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Properties

Label 36.0.90067643300...9616.2
Degree $36$
Signature $[0, 18]$
Discriminant $2^{36}\cdot 3^{54}\cdot 7^{30}$
Root discriminant $52.60$
Ramified primes $2, 3, 7$
Class number $5096$ (GRH)
Class group $[2, 14, 182]$ (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![729, 0, 2187, 0, 5103, 0, 11178, 0, 24057, 0, 51516, 0, 110160, 0, 86994, 0, 57834, 0, 36234, 0, 22032, 0, 12906, 0, 6732, 0, 1728, 0, 441, 0, 111, 0, 27, 0, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 6*x^34 + 27*x^32 + 111*x^30 + 441*x^28 + 1728*x^26 + 6732*x^24 + 12906*x^22 + 22032*x^20 + 36234*x^18 + 57834*x^16 + 86994*x^14 + 110160*x^12 + 51516*x^10 + 24057*x^8 + 11178*x^6 + 5103*x^4 + 2187*x^2 + 729)
 
gp: K = bnfinit(x^36 + 6*x^34 + 27*x^32 + 111*x^30 + 441*x^28 + 1728*x^26 + 6732*x^24 + 12906*x^22 + 22032*x^20 + 36234*x^18 + 57834*x^16 + 86994*x^14 + 110160*x^12 + 51516*x^10 + 24057*x^8 + 11178*x^6 + 5103*x^4 + 2187*x^2 + 729, 1)
 

Normalized defining polynomial

\( x^{36} + 6 x^{34} + 27 x^{32} + 111 x^{30} + 441 x^{28} + 1728 x^{26} + 6732 x^{24} + 12906 x^{22} + 22032 x^{20} + 36234 x^{18} + 57834 x^{16} + 86994 x^{14} + 110160 x^{12} + 51516 x^{10} + 24057 x^{8} + 11178 x^{6} + 5103 x^{4} + 2187 x^{2} + 729 \)

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:  \(90067643300370785938616861622694756230952958181429238736879616=2^{36}\cdot 3^{54}\cdot 7^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.60$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(252=2^{2}\cdot 3^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{252}(1,·)$, $\chi_{252}(131,·)$, $\chi_{252}(11,·)$, $\chi_{252}(13,·)$, $\chi_{252}(143,·)$, $\chi_{252}(145,·)$, $\chi_{252}(23,·)$, $\chi_{252}(25,·)$, $\chi_{252}(155,·)$, $\chi_{252}(157,·)$, $\chi_{252}(37,·)$, $\chi_{252}(167,·)$, $\chi_{252}(169,·)$, $\chi_{252}(47,·)$, $\chi_{252}(179,·)$, $\chi_{252}(181,·)$, $\chi_{252}(59,·)$, $\chi_{252}(61,·)$, $\chi_{252}(191,·)$, $\chi_{252}(193,·)$, $\chi_{252}(71,·)$, $\chi_{252}(73,·)$, $\chi_{252}(205,·)$, $\chi_{252}(83,·)$, $\chi_{252}(85,·)$, $\chi_{252}(215,·)$, $\chi_{252}(95,·)$, $\chi_{252}(97,·)$, $\chi_{252}(227,·)$, $\chi_{252}(229,·)$, $\chi_{252}(107,·)$, $\chi_{252}(109,·)$, $\chi_{252}(239,·)$, $\chi_{252}(241,·)$, $\chi_{252}(121,·)$, $\chi_{252}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6}$, $\frac{1}{3} a^{7}$, $\frac{1}{3} a^{8}$, $\frac{1}{3} a^{9}$, $\frac{1}{3} a^{10}$, $\frac{1}{3} a^{11}$, $\frac{1}{9} a^{12}$, $\frac{1}{9} a^{13}$, $\frac{1}{36} a^{14} + \frac{1}{4}$, $\frac{1}{36} a^{15} + \frac{1}{4} a$, $\frac{1}{36} a^{16} + \frac{1}{4} a^{2}$, $\frac{1}{36} a^{17} + \frac{1}{4} a^{3}$, $\frac{1}{108} a^{18} - \frac{1}{4} a^{4}$, $\frac{1}{108} a^{19} - \frac{1}{4} a^{5}$, $\frac{1}{108} a^{20} + \frac{1}{12} a^{6}$, $\frac{1}{108} a^{21} + \frac{1}{12} a^{7}$, $\frac{1}{108} a^{22} + \frac{1}{12} a^{8}$, $\frac{1}{108} a^{23} + \frac{1}{12} a^{9}$, $\frac{1}{324} a^{24} - \frac{1}{12} a^{10}$, $\frac{1}{324} a^{25} - \frac{1}{12} a^{11}$, $\frac{1}{4531788} a^{26} + \frac{3353}{2265894} a^{24} + \frac{901}{251766} a^{22} - \frac{139}{125883} a^{20} + \frac{998}{377649} a^{18} + \frac{1469}{167844} a^{16} - \frac{1189}{503532} a^{14} - \frac{22139}{503532} a^{12} + \frac{3643}{83922} a^{10} - \frac{1777}{27974} a^{8} + \frac{181}{13987} a^{6} - \frac{3083}{13987} a^{4} + \frac{6739}{55948} a^{2} + \frac{27711}{55948}$, $\frac{1}{4531788} a^{27} + \frac{3353}{2265894} a^{25} + \frac{901}{251766} a^{23} - \frac{139}{125883} a^{21} + \frac{998}{377649} a^{19} + \frac{1469}{167844} a^{17} - \frac{1189}{503532} a^{15} - \frac{22139}{503532} a^{13} + \frac{3643}{83922} a^{11} - \frac{1777}{27974} a^{9} + \frac{181}{13987} a^{7} - \frac{3083}{13987} a^{5} + \frac{6739}{55948} a^{3} + \frac{27711}{55948} a$, $\frac{1}{18127152} a^{28} + \frac{10333}{1007064} a^{14} + \frac{99225}{223792}$, $\frac{1}{18127152} a^{29} + \frac{10333}{1007064} a^{15} + \frac{99225}{223792} a$, $\frac{1}{54381456} a^{30} + \frac{12769}{1007064} a^{16} - \frac{22873}{223792} a^{2}$, $\frac{1}{54381456} a^{31} + \frac{12769}{1007064} a^{17} - \frac{22873}{223792} a^{3}$, $\frac{1}{54381456} a^{32} + \frac{10333}{3021192} a^{18} + \frac{33075}{223792} a^{4}$, $\frac{1}{54381456} a^{33} + \frac{10333}{3021192} a^{19} + \frac{33075}{223792} a^{5}$, $\frac{1}{54381456} a^{34} + \frac{10333}{3021192} a^{20} + \frac{33075}{223792} a^{6}$, $\frac{1}{54381456} a^{35} + \frac{10333}{3021192} a^{21} + \frac{33075}{223792} a^{7}$

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

Class group and class number

$C_{2}\times C_{14}\times C_{182}$, which has order $5096$ (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:  \( -\frac{19}{2014128} a^{32} + \frac{377893}{3021192} a^{18} - \frac{1022295}{223792} a^{4} \) (order $14$)
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:  \( 36543757083175.945 \) (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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{-7}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.1, 3.3.3969.2, \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{3}, \sqrt{-7})\), \(\Q(\zeta_{36})^+\), 6.6.3024568512.1, 6.6.3024568512.2, 6.6.4148928.1, 6.0.432081216.1, 6.0.2250423.1, 6.0.21171979584.2, 6.0.110270727.2, 6.0.21171979584.1, 6.0.110270727.1, 6.0.29042496.1, \(\Q(\zeta_{7})\), 9.9.62523502209.1, 12.0.186694177220038656.2, 12.0.448252719505312813056.3, 12.0.448252719505312813056.4, 12.0.843466573910016.3, 18.18.27668797159880354103659593728.1, 18.0.9490397425838961457555240648704.1, 18.0.1340851596668237962730583.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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$2.6.6.5$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$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$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
7Data not computed