{15} - \frac{424443804700432495311956859179585518305120391149135510884962}{2370323133733829005531308604060670806104271856063017865901443} a^{14} - \frac{390697756110481611597575310890037689157773024315902395014870}{2370323133733829005531308604060670806104271856063017865901443} a^{13} - \frac{176541682169448505522121785438061421896614771283997412769957}{4740646267467658011062617208121341612208543712126035731802886} a^{12} - \frac{152987657682771551636182691835662747126219830423202384837317}{2370323133733829005531308604060670806104271856063017865901443} a^{11} + \frac{32700015458690601824666271603099562316307817981246921545232}{2370323133733829005531308604060670806104271856063017865901443} a^{10} + \frac{297372041651776717262848522433755597107422342422970541847111}{2370323133733829005531308604060670806104271856063017865901443} a^{9} - \frac{518115319368477844562656360226610772166455053601161805570000}{2370323133733829005531308604060670806104271856063017865901443} a^{8} - \frac{1767630109751137477019383911316550956880879961139852632251361}{4740646267467658011062617208121341612208543712126035731802886} a^{7} - \frac{1464767365547544963418183914624058193586106129120394196672291}{4740646267467658011062617208121341612208543712126035731802886} a^{6} + \frac{6676158917350248789686069956973741550616050833007113841449}{103057527553644739370926461046116122004533558959261646343541} a^{5} + \frac{1062376021621825827049641356746092790927923184613785821232235}{4740646267467658011062617208121341612208543712126035731802886} a^{4} + \frac{144973462769040554830372217923913682313345867390270232338297}{2370323133733829005531308604060670806104271856063017865901443} a^{3} - \frac{1714747878335859578800304553877459524230573547208304021054165}{4740646267467658011062617208121341612208543712126035731802886} a^{2} + \frac{752084277059650823249998231319742671677583902237225513538025}{4740646267467658011062617208121341612208543712126035731802886} a - \frac{1155808407107717198140418369352860408612685491477150069352486}{2370323133733829005531308604060670806104271856063017865901443}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{196664}$, which has order $1573312$ (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:  \( 615797.1340659427 \) (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

\(\Q(\sqrt{-13}) \), 3.3.1369.1, 3.3.148.1, 6.0.263522029888.2, 6.0.769969408.1, 9.9.6075640136512.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$13$13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.12.6.1$x^{12} + 338 x^{8} + 8788 x^{6} + 28561 x^{4} + 19307236$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$37$37.6.4.1$x^{6} + 518 x^{3} + 171125$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
37.12.10.1$x^{12} + 1998 x^{6} + 21390625$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$