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Properties

Label 36.0.13401573218...4769.1
Degree $36$
Signature $[0, 18]$
Discriminant $3^{54}\cdot 19^{30}$
Root discriminant $60.44$
Ramified primes $3, 19$
Class number $756$ (GRH)
Class group $[3, 6, 42]$ (GRH)
Galois group $C_6^2$ (as 36T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![117649, 0, 0, -324478, 0, 0, 395165, 0, 0, -870682, 0, 0, 1334658, 0, 0, -601883, 0, 0, 146403, 0, 0, -106620, 0, 0, 43132, 0, 0, -5973, 0, 0, 584, 0, 0, -29, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 29*x^33 + 584*x^30 - 5973*x^27 + 43132*x^24 - 106620*x^21 + 146403*x^18 - 601883*x^15 + 1334658*x^12 - 870682*x^9 + 395165*x^6 - 324478*x^3 + 117649)
 
gp: K = bnfinit(x^36 - 29*x^33 + 584*x^30 - 5973*x^27 + 43132*x^24 - 106620*x^21 + 146403*x^18 - 601883*x^15 + 1334658*x^12 - 870682*x^9 + 395165*x^6 - 324478*x^3 + 117649, 1)
 

Normalized defining polynomial

\( x^{36} - 29 x^{33} + 584 x^{30} - 5973 x^{27} + 43132 x^{24} - 106620 x^{21} + 146403 x^{18} - 601883 x^{15} + 1334658 x^{12} - 870682 x^{9} + 395165 x^{6} - 324478 x^{3} + 117649 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 18]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(13401573218702027067604638251610483016128327492774416692842974769=3^{54}\cdot 19^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.44$
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:  $3, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(171=3^{2}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{171}(1,·)$, $\chi_{171}(134,·)$, $\chi_{171}(7,·)$, $\chi_{171}(8,·)$, $\chi_{171}(11,·)$, $\chi_{171}(140,·)$, $\chi_{171}(145,·)$, $\chi_{171}(20,·)$, $\chi_{171}(151,·)$, $\chi_{171}(26,·)$, $\chi_{171}(31,·)$, $\chi_{171}(160,·)$, $\chi_{171}(163,·)$, $\chi_{171}(164,·)$, $\chi_{171}(37,·)$, $\chi_{171}(170,·)$, $\chi_{171}(46,·)$, $\chi_{171}(49,·)$, $\chi_{171}(50,·)$, $\chi_{171}(56,·)$, $\chi_{171}(58,·)$, $\chi_{171}(64,·)$, $\chi_{171}(65,·)$, $\chi_{171}(68,·)$, $\chi_{171}(77,·)$, $\chi_{171}(83,·)$, $\chi_{171}(88,·)$, $\chi_{171}(94,·)$, $\chi_{171}(103,·)$, $\chi_{171}(106,·)$, $\chi_{171}(107,·)$, $\chi_{171}(113,·)$, $\chi_{171}(115,·)$, $\chi_{171}(121,·)$, $\chi_{171}(122,·)$, $\chi_{171}(125,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7} a^{13} + \frac{1}{7} a^{7} + \frac{1}{7} a$, $\frac{1}{7} a^{14} + \frac{1}{7} a^{8} + \frac{1}{7} a^{2}$, $\frac{1}{7} a^{15} + \frac{1}{7} a^{9} + \frac{1}{7} a^{3}$, $\frac{1}{7} a^{16} + \frac{1}{7} a^{10} + \frac{1}{7} a^{4}$, $\frac{1}{7} a^{17} + \frac{1}{7} a^{11} + \frac{1}{7} a^{5}$, $\frac{1}{7} a^{18} + \frac{1}{7} a^{12} + \frac{1}{7} a^{6}$, $\frac{1}{7} a^{19} - \frac{1}{7} a$, $\frac{1}{7} a^{20} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{21} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{22} - \frac{1}{7} a^{4}$, $\frac{1}{7} a^{23} - \frac{1}{7} a^{5}$, $\frac{1}{7} a^{24} - \frac{1}{7} a^{6}$, $\frac{1}{7} a^{25} - \frac{1}{7} a^{7}$, $\frac{1}{49} a^{26} + \frac{2}{49} a^{20} + \frac{3}{49} a^{14} + \frac{2}{49} a^{8} + \frac{1}{49} a^{2}$, $\frac{1}{111818} a^{27} - \frac{342}{7987} a^{24} + \frac{2143}{55909} a^{21} + \frac{15}{1141} a^{18} - \frac{2711}{55909} a^{15} + \frac{526}{1141} a^{12} + \frac{52103}{111818} a^{9} - \frac{834}{7987} a^{6} + \frac{5387}{55909} a^{3} + \frac{17}{326}$, $\frac{1}{111818} a^{28} - \frac{342}{7987} a^{25} + \frac{2143}{55909} a^{22} + \frac{15}{1141} a^{19} - \frac{2711}{55909} a^{16} + \frac{37}{1141} a^{13} + \frac{52103}{111818} a^{10} + \frac{3730}{7987} a^{7} + \frac{5387}{55909} a^{4} - \frac{859}{2282} a$, $\frac{1}{111818} a^{29} - \frac{16}{7987} a^{26} + \frac{2143}{55909} a^{23} - \frac{384}{7987} a^{20} - \frac{2711}{55909} a^{17} + \frac{96}{7987} a^{14} + \frac{52103}{111818} a^{11} + \frac{463}{1141} a^{8} + \frac{5387}{55909} a^{5} - \frac{5361}{15974} a^{2}$, $\frac{1}{1229998} a^{30} - \frac{1}{1229998} a^{27} - \frac{28538}{614999} a^{24} - \frac{19993}{614999} a^{21} + \frac{17428}{614999} a^{18} - \frac{12843}{614999} a^{15} + \frac{77975}{1229998} a^{12} + \frac{210985}{1229998} a^{9} - \frac{98437}{614999} a^{6} + \frac{176637}{1229998} a^{3} - \frac{773}{3586}$, $\frac{1}{1229998} a^{31} - \frac{1}{1229998} a^{28} - \frac{28538}{614999} a^{25} - \frac{19993}{614999} a^{22} + \frac{17428}{614999} a^{19} - \frac{12843}{614999} a^{16} + \frac{77975}{1229998} a^{13} + \frac{210985}{1229998} a^{10} - \frac{98437}{614999} a^{7} + \frac{176637}{1229998} a^{4} - \frac{773}{3586} a$, $\frac{1}{1229998} a^{32} - \frac{1}{1229998} a^{29} - \frac{3436}{614999} a^{26} - \frac{19993}{614999} a^{23} - \frac{20225}{614999} a^{20} - \frac{12843}{614999} a^{17} + \frac{52873}{1229998} a^{14} + \frac{210985}{1229998} a^{11} - \frac{136090}{614999} a^{8} + \frac{176637}{1229998} a^{5} - \frac{30705}{175714} a^{2}$, $\frac{1}{10003091365956122868505918} a^{33} - \frac{683584727500597765}{5001545682978061434252959} a^{30} - \frac{25200907808937613}{102072360877103294576591} a^{27} - \frac{355319694561006907468680}{5001545682978061434252959} a^{24} + \frac{17218932248458041171339}{5001545682978061434252959} a^{21} - \frac{279797113814050771078429}{5001545682978061434252959} a^{18} - \frac{714005405218866299399445}{10003091365956122868505918} a^{15} + \frac{1421198072042598669190064}{5001545682978061434252959} a^{12} + \frac{804773194728884013206527}{5001545682978061434252959} a^{9} + \frac{4424990042576261477348779}{10003091365956122868505918} a^{6} - \frac{675600328660426578956065}{5001545682978061434252959} a^{3} + \frac{666463059704291329351}{14581765839586184939513}$, $\frac{1}{70021639561692860079541426} a^{34} - \frac{863616116838377961}{6365603596517532734503766} a^{31} + \frac{47560802498687829209}{35010819780846430039770713} a^{28} + \frac{377111099236523154905821}{35010819780846430039770713} a^{25} - \frac{2486501979579027774695003}{35010819780846430039770713} a^{22} + \frac{1787739509647363966680919}{35010819780846430039770713} a^{19} + \frac{4725895972415935209825455}{70021639561692860079541426} a^{16} + \frac{1103620192160464991128205}{70021639561692860079541426} a^{13} - \frac{4867688236418916126542809}{35010819780846430039770713} a^{10} - \frac{22169678536048552679862469}{70021639561692860079541426} a^{7} - \frac{30404153267825708616066653}{70021639561692860079541426} a^{4} - \frac{39175182700548201709508}{102072360877103294576591} a$, $\frac{1}{490151476931850020556789982} a^{35} - \frac{863616116838377961}{44559225175622729141526362} a^{32} - \frac{578650000428326247948}{245075738465925010278394991} a^{29} - \frac{1626137259326994877919422}{245075738465925010278394991} a^{26} - \frac{168895797946148675136946}{245075738465925010278394991} a^{23} + \frac{15871846678278837576019006}{245075738465925010278394991} a^{20} + \frac{21519617285312598731021881}{490151476931850020556789982} a^{17} + \frac{26571613547202127509103395}{490151476931850020556789982} a^{14} - \frac{102515243580039929233942447}{245075738465925010278394991} a^{11} + \frac{232527788916850028894049827}{490151476931850020556789982} a^{8} + \frac{236188814597474432367520015}{490151476931850020556789982} a^{5} - \frac{166474893066996920537281}{714506526139723062036137} a^{2}$

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

Class group and class number

$C_{3}\times C_{6}\times C_{42}$, which has order $756$ (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:  $17$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{923190657524195357}{64955138739974823821467} a^{35} - \frac{376216007488709476203}{909371942359647533500538} a^{32} + \frac{540067001267869045925}{64955138739974823821467} a^{29} - \frac{38645480084335274688870}{454685971179823766750269} a^{26} + \frac{39407404381382012210629}{64955138739974823821467} a^{23} - \frac{656462217340336671701788}{454685971179823766750269} a^{20} + \frac{86292983585252471704543}{64955138739974823821467} a^{17} - \frac{7454330131825724013869939}{909371942359647533500538} a^{14} + \frac{1159233791555800453201708}{64955138739974823821467} a^{11} - \frac{1021065831999859772015020}{454685971179823766750269} a^{8} + \frac{5889749282309807780379}{2651230152652033625366} a^{5} - \frac{205446867111044266614083}{64955138739974823821467} a^{2} \) (order $18$)
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:  \( 2007721725946302.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6^2$ (as 36T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 36
The 36 conjugacy class representatives for $C_6^2$
Character table for $C_6^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{57}) \), \(\Q(\sqrt{-19}) \), \(\Q(\zeta_{9})^+\), 3.3.29241.1, 3.3.361.1, 3.3.29241.2, \(\Q(\sqrt{-3}, \sqrt{-19})\), \(\Q(\zeta_{9})\), 6.0.2565108243.1, 6.0.3518667.1, 6.0.2565108243.2, 6.6.135005697.1, 6.0.45001899.1, 6.6.48737056617.2, 6.0.16245685539.2, 6.6.66854673.1, 6.0.2476099.1, 6.6.48737056617.1, 6.0.16245685539.1, 9.9.25002110044521.1, 12.0.18226538222455809.1, 12.0.2375300687688663484689.1, 12.0.4469547301936929.1, 12.0.2375300687688663484689.2, 18.0.16877848680315122776257224907.4, 18.18.115765164098281427122348305637113.1, 18.0.4287598670306719523049937245819.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{18}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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
3Data not computed
$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$$[\ ]_{6}$
19.6.5.5$x^{6} + 1216$$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}$