↑

Properties

Label 18.0.69609534401...1552.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 7^{15}\cdot 11^{9}\cdot 19^{15}$
Root discriminant $309.93$
Ramified primes $2, 7, 11, 19$
Class number $835580928$ (GRH)
Class group $[2, 2, 2, 56, 1865136]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5260791417344, 681176924480, 1285667140736, 186897927432, 184580325172, 14640502408, 11617030252, 428932198, 556579681, -34844136, 21825209, -1015636, 393586, -2840, 5594, -218, 121, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 + 121*x^16 - 218*x^15 + 5594*x^14 - 2840*x^13 + 393586*x^12 - 1015636*x^11 + 21825209*x^10 - 34844136*x^9 + 556579681*x^8 + 428932198*x^7 + 11617030252*x^6 + 14640502408*x^5 + 184580325172*x^4 + 186897927432*x^3 + 1285667140736*x^2 + 681176924480*x + 5260791417344)
 
gp: K = bnfinit(x^18 - 8*x^17 + 121*x^16 - 218*x^15 + 5594*x^14 - 2840*x^13 + 393586*x^12 - 1015636*x^11 + 21825209*x^10 - 34844136*x^9 + 556579681*x^8 + 428932198*x^7 + 11617030252*x^6 + 14640502408*x^5 + 184580325172*x^4 + 186897927432*x^3 + 1285667140736*x^2 + 681176924480*x + 5260791417344, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{17} + 121 x^{16} - 218 x^{15} + 5594 x^{14} - 2840 x^{13} + 393586 x^{12} - 1015636 x^{11} + 21825209 x^{10} - 34844136 x^{9} + 556579681 x^{8} + 428932198 x^{7} + 11617030252 x^{6} + 14640502408 x^{5} + 184580325172 x^{4} + 186897927432 x^{3} + 1285667140736 x^{2} + 681176924480 x + 5260791417344 \)

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:  \(-696095344015988004005130387702332687426711552=-\,2^{12}\cdot 7^{15}\cdot 11^{9}\cdot 19^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $309.93$
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:  $2, 7, 11, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{56} a^{9} + \frac{3}{56} a^{8} - \frac{1}{14} a^{6} + \frac{9}{56} a^{5} - \frac{5}{56} a^{4} - \frac{13}{28} a^{3} - \frac{11}{28} a^{2} - \frac{3}{14} a - \frac{1}{7}$, $\frac{1}{224} a^{10} - \frac{1}{224} a^{9} + \frac{1}{112} a^{8} - \frac{9}{112} a^{7} - \frac{17}{224} a^{6} - \frac{27}{224} a^{5} + \frac{25}{112} a^{4} + \frac{55}{112} a^{3} - \frac{9}{56} a^{2} + \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{224} a^{11} + \frac{1}{224} a^{9} + \frac{3}{56} a^{8} + \frac{3}{32} a^{7} + \frac{3}{56} a^{6} + \frac{23}{224} a^{5} + \frac{5}{56} a^{4} + \frac{9}{112} a^{3} - \frac{27}{56} a^{2} - \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{224} a^{12} + \frac{1}{224} a^{9} + \frac{11}{224} a^{8} - \frac{13}{112} a^{7} - \frac{3}{28} a^{6} - \frac{5}{224} a^{5} - \frac{1}{4} a^{4} + \frac{47}{112} a^{3} + \frac{9}{56} a^{2} - \frac{1}{7} a + \frac{2}{7}$, $\frac{1}{224} a^{13} - \frac{1}{28} a^{8} - \frac{3}{112} a^{7} + \frac{1}{56} a^{6} + \frac{31}{224} a^{5} - \frac{1}{28} a^{4} + \frac{5}{16} a^{3} + \frac{11}{56} a^{2} + \frac{2}{7}$, $\frac{1}{448} a^{14} - \frac{1}{448} a^{13} - \frac{1}{448} a^{12} - \frac{1}{448} a^{11} - \frac{1}{448} a^{10} + \frac{3}{448} a^{9} - \frac{15}{448} a^{8} - \frac{23}{448} a^{7} + \frac{1}{56} a^{6} + \frac{11}{224} a^{5} + \frac{11}{56} a^{4} - \frac{9}{28} a^{2} + \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{448} a^{15} + \frac{1}{224} a^{9} + \frac{3}{112} a^{8} - \frac{1}{448} a^{7} + \frac{1}{28} a^{6} + \frac{55}{224} a^{5} + \frac{3}{16} a^{4} + \frac{1}{28} a^{3} - \frac{1}{28} a^{2} - \frac{3}{14} a + \frac{2}{7}$, $\frac{1}{896} a^{16} - \frac{1}{448} a^{12} + \frac{1}{224} a^{9} + \frac{5}{896} a^{8} + \frac{13}{112} a^{7} - \frac{1}{8} a^{6} + \frac{19}{224} a^{5} + \frac{3}{224} a^{4} + \frac{1}{112} a^{3} + \frac{13}{28} a^{2} - \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{151595431079284393606884487091216931883711974359878466874198171778448878101399620096} a^{17} + \frac{37950645467899186309847257110595977472459993317501172512055914648409439458103149}{151595431079284393606884487091216931883711974359878466874198171778448878101399620096} a^{16} + \frac{52011018194695461703252866612823185747203890193048434864823654569176942572460077}{75797715539642196803442243545608465941855987179939233437099085889224439050699810048} a^{15} - \frac{4330201290364522387893410926624109503718863411307210589364942183216398141764213}{18949428884910549200860560886402116485463996794984808359274771472306109762674952512} a^{14} + \frac{140786027001431855290502309548802767589646034048449764689576515645347754487172489}{75797715539642196803442243545608465941855987179939233437099085889224439050699810048} a^{13} - \frac{79299324999126695916059274113757560365521042024249916141160127665444771803769671}{75797715539642196803442243545608465941855987179939233437099085889224439050699810048} a^{12} + \frac{44890695190995421239879284521502296618017873431340871211335795567581410821209587}{37898857769821098401721121772804232970927993589969616718549542944612219525349905024} a^{11} - \frac{614846129965662891763851686478561514065883454949222602240590377531594451631409}{18949428884910549200860560886402116485463996794984808359274771472306109762674952512} a^{10} + \frac{334699119402476980243286439096298250017497599592110213521634634933743742028166225}{151595431079284393606884487091216931883711974359878466874198171778448878101399620096} a^{9} - \frac{8082583776439646573881367681296723401554411748364778573820660622445660338434993619}{151595431079284393606884487091216931883711974359878466874198171778448878101399620096} a^{8} + \frac{7776846448131097956006225672678751823157226680678799230460296383029246850715016593}{75797715539642196803442243545608465941855987179939233437099085889224439050699810048} a^{7} - \frac{780932814871960303757194513386211312171782077372464437888946349272269780691467055}{9474714442455274600430280443201058242731998397492404179637385736153054881337476256} a^{6} + \frac{6570467869495898682680600870476712287535921531058273580239434347671753814864584479}{37898857769821098401721121772804232970927993589969616718549542944612219525349905024} a^{5} + \frac{9119932101550731211764361962670734804089911241710403137703197267055154715830665653}{37898857769821098401721121772804232970927993589969616718549542944612219525349905024} a^{4} - \frac{2435512097514886554140618959687229894952687258109850456837323688804976685046020893}{18949428884910549200860560886402116485463996794984808359274771472306109762674952512} a^{3} + \frac{495477227070081397012666671813882176572364403017811224530062349040785038587109509}{2368678610613818650107570110800264560682999599373101044909346434038263720334369064} a^{2} + \frac{99453175119573516670875395323788040150311239680121580643990051718072524775107811}{2368678610613818650107570110800264560682999599373101044909346434038263720334369064} a + \frac{124412512165284539623011536296343545201932236405917834513053742359974323922981833}{296084826326727331263446263850033070085374949921637630613668304254782965041796133}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{56}\times C_{1865136}$, which has order $835580928$ (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:  \( 2494653063.4840164 \) (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

\(\Q(\sqrt{-1463}) \), 3.3.17689.2, 3.3.15884.1, Deg 6, 6.0.55390624333583.2, 9.9.471484994327458496.3

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R R ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$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$$0$Trivial$[\ ]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
7Data not computed
$11$11.6.3.2$x^{6} - 121 x^{2} + 3993$$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$$[\ ]_{2}^{6}$
$19$19.6.5.4$x^{6} + 76$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.4$x^{6} + 76$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.4$x^{6} + 76$$6$$1$$5$$C_6$$[\ ]_{6}$