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Properties

Label 18.6.75345346281...0000.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{12}\cdot 5^{11}\cdot 37^{6}\cdot 59^{8}$
Root discriminant $86.62$
Ramified primes $2, 5, 37, 59$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times (C_3\times A_4):S_3$ (as 18T156)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-99042304, 214803968, -197999872, -5419904, 107242976, -43606048, -11983740, 10748536, -284084, -1214856, 155150, 89776, -14811, -5401, 840, 287, -50, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 - 50*x^16 + 287*x^15 + 840*x^14 - 5401*x^13 - 14811*x^12 + 89776*x^11 + 155150*x^10 - 1214856*x^9 - 284084*x^8 + 10748536*x^7 - 11983740*x^6 - 43606048*x^5 + 107242976*x^4 - 5419904*x^3 - 197999872*x^2 + 214803968*x - 99042304)
 
gp: K = bnfinit(x^18 - 5*x^17 - 50*x^16 + 287*x^15 + 840*x^14 - 5401*x^13 - 14811*x^12 + 89776*x^11 + 155150*x^10 - 1214856*x^9 - 284084*x^8 + 10748536*x^7 - 11983740*x^6 - 43606048*x^5 + 107242976*x^4 - 5419904*x^3 - 197999872*x^2 + 214803968*x - 99042304, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} - 50 x^{16} + 287 x^{15} + 840 x^{14} - 5401 x^{13} - 14811 x^{12} + 89776 x^{11} + 155150 x^{10} - 1214856 x^{9} - 284084 x^{8} + 10748536 x^{7} - 11983740 x^{6} - 43606048 x^{5} + 107242976 x^{4} - 5419904 x^{3} - 197999872 x^{2} + 214803968 x - 99042304 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(75345346281286616442657800000000000=2^{12}\cdot 5^{11}\cdot 37^{6}\cdot 59^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.62$
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:  $2, 5, 37, 59$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{12} + \frac{1}{8} a^{10} - \frac{3}{8} a^{8} - \frac{3}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{13} - \frac{1}{8} a^{12} - \frac{1}{16} a^{11} + \frac{3}{16} a^{9} + \frac{5}{16} a^{8} - \frac{1}{2} a^{7} - \frac{1}{8} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{64} a^{15} - \frac{1}{64} a^{14} + \frac{1}{32} a^{13} - \frac{1}{64} a^{12} + \frac{1}{16} a^{11} - \frac{1}{64} a^{10} - \frac{15}{64} a^{9} + \frac{7}{16} a^{8} - \frac{13}{32} a^{7} - \frac{1}{2} a^{6} - \frac{1}{16} a^{5} + \frac{1}{8} a^{4} + \frac{1}{16} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{1664} a^{16} + \frac{5}{1664} a^{15} - \frac{5}{416} a^{14} + \frac{7}{128} a^{13} - \frac{17}{832} a^{12} + \frac{135}{1664} a^{11} + \frac{11}{1664} a^{10} - \frac{87}{832} a^{9} + \frac{207}{832} a^{8} - \frac{79}{416} a^{7} - \frac{145}{416} a^{6} + \frac{41}{104} a^{5} + \frac{61}{416} a^{4} - \frac{47}{208} a^{3} - \frac{15}{52} a^{2} - \frac{7}{26} a - \frac{3}{13}$, $\frac{1}{21969709146030018008310362891229310765241290827764562727424} a^{17} + \frac{2726950222913660433783305469984609565305391130505372127}{21969709146030018008310362891229310765241290827764562727424} a^{16} - \frac{70399259712096137604851591569478342423549878979605528475}{10984854573015009004155181445614655382620645413882281363712} a^{15} + \frac{265958863111881525499811715330003983489708298351912790039}{21969709146030018008310362891229310765241290827764562727424} a^{14} - \frac{259964631594898688019018297260882578686489922901554092699}{5492427286507504502077590722807327691310322706941140681856} a^{13} - \frac{2259626906962913402849842277000346097280554135900029900105}{21969709146030018008310362891229310765241290827764562727424} a^{12} + \frac{2689761097270112054708222141399998893673967328661615076785}{21969709146030018008310362891229310765241290827764562727424} a^{11} + \frac{79774438078382164061821221520466024107255958234916289581}{5492427286507504502077590722807327691310322706941140681856} a^{10} + \frac{1617977853939629658469537498790988941557355855491202563}{844988813308846846473475495816511952509280416452483181824} a^{9} - \frac{13244712618067516995969881683420754173910842935894341145}{686553410813438062759698840350915961413790338367642585232} a^{8} - \frac{225722729511295738586012398270811546380932490403887645133}{5492427286507504502077590722807327691310322706941140681856} a^{7} - \frac{830048702263591218792392460645703389959370550098529123063}{2746213643253752251038795361403663845655161353470570340928} a^{6} + \frac{1180027547375322105975394360211492887713987741299813622505}{5492427286507504502077590722807327691310322706941140681856} a^{5} - \frac{286910338178253545589578934201718659410215094490854452221}{1373106821626876125519397680701831922827580676735285170464} a^{4} - \frac{112113296771076182297817812734639500528870432330010206875}{686553410813438062759698840350915961413790338367642585232} a^{3} + \frac{157770299290074801549036511326295093770659992708889701039}{343276705406719031379849420175457980706895169183821292616} a^{2} - \frac{10027466120198189673494532999709435843586592727988954579}{85819176351679757844962355043864495176723792295955323154} a - \frac{4729620379138728060701862206824587170687335452386346}{137972952333890285924376776597852886136211884720185407}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 68983392448.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times (C_3\times A_4):S_3$ (as 18T156):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 432
The 38 conjugacy class representatives for $C_2\times (C_3\times A_4):S_3$
Character table for $C_2\times (C_3\times A_4):S_3$ is not computed

Intermediate fields

3.3.148.1, 6.2.109520.1, 9.9.24551227469320000.1

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

Sibling fields

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ R

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.12.8.2$x^{12} + 25 x^{6} - 250 x^{3} + 1250$$3$$4$$8$$C_3\times (C_3 : C_4)$$[\ ]_{3}^{12}$
$37$37.6.0.1$x^{6} - x + 20$$1$$6$$0$$C_6$$[\ ]^{6}$
37.6.3.1$x^{6} - 74 x^{4} + 1369 x^{2} - 202612$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
37.6.3.1$x^{6} - 74 x^{4} + 1369 x^{2} - 202612$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$59$59.3.0.1$x^{3} - x + 17$$1$$3$$0$$C_3$$[\ ]^{3}$
59.3.0.1$x^{3} - x + 17$$1$$3$$0$$C_3$$[\ ]^{3}$
59.6.4.2$x^{6} - 59 x^{3} + 6962$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
59.6.4.2$x^{6} - 59 x^{3} + 6962$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$