↑

Properties

Label 18.0.11325865496...7424.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{33}\cdot 3^{9}\cdot 7^{14}\cdot 61^{14}$
Root discriminant $686.02$
Ramified primes $2, 3, 7, 61$
Class number $1603613376$ (GRH)
Class group $[6, 18, 108, 137484]$ (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![222285198247, 153363319298, 110400751965, 51046721698, 22847752382, 7978959520, 2546248545, 695406524, 180850253, 36876810, 4974933, 659382, 359127, 124802, 17550, -6, -206, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 - 206*x^16 - 6*x^15 + 17550*x^14 + 124802*x^13 + 359127*x^12 + 659382*x^11 + 4974933*x^10 + 36876810*x^9 + 180850253*x^8 + 695406524*x^7 + 2546248545*x^6 + 7978959520*x^5 + 22847752382*x^4 + 51046721698*x^3 + 110400751965*x^2 + 153363319298*x + 222285198247)
 
gp: K = bnfinit(x^18 - 8*x^17 - 206*x^16 - 6*x^15 + 17550*x^14 + 124802*x^13 + 359127*x^12 + 659382*x^11 + 4974933*x^10 + 36876810*x^9 + 180850253*x^8 + 695406524*x^7 + 2546248545*x^6 + 7978959520*x^5 + 22847752382*x^4 + 51046721698*x^3 + 110400751965*x^2 + 153363319298*x + 222285198247, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{17} - 206 x^{16} - 6 x^{15} + 17550 x^{14} + 124802 x^{13} + 359127 x^{12} + 659382 x^{11} + 4974933 x^{10} + 36876810 x^{9} + 180850253 x^{8} + 695406524 x^{7} + 2546248545 x^{6} + 7978959520 x^{5} + 22847752382 x^{4} + 51046721698 x^{3} + 110400751965 x^{2} + 153363319298 x + 222285198247 \)

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:  \(-1132586549659813618524246198887307765542862584807424=-\,2^{33}\cdot 3^{9}\cdot 7^{14}\cdot 61^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $686.02$
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:  $2, 3, 7, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6} a^{12} + \frac{1}{6} a^{10} + \frac{1}{3} a^{9} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{6} a^{6} - \frac{1}{3} a^{5} - \frac{1}{6} a^{4} + \frac{1}{3} a^{3} + \frac{1}{6} a^{2} + \frac{1}{6}$, $\frac{1}{6} a^{13} + \frac{1}{6} a^{11} + \frac{1}{3} a^{10} - \frac{1}{6} a^{9} - \frac{1}{3} a^{8} - \frac{1}{6} a^{7} - \frac{1}{3} a^{6} - \frac{1}{6} a^{5} + \frac{1}{3} a^{4} + \frac{1}{6} a^{3} + \frac{1}{6} a$, $\frac{1}{6} a^{14} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{6}$, $\frac{1}{36} a^{15} - \frac{1}{36} a^{14} - \frac{1}{36} a^{13} + \frac{1}{18} a^{12} + \frac{1}{36} a^{11} + \frac{1}{18} a^{10} + \frac{11}{36} a^{9} - \frac{5}{18} a^{8} + \frac{13}{36} a^{7} + \frac{1}{6} a^{6} - \frac{7}{36} a^{5} - \frac{1}{6} a^{4} - \frac{11}{36} a^{3} + \frac{1}{6} a^{2} + \frac{5}{18} a - \frac{5}{36}$, $\frac{1}{26058751102476} a^{16} - \frac{29571792130}{6514687775619} a^{15} - \frac{336540483724}{6514687775619} a^{14} + \frac{7812043643}{26058751102476} a^{13} - \frac{989130285575}{26058751102476} a^{12} - \frac{2302905155251}{26058751102476} a^{11} + \frac{10324248717407}{26058751102476} a^{10} + \frac{8018442729971}{26058751102476} a^{9} + \frac{2897299344949}{26058751102476} a^{8} + \frac{1067105919295}{8686250367492} a^{7} - \frac{2069464460551}{26058751102476} a^{6} + \frac{2828834616059}{8686250367492} a^{5} + \frac{10285297010905}{26058751102476} a^{4} - \frac{389838671009}{8686250367492} a^{3} - \frac{689032147393}{13029375551238} a^{2} - \frac{10013525684561}{26058751102476} a - \frac{3241270470851}{8686250367492}$, $\frac{1}{195394640272531013594175671509232040611953920286497990181619103036211380036} a^{17} + \frac{1733250782734489261856878949680161735149733390573669675504553}{195394640272531013594175671509232040611953920286497990181619103036211380036} a^{16} - \frac{488929413211733268141797119955545777050982642038766995264785389728788909}{195394640272531013594175671509232040611953920286497990181619103036211380036} a^{15} - \frac{1994102894598549411644597283737389556149055800060259189236672361112362215}{48848660068132753398543917877308010152988480071624497545404775759052845009} a^{14} - \frac{15133325169183345858993542714024162986643861852996210856289523065788164009}{195394640272531013594175671509232040611953920286497990181619103036211380036} a^{13} + \frac{727514777450099562679888223267317036126033903101039642005382501111914353}{32565773378755168932362611918205340101992320047749665030269850506035230006} a^{12} - \frac{53380843332339329432930212159489721900238529773606539934857792497645476047}{195394640272531013594175671509232040611953920286497990181619103036211380036} a^{11} + \frac{6366952043579916272659269489571189386131946278951950902139611678504064723}{97697320136265506797087835754616020305976960143248995090809551518105690018} a^{10} + \frac{64935603512395156858678917127512814627724550482377147401364924372462776571}{195394640272531013594175671509232040611953920286497990181619103036211380036} a^{9} + \frac{48426418554147249336268919781726052974671837059334303471353436753618729669}{97697320136265506797087835754616020305976960143248995090809551518105690018} a^{8} - \frac{19094385269560180267562909184874915705662436829813306982013050704182875187}{65131546757510337864725223836410680203984640095499330060539701012070460012} a^{7} - \frac{7279462441988404457940555068618420888508590899413257040403733198253640579}{97697320136265506797087835754616020305976960143248995090809551518105690018} a^{6} + \frac{31840786262407726260055721182272124332498240291494586908904199076772458365}{65131546757510337864725223836410680203984640095499330060539701012070460012} a^{5} + \frac{190422412866001199057275199020388839664701344485703054864357690056011465}{948517671225878706767843065578796313650261748963582476609801471049569806} a^{4} - \frac{10057518300572647565942387891514862408876118914173433586386376092342671721}{48848660068132753398543917877308010152988480071624497545404775759052845009} a^{3} + \frac{1118020666255066256280148967251403995677674904795985279332161654257498481}{21710515585836779288241741278803560067994880031833110020179900337356820004} a^{2} - \frac{7509261910474412391130668484125651363445999235409045172356253139042974687}{32565773378755168932362611918205340101992320047749665030269850506035230006} a - \frac{185333262362118251764558668806010884598918916136773024696638520820410463}{564724393851245704029409455228994337028768555741323671045141916289628266}$

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

Class group and class number

$C_{6}\times C_{18}\times C_{108}\times C_{137484}$, which has order $1603613376$ (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:  \( 1363491204.97286 \) (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

\(\Q(\sqrt{-6}) \), 3.3.1708.1, 3.3.182329.2, 6.0.10082064384.5, 6.0.459563179267584.2, 9.9.165643767257935513792.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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$2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
2.12.24.307$x^{12} + 28 x^{11} - 2 x^{10} + 16 x^{9} + 26 x^{8} + 8 x^{7} + 20 x^{6} - 24 x^{5} - 8 x^{4} + 32 x^{3} + 32 x^{2} + 32 x + 24$$4$$3$$24$$C_6\times C_2$$[2, 3]^{3}$
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.4$x^{6} + 14$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.4$x^{6} + 14$$6$$1$$5$$C_6$$[\ ]_{6}$
$61$61.6.4.3$x^{6} + 183 x^{3} + 14884$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
61.12.10.2$x^{12} + 183 x^{6} + 14884$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$