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Properties

Label 36.0.76556233627...7984.1
Degree $36$
Signature $[0, 18]$
Discriminant $2^{36}\cdot 3^{54}\cdot 7^{24}$
Root discriminant $38.03$
Ramified primes $2, 3, 7$
Class number $28$ (GRH)
Class group $[2, 14]$ (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![1, 0, 0, 0, 0, 0, -129, 0, 0, 0, 0, 0, 16603, 0, 0, 0, 0, 0, -4900, 0, 0, 0, 0, 0, 1315, 0, 0, 0, 0, 0, -38, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 38*x^30 + 1315*x^24 - 4900*x^18 + 16603*x^12 - 129*x^6 + 1)
 
gp: K = bnfinit(x^36 - 38*x^30 + 1315*x^24 - 4900*x^18 + 16603*x^12 - 129*x^6 + 1, 1)
 

Normalized defining polynomial

\( x^{36} - 38 x^{30} + 1315 x^{24} - 4900 x^{18} + 16603 x^{12} - 129 x^{6} + 1 \)

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:  \(765562336274603149526276140236591524202950795854016937984=2^{36}\cdot 3^{54}\cdot 7^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.03$
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$$\chi_{252}(1,·)$, $\chi_{252}(107,·)$, $\chi_{252}(65,·)$, $\chi_{252}(137,·)$, $\chi_{252}(11,·)$, $\chi_{252}(149,·)$, $\chi_{252}(23,·)$, $\chi_{252}(25,·)$, $\chi_{252}(155,·)$, $\chi_{252}(29,·)$, $\chi_{252}(163,·)$, $\chi_{252}(37,·)$, $\chi_{252}(169,·)$, $\chi_{252}(43,·)$, $\chi_{252}(179,·)$, $\chi_{252}(53,·)$, $\chi_{252}(151,·)$, $\chi_{252}(191,·)$, $\chi_{252}(193,·)$, $\chi_{252}(67,·)$, $\chi_{252}(197,·)$, $\chi_{252}(71,·)$, $\chi_{252}(205,·)$, $\chi_{252}(79,·)$, $\chi_{252}(211,·)$, $\chi_{252}(85,·)$, $\chi_{252}(221,·)$, $\chi_{252}(95,·)$, $\chi_{252}(233,·)$, $\chi_{252}(235,·)$, $\chi_{252}(109,·)$, $\chi_{252}(239,·)$, $\chi_{252}(113,·)$, $\chi_{252}(247,·)$, $\chi_{252}(121,·)$, $\chi_{252}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{559} a^{24} - \frac{12}{43} a^{18} + \frac{209}{559} a^{12} - \frac{5}{43} a^{6} + \frac{274}{559}$, $\frac{1}{559} a^{25} - \frac{12}{43} a^{19} + \frac{209}{559} a^{13} - \frac{5}{43} a^{7} + \frac{274}{559} a$, $\frac{1}{559} a^{26} - \frac{12}{43} a^{20} + \frac{209}{559} a^{14} - \frac{5}{43} a^{8} + \frac{274}{559} a^{2}$, $\frac{1}{559} a^{27} - \frac{12}{43} a^{21} + \frac{209}{559} a^{15} - \frac{5}{43} a^{9} + \frac{274}{559} a^{3}$, $\frac{1}{559} a^{28} - \frac{12}{43} a^{22} + \frac{209}{559} a^{16} - \frac{5}{43} a^{10} + \frac{274}{559} a^{4}$, $\frac{1}{559} a^{29} - \frac{12}{43} a^{23} + \frac{209}{559} a^{17} - \frac{5}{43} a^{11} + \frac{274}{559} a^{5}$, $\frac{1}{12201876037} a^{30} - \frac{7467523}{12201876037} a^{24} - \frac{114384399}{12201876037} a^{18} + \frac{5503538510}{12201876037} a^{12} + \frac{1127590751}{12201876037} a^{6} + \frac{5582733509}{12201876037}$, $\frac{1}{12201876037} a^{31} - \frac{7467523}{12201876037} a^{25} - \frac{114384399}{12201876037} a^{19} + \frac{5503538510}{12201876037} a^{13} + \frac{1127590751}{12201876037} a^{7} + \frac{5582733509}{12201876037} a$, $\frac{1}{12201876037} a^{32} - \frac{7467523}{12201876037} a^{26} - \frac{114384399}{12201876037} a^{20} + \frac{5503538510}{12201876037} a^{14} + \frac{1127590751}{12201876037} a^{8} + \frac{5582733509}{12201876037} a^{2}$, $\frac{1}{12201876037} a^{33} - \frac{7467523}{12201876037} a^{27} - \frac{114384399}{12201876037} a^{21} + \frac{5503538510}{12201876037} a^{15} + \frac{1127590751}{12201876037} a^{9} + \frac{5582733509}{12201876037} a^{3}$, $\frac{1}{12201876037} a^{34} - \frac{7467523}{12201876037} a^{28} - \frac{114384399}{12201876037} a^{22} + \frac{5503538510}{12201876037} a^{16} + \frac{1127590751}{12201876037} a^{10} + \frac{5582733509}{12201876037} a^{4}$, $\frac{1}{12201876037} a^{35} - \frac{7467523}{12201876037} a^{29} - \frac{114384399}{12201876037} a^{23} + \frac{5503538510}{12201876037} a^{17} + \frac{1127590751}{12201876037} a^{11} + \frac{5582733509}{12201876037} a^{5}$

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

Class group and class number

$C_{2}\times C_{14}$, which has order $28$ (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:  \( -\frac{5430336179}{12201876037} a^{35} + \frac{206353143668}{12201876037} a^{29} - \frac{7140904840090}{12201876037} a^{23} + \frac{26609083833276}{12201876037} a^{17} - \frac{90160032745258}{12201876037} a^{11} + \frac{16291037658}{283764559} a^{5} \) (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:  \( 18271878541587.973 \) (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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.2, 3.3.3969.1, \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{12})\), \(\Q(\zeta_{9})\), 6.0.47258883.1, 6.0.47258883.2, 6.0.64827.1, \(\Q(\zeta_{36})^+\), 6.0.419904.1, 6.6.3024568512.2, 6.0.1008189504.2, 6.6.3024568512.1, 6.0.1008189504.1, 6.6.4148928.1, 6.0.153664.1, 9.9.62523502209.1, \(\Q(\zeta_{36})\), 12.0.9148014683781894144.2, 12.0.9148014683781894144.1, 12.0.17213603549184.1, 18.0.105548084868928352751387.1, 18.18.27668797159880354103659593728.1, 18.0.1024770265180753855691096064.1

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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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$2.12.12.26$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
7Data not computed