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Properties

Label 18.0.22497960113...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{27}\cdot 5^{9}\cdot 79^{14}$
Root discriminant $551.81$
Ramified primes $2, 3, 5, 79$
Class number $22706193888$ (GRH)
Class group $[6, 84, 45051972]$ (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![15157286557696, 24153643711488, 19470659444736, 9516122868224, 2638164247776, 185836782000, -128655698368, -39884181384, -1050471018, 1433969187, 205547661, -17815248, -5537100, -10914, 69018, 1840, -402, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 - 402*x^16 + 1840*x^15 + 69018*x^14 - 10914*x^13 - 5537100*x^12 - 17815248*x^11 + 205547661*x^10 + 1433969187*x^9 - 1050471018*x^8 - 39884181384*x^7 - 128655698368*x^6 + 185836782000*x^5 + 2638164247776*x^4 + 9516122868224*x^3 + 19470659444736*x^2 + 24153643711488*x + 15157286557696)
 
gp: K = bnfinit(x^18 - 9*x^17 - 402*x^16 + 1840*x^15 + 69018*x^14 - 10914*x^13 - 5537100*x^12 - 17815248*x^11 + 205547661*x^10 + 1433969187*x^9 - 1050471018*x^8 - 39884181384*x^7 - 128655698368*x^6 + 185836782000*x^5 + 2638164247776*x^4 + 9516122868224*x^3 + 19470659444736*x^2 + 24153643711488*x + 15157286557696, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} - 402 x^{16} + 1840 x^{15} + 69018 x^{14} - 10914 x^{13} - 5537100 x^{12} - 17815248 x^{11} + 205547661 x^{10} + 1433969187 x^{9} - 1050471018 x^{8} - 39884181384 x^{7} - 128655698368 x^{6} + 185836782000 x^{5} + 2638164247776 x^{4} + 9516122868224 x^{3} + 19470659444736 x^{2} + 24153643711488 x + 15157286557696 \)

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:  \(-22497960113751479934820492718164014658776000000000=-\,2^{12}\cdot 3^{27}\cdot 5^{9}\cdot 79^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $551.81$
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:  $2, 3, 5, 79$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{7} - \frac{1}{8} a^{4} - \frac{1}{16} a^{3} + \frac{1}{8} a^{2}$, $\frac{1}{32} a^{8} - \frac{1}{16} a^{6} - \frac{1}{8} a^{5} - \frac{3}{32} a^{4} + \frac{1}{8} a^{3} + \frac{1}{8} a^{2}$, $\frac{1}{192} a^{9} - \frac{1}{64} a^{8} - \frac{1}{32} a^{7} + \frac{1}{96} a^{6} + \frac{3}{64} a^{5} + \frac{7}{64} a^{4} - \frac{5}{48} a^{3} + \frac{1}{16} a^{2} - \frac{1}{4} a - \frac{1}{6}$, $\frac{1}{192} a^{10} - \frac{1}{64} a^{8} - \frac{1}{48} a^{7} - \frac{3}{64} a^{6} - \frac{17}{192} a^{4} - \frac{1}{16} a^{3} - \frac{3}{16} a^{2} - \frac{5}{12} a - \frac{1}{2}$, $\frac{1}{768} a^{11} + \frac{1}{768} a^{9} + \frac{1}{96} a^{8} - \frac{3}{256} a^{7} - \frac{1}{48} a^{6} + \frac{43}{768} a^{5} - \frac{1}{32} a^{4} + \frac{31}{192} a^{3} + \frac{5}{24} a^{2} + \frac{1}{8} a - \frac{1}{6}$, $\frac{1}{60672} a^{12} - \frac{1}{10112} a^{11} - \frac{119}{60672} a^{10} + \frac{3}{10112} a^{9} + \frac{165}{20224} a^{8} + \frac{187}{30336} a^{7} - \frac{159}{20224} a^{6} + \frac{697}{10112} a^{5} + \frac{1585}{15168} a^{4} + \frac{1645}{7584} a^{3} - \frac{133}{632} a^{2} + \frac{457}{948} a + \frac{16}{237}$, $\frac{1}{121344} a^{13} - \frac{1}{121344} a^{12} + \frac{3}{40448} a^{11} + \frac{55}{121344} a^{10} + \frac{37}{40448} a^{9} + \frac{107}{40448} a^{8} - \frac{2557}{121344} a^{7} + \frac{2495}{40448} a^{6} + \frac{467}{10112} a^{5} - \frac{159}{10112} a^{4} + \frac{413}{3792} a^{3} + \frac{149}{632} a^{2} + \frac{106}{237} a - \frac{157}{474}$, $\frac{1}{970752} a^{14} - \frac{1}{323584} a^{13} + \frac{7}{970752} a^{12} + \frac{61}{970752} a^{11} - \frac{155}{970752} a^{10} - \frac{605}{970752} a^{9} + \frac{13781}{970752} a^{8} - \frac{20497}{970752} a^{7} - \frac{9101}{485376} a^{6} - \frac{1697}{80896} a^{5} - \frac{3253}{121344} a^{4} + \frac{593}{5056} a^{3} - \frac{1757}{7584} a^{2} - \frac{11}{3792} a + \frac{283}{1896}$, $\frac{1}{5824512} a^{15} - \frac{1}{2912256} a^{14} - \frac{1}{485376} a^{13} + \frac{5}{1456128} a^{12} + \frac{1337}{2912256} a^{11} - \frac{1781}{728064} a^{10} + \frac{1597}{728064} a^{9} - \frac{20995}{1456128} a^{8} + \frac{36421}{5824512} a^{7} - \frac{38153}{970752} a^{6} - \frac{1453}{161792} a^{5} - \frac{46037}{728064} a^{4} - \frac{20885}{91008} a^{3} - \frac{7187}{45504} a^{2} + \frac{6599}{22752} a - \frac{67}{144}$, $\frac{1}{93192192} a^{16} - \frac{1}{15532032} a^{15} + \frac{19}{46596096} a^{14} + \frac{115}{46596096} a^{13} + \frac{73}{11649024} a^{12} + \frac{18905}{46596096} a^{11} + \frac{481}{196608} a^{10} + \frac{35731}{15532032} a^{9} + \frac{1137767}{93192192} a^{8} + \frac{444755}{23298048} a^{7} - \frac{24803}{485376} a^{6} - \frac{452485}{5824512} a^{5} - \frac{608563}{5824512} a^{4} - \frac{15587}{364032} a^{3} + \frac{419}{7584} a^{2} + \frac{9467}{91008} a - \frac{16111}{91008}$, $\frac{1}{35728077418013854491087937451558898461966615552012845056} a^{17} - \frac{127639953634274406153923426898162033902162263849}{35728077418013854491087937451558898461966615552012845056} a^{16} + \frac{60634114659664683389363846281818461105237149391}{4466009677251731811385992181444862307745826944001605632} a^{15} - \frac{2886726890451432974275745984964661846455146953123}{8932019354503463622771984362889724615491653888003211264} a^{14} - \frac{14155092889218355654907220936328291399922995094503}{5954679569668975748514656241926483076994435925335474176} a^{13} + \frac{8919150394265247084041960344420937424522157442917}{1984893189889658582838218747308827692331478641778491392} a^{12} - \frac{5749914483750296875136905413549618089136789481229951}{8932019354503463622771984362889724615491653888003211264} a^{11} + \frac{7783382939473687412626501788267288594190297530361225}{8932019354503463622771984362889724615491653888003211264} a^{10} + \frac{16953120678410164503653772731660256483162416956086619}{11909359139337951497029312483852966153988871850670948352} a^{9} + \frac{157186027392484959077678601455762911161077127833108983}{35728077418013854491087937451558898461966615552012845056} a^{8} + \frac{13464219728323131194656947959328257446660929627586383}{992446594944829291419109373654413846165739320889245696} a^{7} - \frac{62064025508856488833509315743835412708465421425653793}{2233004838625865905692996090722431153872913472000802816} a^{6} - \frac{1561784969665031167280475257289315467103292378085117}{62027912184051830713694335853400865385358707555577856} a^{5} + \frac{3184963523868642913808080518620349457949356513437857}{248111648736207322854777343413603461541434830222311424} a^{4} + \frac{6379922340406023913510881231135422213816292850230433}{139562802414116619105812255670151947117057092000050176} a^{3} - \frac{1119775071682202858533967523008149578705690435838123}{11630233534509718258817687972512662259754757666670848} a^{2} + \frac{523897218495617149602342534400700295977863144533497}{1090334393860286086764158247423062086852008531250392} a + \frac{221911727229566639412169190304246717667352489395247}{11630233534509718258817687972512662259754757666670848}$

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

Class group and class number

$C_{6}\times C_{84}\times C_{45051972}$, which has order $22706193888$ (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:  \( 32038743174.66701 \) (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

\(\Q(\sqrt{-15}) \), 3.3.505521.1, 3.3.316.1, 6.0.337014000.4, 6.0.95831805540375.1, 9.9.653167654112852807616.1

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 R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$$0$Trivial$[\ ]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
3Data not computed
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$79$79.3.2.3$x^{3} - 316$$3$$1$$2$$C_3$$[\ ]_{3}$
79.3.2.3$x^{3} - 316$$3$$1$$2$$C_3$$[\ ]_{3}$
79.6.5.2$x^{6} - 316$$6$$1$$5$$C_6$$[\ ]_{6}$
79.6.5.2$x^{6} - 316$$6$$1$$5$$C_6$$[\ ]_{6}$