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Properties

Label 18.0.45356173679...1152.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{24}\cdot 7^{9}\cdot 19^{15}$
Root discriminant $266.30$
Ramified primes $2, 3, 7, 19$
Class number $168942592$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 28, 94276]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1160832069377, 154616908968, 534053961429, 28913721202, 112232882907, 2149578306, 15665949557, 107838672, 1454420187, -4593018, 88101033, -129816, 3360784, -708, 54000, -6, 381, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 381*x^16 - 6*x^15 + 54000*x^14 - 708*x^13 + 3360784*x^12 - 129816*x^11 + 88101033*x^10 - 4593018*x^9 + 1454420187*x^8 + 107838672*x^7 + 15665949557*x^6 + 2149578306*x^5 + 112232882907*x^4 + 28913721202*x^3 + 534053961429*x^2 + 154616908968*x + 1160832069377)
 
gp: K = bnfinit(x^18 + 381*x^16 - 6*x^15 + 54000*x^14 - 708*x^13 + 3360784*x^12 - 129816*x^11 + 88101033*x^10 - 4593018*x^9 + 1454420187*x^8 + 107838672*x^7 + 15665949557*x^6 + 2149578306*x^5 + 112232882907*x^4 + 28913721202*x^3 + 534053961429*x^2 + 154616908968*x + 1160832069377, 1)
 

Normalized defining polynomial

\( x^{18} + 381 x^{16} - 6 x^{15} + 54000 x^{14} - 708 x^{13} + 3360784 x^{12} - 129816 x^{11} + 88101033 x^{10} - 4593018 x^{9} + 1454420187 x^{8} + 107838672 x^{7} + 15665949557 x^{6} + 2149578306 x^{5} + 112232882907 x^{4} + 28913721202 x^{3} + 534053961429 x^{2} + 154616908968 x + 1160832069377 \)

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:  \(-45356173679730341731994070939714799306801152=-\,2^{18}\cdot 3^{24}\cdot 7^{9}\cdot 19^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $266.30$
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:  $2, 3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4788=2^{2}\cdot 3^{2}\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{4788}(1,·)$, $\chi_{4788}(3193,·)$, $\chi_{4788}(4591,·)$, $\chi_{4788}(1063,·)$, $\chi_{4788}(2659,·)$, $\chi_{4788}(4453,·)$, $\chi_{4788}(3751,·)$, $\chi_{4788}(2857,·)$, $\chi_{4788}(2155,·)$, $\chi_{4788}(1261,·)$, $\chi_{4788}(559,·)$, $\chi_{4788}(3697,·)$, $\chi_{4788}(2995,·)$, $\chi_{4788}(2101,·)$, $\chi_{4788}(1399,·)$, $\chi_{4788}(505,·)$, $\chi_{4788}(4255,·)$, $\chi_{4788}(1597,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} + \frac{1}{7} a^{4} - \frac{2}{7} a^{3} + \frac{2}{7} a^{2} - \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{7} + \frac{1}{7} a^{5} - \frac{2}{7} a^{4} + \frac{2}{7} a^{3} - \frac{1}{7} a^{2} + \frac{1}{7} a$, $\frac{1}{14} a^{8} - \frac{1}{7} a^{5} + \frac{1}{14} a^{4} - \frac{3}{7} a^{3} - \frac{1}{14} a^{2} - \frac{3}{7} a - \frac{1}{14}$, $\frac{1}{14} a^{9} + \frac{1}{14} a^{5} - \frac{2}{7} a^{4} - \frac{5}{14} a^{3} - \frac{1}{7} a^{2} - \frac{3}{14} a + \frac{1}{7}$, $\frac{1}{14} a^{10} - \frac{1}{14} a^{6} - \frac{2}{7} a^{5} - \frac{1}{2} a^{4} + \frac{1}{7} a^{3} - \frac{1}{2} a^{2} + \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{14} a^{11} - \frac{1}{14} a^{7} - \frac{1}{2} a^{5} + \frac{3}{7} a^{4} - \frac{1}{14} a^{3} - \frac{1}{7} a^{2} - \frac{3}{7} a + \frac{2}{7}$, $\frac{1}{686} a^{12} - \frac{19}{686} a^{10} - \frac{2}{343} a^{9} - \frac{1}{343} a^{8} + \frac{18}{343} a^{7} + \frac{5}{343} a^{6} - \frac{131}{343} a^{5} - \frac{151}{686} a^{4} - \frac{88}{343} a^{3} - \frac{85}{343} a^{2} - \frac{113}{343} a - \frac{265}{686}$, $\frac{1}{686} a^{13} - \frac{19}{686} a^{11} - \frac{2}{343} a^{10} - \frac{1}{343} a^{9} - \frac{13}{686} a^{8} + \frac{5}{343} a^{7} + \frac{16}{343} a^{6} - \frac{53}{686} a^{5} + \frac{69}{686} a^{4} + \frac{111}{343} a^{3} - \frac{275}{686} a^{2} - \frac{265}{686} a - \frac{1}{2}$, $\frac{1}{686} a^{14} - \frac{2}{343} a^{11} - \frac{10}{343} a^{10} + \frac{9}{686} a^{9} + \frac{3}{98} a^{8} + \frac{15}{343} a^{7} - \frac{5}{343} a^{6} - \frac{107}{686} a^{5} + \frac{293}{686} a^{4} + \frac{1}{98} a^{3} - \frac{261}{686} a^{2} + \frac{67}{686} a + \frac{55}{343}$, $\frac{1}{686} a^{15} - \frac{10}{343} a^{11} - \frac{9}{343} a^{10} + \frac{5}{686} a^{9} + \frac{11}{343} a^{8} + \frac{18}{343} a^{7} - \frac{9}{343} a^{6} + \frac{323}{686} a^{5} + \frac{20}{343} a^{4} + \frac{113}{686} a^{3} + \frac{12}{343} a^{2} - \frac{54}{343} a + \frac{156}{343}$, $\frac{1}{780817258706036241193462852446028} a^{16} - \frac{81395515194059297762424768312}{195204314676509060298365713111507} a^{15} - \frac{3914128687032693636371699253}{27886330668072722899766530444501} a^{14} + \frac{5431188216306168118760045007}{390408629353018120596731426223014} a^{13} + \frac{21281395380735522188014902585}{390408629353018120596731426223014} a^{12} - \frac{10446177363483298944188285890437}{390408629353018120596731426223014} a^{11} - \frac{2483492921335383553436372043017}{195204314676509060298365713111507} a^{10} - \frac{4402999427315159559081601197508}{195204314676509060298365713111507} a^{9} + \frac{2679983079226034922120370823879}{780817258706036241193462852446028} a^{8} - \frac{3592160315765480097172808106127}{195204314676509060298365713111507} a^{7} + \frac{6481270641127936111304473394416}{195204314676509060298365713111507} a^{6} + \frac{87732620478691388744240304888479}{390408629353018120596731426223014} a^{5} - \frac{43269904307820401169289695901105}{111545322672290891599066121778004} a^{4} - \frac{62546894649797301877322749170461}{195204314676509060298365713111507} a^{3} - \frac{60026440377420249297119222464411}{195204314676509060298365713111507} a^{2} - \frac{151718719611214898187289330320105}{390408629353018120596731426223014} a + \frac{349954406328455822983202055992137}{780817258706036241193462852446028}$, $\frac{1}{173841400332076261788912558127401577283304814513276} a^{17} + \frac{17946247565513508}{43460350083019065447228139531850394320826203628319} a^{16} + \frac{3093539087485681332518991546457738388513186659}{12417242880862590127779468437671541234521772465234} a^{15} - \frac{29082293436284623247444076865511896521567376203}{86920700166038130894456279063700788641652407256638} a^{14} + \frac{10350773716036246621716617290209800632973448262}{43460350083019065447228139531850394320826203628319} a^{13} - \frac{27623179100925141229748005151133664197649685367}{86920700166038130894456279063700788641652407256638} a^{12} - \frac{2783325374280607191285527982438102613624949039315}{86920700166038130894456279063700788641652407256638} a^{11} + \frac{184938213260554304623014039296751886674448105222}{43460350083019065447228139531850394320826203628319} a^{10} + \frac{2565391047297915340688741592741352903260557610121}{173841400332076261788912558127401577283304814513276} a^{9} + \frac{2266671736068292885361197925627902294626050903471}{86920700166038130894456279063700788641652407256638} a^{8} - \frac{20673311376373832093567143669968547241348158007}{43460350083019065447228139531850394320826203628319} a^{7} + \frac{1517302357490024404249505766747816189978166499887}{86920700166038130894456279063700788641652407256638} a^{6} + \frac{29812559364055013977096633231275997729199391956165}{173841400332076261788912558127401577283304814513276} a^{5} - \frac{32107539572919508755029436179074997051897911880275}{86920700166038130894456279063700788641652407256638} a^{4} + \frac{2199067027839709497187930547736059300446326610709}{12417242880862590127779468437671541234521772465234} a^{3} - \frac{16859455779222875301464673779115156041719612375133}{43460350083019065447228139531850394320826203628319} a^{2} + \frac{79754041974115567108593354877522823190704768614395}{173841400332076261788912558127401577283304814513276} a + \frac{14909426500991293116866990283608513297294655516265}{86920700166038130894456279063700788641652407256638}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{28}\times C_{94276}$, which has order $168942592$ (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:  \( 1472619.082400847 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-133}) \), \(\Q(\zeta_{9})^+\), 3.3.361.1, 3.3.29241.1, 3.3.29241.2, 6.0.987881686848.8, 6.0.54355325248.1, 6.0.356625288952128.2, 6.0.356625288952128.1, 9.9.25002110044521.1

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

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
3Data not computed
$7$7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$19$19.6.5.5$x^{6} + 1216$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.5$x^{6} + 1216$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.5$x^{6} + 1216$$6$$1$$5$$C_6$$[\ ]_{6}$