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Properties

Label 36.0.19575552194...9664.1
Degree $36$
Signature $[0, 18]$
Discriminant $2^{36}\cdot 3^{54}\cdot 19^{24}$
Root discriminant $74.00$
Ramified primes $2, 3, 19$
Class number $15984$ (GRH)
Class group $[3, 12, 444]$ (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![13841287201, 0, 0, 0, 0, 0, -4312188797, 0, 0, 0, 0, 0, 1297088703, 0, 0, 0, 0, 0, -14205984, 0, 0, 0, 0, 0, 118583, 0, 0, 0, 0, 0, -394, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 394*x^30 + 118583*x^24 - 14205984*x^18 + 1297088703*x^12 - 4312188797*x^6 + 13841287201)
 
gp: K = bnfinit(x^36 - 394*x^30 + 118583*x^24 - 14205984*x^18 + 1297088703*x^12 - 4312188797*x^6 + 13841287201, 1)
 

Normalized defining polynomial

\( x^{36} - 394 x^{30} + 118583 x^{24} - 14205984 x^{18} + 1297088703 x^{12} - 4312188797 x^{6} + 13841287201 \)

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:  \(19575552194003861659818831697024491945650963873549328070183080689664=2^{36}\cdot 3^{54}\cdot 19^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.00$
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:  $2, 3, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(684=2^{2}\cdot 3^{2}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{684}(1,·)$, $\chi_{684}(235,·)$, $\chi_{684}(391,·)$, $\chi_{684}(11,·)$, $\chi_{684}(653,·)$, $\chi_{684}(121,·)$, $\chi_{684}(277,·)$, $\chi_{684}(539,·)$, $\chi_{684}(197,·)$, $\chi_{684}(163,·)$, $\chi_{684}(305,·)$, $\chi_{684}(425,·)$, $\chi_{684}(7,·)$, $\chi_{684}(49,·)$, $\chi_{684}(115,·)$, $\chi_{684}(311,·)$, $\chi_{684}(571,·)$, $\chi_{684}(191,·)$, $\chi_{684}(577,·)$, $\chi_{684}(83,·)$, $\chi_{684}(581,·)$, $\chi_{684}(457,·)$, $\chi_{684}(77,·)$, $\chi_{684}(463,·)$, $\chi_{684}(419,·)$, $\chi_{684}(343,·)$, $\chi_{684}(349,·)$, $\chi_{684}(353,·)$, $\chi_{684}(229,·)$, $\chi_{684}(619,·)$, $\chi_{684}(647,·)$, $\chi_{684}(239,·)$, $\chi_{684}(467,·)$, $\chi_{684}(505,·)$, $\chi_{684}(125,·)$, $\chi_{684}(533,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7} a^{13} - \frac{1}{7} a^{7} + \frac{1}{7} a$, $\frac{1}{7} a^{14} - \frac{1}{7} a^{8} + \frac{1}{7} a^{2}$, $\frac{1}{7} a^{15} - \frac{1}{7} a^{9} + \frac{1}{7} a^{3}$, $\frac{1}{7} a^{16} - \frac{1}{7} a^{10} + \frac{1}{7} a^{4}$, $\frac{1}{7} a^{17} - \frac{1}{7} a^{11} + \frac{1}{7} a^{5}$, $\frac{1}{7} a^{18} - \frac{1}{7} a^{12} + \frac{1}{7} a^{6}$, $\frac{1}{7} a^{19} + \frac{1}{7} a$, $\frac{1}{7} a^{20} + \frac{1}{7} a^{2}$, $\frac{1}{7} a^{21} + \frac{1}{7} a^{3}$, $\frac{1}{7} a^{22} + \frac{1}{7} a^{4}$, $\frac{1}{7} a^{23} + \frac{1}{7} a^{5}$, $\frac{1}{189847} a^{24} - \frac{6796}{189847} a^{18} + \frac{64476}{189847} a^{12} + \frac{28030}{189847} a^{6} - \frac{3873}{27121}$, $\frac{1}{189847} a^{25} - \frac{6796}{189847} a^{19} + \frac{1462}{27121} a^{13} + \frac{82272}{189847} a^{7} - \frac{81353}{189847} a$, $\frac{1}{1328929} a^{26} - \frac{33917}{1328929} a^{20} + \frac{37355}{1328929} a^{14} - \frac{324543}{1328929} a^{8} + \frac{108494}{1328929} a^{2}$, $\frac{1}{9302503} a^{27} - \frac{413611}{9302503} a^{21} - \frac{532186}{9302503} a^{15} - \frac{2412860}{9302503} a^{9} + \frac{1817117}{9302503} a^{3}$, $\frac{1}{65117521} a^{28} - \frac{4400398}{65117521} a^{22} + \frac{3454601}{65117521} a^{16} + \frac{21507862}{65117521} a^{10} - \frac{7485386}{65117521} a^{4}$, $\frac{1}{455822647} a^{29} - \frac{13702901}{455822647} a^{23} - \frac{5847902}{455822647} a^{17} - \frac{34307156}{455822647} a^{11} - \frac{91207913}{455822647} a^{5}$, $\frac{1}{485357981256970509986599} a^{30} + \frac{248754877568597426}{485357981256970509986599} a^{24} - \frac{14218616539342088106811}{485357981256970509986599} a^{18} + \frac{236667550062985638885405}{485357981256970509986599} a^{12} + \frac{3687747521471003448410}{485357981256970509986599} a^{6} - \frac{1268818109576643066}{4125474770350538551}$, $\frac{1}{3397505868798793569906193} a^{31} + \frac{5361903689580064660}{3397505868798793569906193} a^{25} - \frac{187641284796335022282389}{3397505868798793569906193} a^{19} + \frac{5936720555130526841392}{91824482940507934321789} a^{13} - \frac{962380362722351592809082}{3397505868798793569906193} a^{7} - \frac{10698040117042783094}{28878323392453769857} a$, $\frac{1}{23782541081591554989343351} a^{32} + \frac{5361903689580064660}{23782541081591554989343351} a^{26} - \frac{672999266053305532268988}{23782541081591554989343351} a^{20} + \frac{32172287109561365219046}{642771380583555540252523} a^{14} + \frac{1464409543562500957123913}{23782541081591554989343351} a^{8} - \frac{938937906670320649}{28878323392453769857} a^{2}$, $\frac{1}{166477787571140884925403457} a^{33} + \frac{5361903689580064660}{166477787571140884925403457} a^{27} + \frac{2724506602745488037637205}{166477787571140884925403457} a^{21} + \frac{307645735931085168184413}{4499399664084888781767661} a^{15} + \frac{62619515181940785215435387}{166477787571140884925403457} a^{9} - \frac{10153144143232782288}{28878323392453769857} a^{3}$, $\frac{1}{1165344512997986194477824199} a^{34} + \frac{5361903689580064660}{1165344512997986194477824199} a^{28} + \frac{74072129847520153005667258}{1165344512997986194477824199} a^{22} + \frac{950417116514640708436936}{31495797648594221472373627} a^{16} + \frac{371792549242631000076898950}{1165344512997986194477824199} a^{10} + \frac{64105401723076911630}{202148263747176388999} a^{4}$, $\frac{1}{8157411590985903361344769393} a^{35} + \frac{5361903689580064660}{8157411590985903361344769393} a^{29} + \frac{573505492560942807781877629}{8157411590985903361344769393} a^{23} + \frac{14448616108769307053739919}{220470583540159550306615389} a^{17} - \frac{127640813470791654699311421}{8157411590985903361344769393} a^{11} - \frac{166921185416553247226}{1415037846230234722993} a^{5}$

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

Class group and class number

$C_{3}\times C_{12}\times C_{444}$, which has order $15984$ (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:  \( \frac{745606568130638905}{8157411590985903361344769393} a^{35} - \frac{291275732977210271517}{8157411590985903361344769393} a^{29} + \frac{87473322363672887039805}{8157411590985903361344769393} a^{23} - \frac{10308275970137477235408725}{8157411590985903361344769393} a^{17} + \frac{934315988642476886763757963}{8157411590985903361344769393} a^{11} - \frac{56104081927926830}{4125474770350538551} a^{5} \) (order $36$)
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:  \( 10470974244023312 \) (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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{9})^+\), 3.3.29241.2, 3.3.361.1, 3.3.29241.1, \(\Q(\zeta_{12})\), \(\Q(\zeta_{9})\), 6.0.2565108243.2, 6.0.3518667.1, 6.0.2565108243.1, \(\Q(\zeta_{36})^+\), 6.0.419904.1, 6.6.164166927552.1, 6.0.54722309184.1, 6.6.225194688.1, 6.0.8340544.1, 6.6.164166927552.2, 6.0.54722309184.5, 9.9.25002110044521.1, \(\Q(\zeta_{36})\), 12.0.26950780101863616712704.1, 12.0.50712647503417344.1, 12.0.26950780101863616712704.2, 18.0.16877848680315122776257224907.4, 18.18.4424426764452527545059173966020608.1, 18.0.163867657942686205372561998741504.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{36}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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
$2$2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
3Data not computed
$19$19.6.4.3$x^{6} + 95 x^{3} + 2888$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
19.6.4.3$x^{6} + 95 x^{3} + 2888$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
19.6.4.3$x^{6} + 95 x^{3} + 2888$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
19.6.4.3$x^{6} + 95 x^{3} + 2888$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
19.6.4.3$x^{6} + 95 x^{3} + 2888$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
19.6.4.3$x^{6} + 95 x^{3} + 2888$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$