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Properties

Label 36.36.3347174706...2357.1
Degree $36$
Signature $[36, 0]$
Discriminant $3^{54}\cdot 13^{33}$
Root discriminant $54.55$
Ramified primes $3, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times C_{12}$ (as 36T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -36, -108, 2145, 6471, -27162, -83631, 158337, 502173, -534544, -1761969, 1165464, 4030936, -1746123, -6403833, 1867585, 7348878, -1459998, -6247579, 845295, 3995883, -364067, -1937496, 116154, 712529, -27054, -197316, 4467, 40455, -495, -5952, 33, 594, -1, -36, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 36*x^34 - x^33 + 594*x^32 + 33*x^31 - 5952*x^30 - 495*x^29 + 40455*x^28 + 4467*x^27 - 197316*x^26 - 27054*x^25 + 712529*x^24 + 116154*x^23 - 1937496*x^22 - 364067*x^21 + 3995883*x^20 + 845295*x^19 - 6247579*x^18 - 1459998*x^17 + 7348878*x^16 + 1867585*x^15 - 6403833*x^14 - 1746123*x^13 + 4030936*x^12 + 1165464*x^11 - 1761969*x^10 - 534544*x^9 + 502173*x^8 + 158337*x^7 - 83631*x^6 - 27162*x^5 + 6471*x^4 + 2145*x^3 - 108*x^2 - 36*x + 1)
 
gp: K = bnfinit(x^36 - 36*x^34 - x^33 + 594*x^32 + 33*x^31 - 5952*x^30 - 495*x^29 + 40455*x^28 + 4467*x^27 - 197316*x^26 - 27054*x^25 + 712529*x^24 + 116154*x^23 - 1937496*x^22 - 364067*x^21 + 3995883*x^20 + 845295*x^19 - 6247579*x^18 - 1459998*x^17 + 7348878*x^16 + 1867585*x^15 - 6403833*x^14 - 1746123*x^13 + 4030936*x^12 + 1165464*x^11 - 1761969*x^10 - 534544*x^9 + 502173*x^8 + 158337*x^7 - 83631*x^6 - 27162*x^5 + 6471*x^4 + 2145*x^3 - 108*x^2 - 36*x + 1, 1)
 

Normalized defining polynomial

\( x^{36} - 36 x^{34} - x^{33} + 594 x^{32} + 33 x^{31} - 5952 x^{30} - 495 x^{29} + 40455 x^{28} + 4467 x^{27} - 197316 x^{26} - 27054 x^{25} + 712529 x^{24} + 116154 x^{23} - 1937496 x^{22} - 364067 x^{21} + 3995883 x^{20} + 845295 x^{19} - 6247579 x^{18} - 1459998 x^{17} + 7348878 x^{16} + 1867585 x^{15} - 6403833 x^{14} - 1746123 x^{13} + 4030936 x^{12} + 1165464 x^{11} - 1761969 x^{10} - 534544 x^{9} + 502173 x^{8} + 158337 x^{7} - 83631 x^{6} - 27162 x^{5} + 6471 x^{4} + 2145 x^{3} - 108 x^{2} - 36 x + 1 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[36, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(334717470607298852954929976123524497086119009458311989825932357=3^{54}\cdot 13^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.55$
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:  $3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(117=3^{2}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{117}(1,·)$, $\chi_{117}(2,·)$, $\chi_{117}(4,·)$, $\chi_{117}(5,·)$, $\chi_{117}(8,·)$, $\chi_{117}(10,·)$, $\chi_{117}(11,·)$, $\chi_{117}(16,·)$, $\chi_{117}(20,·)$, $\chi_{117}(22,·)$, $\chi_{117}(25,·)$, $\chi_{117}(32,·)$, $\chi_{117}(40,·)$, $\chi_{117}(41,·)$, $\chi_{117}(43,·)$, $\chi_{117}(44,·)$, $\chi_{117}(47,·)$, $\chi_{117}(49,·)$, $\chi_{117}(50,·)$, $\chi_{117}(55,·)$, $\chi_{117}(59,·)$, $\chi_{117}(61,·)$, $\chi_{117}(64,·)$, $\chi_{117}(71,·)$, $\chi_{117}(79,·)$, $\chi_{117}(80,·)$, $\chi_{117}(82,·)$, $\chi_{117}(83,·)$, $\chi_{117}(86,·)$, $\chi_{117}(88,·)$, $\chi_{117}(89,·)$, $\chi_{117}(94,·)$, $\chi_{117}(98,·)$, $\chi_{117}(100,·)$, $\chi_{117}(103,·)$, $\chi_{117}(110,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $35$
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:  \( 63356932035659470000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_{12}$ (as 36T3):

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 $C_3\times C_{12}$
Character table for $C_3\times C_{12}$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\zeta_{9})^+\), 3.3.13689.2, 3.3.13689.1, 3.3.169.1, 4.4.19773.1, 6.6.14414517.1, 6.6.2436053373.2, 6.6.2436053373.1, \(\Q(\zeta_{13})^+\), 9.9.2565164201769.1, 12.12.4108400332687853397.1, 12.12.694319656224247224093.1, 12.12.694319656224247224093.2, \(\Q(\zeta_{39})^+\), 18.18.14456408038335708501176406117.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/5.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/7.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/11.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{18}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{3}$

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
3Data not computed
13Data not computed