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Properties

Label 36.0.33294538757...0625.1
Degree $36$
Signature $[0, 18]$
Discriminant $3^{90}\cdot 5^{18}$
Root discriminant $34.86$
Ramified primes $3, 5$
Class number $37$ (GRH)
Class group $[37]$ (GRH)
Galois group $C_2\times C_{18}$ (as 36T2)

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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, 0, 0, 0, 76, 0, 0, 0, 0, 0, 0, 0, 0, 5777, 0, 0, 0, 0, 0, 0, 0, 0, -76, 0, 0, 0, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 76*x^27 + 5777*x^18 + 76*x^9 + 1)
 
gp: K = bnfinit(x^36 - 76*x^27 + 5777*x^18 + 76*x^9 + 1, 1)
 

Normalized defining polynomial

\( x^{36} - 76 x^{27} + 5777 x^{18} + 76 x^{9} + 1 \)

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:  \(33294538757658815101209249418169888839879795074462890625=3^{90}\cdot 5^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.86$
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:  $3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(135=3^{3}\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{135}(1,·)$, $\chi_{135}(131,·)$, $\chi_{135}(4,·)$, $\chi_{135}(134,·)$, $\chi_{135}(11,·)$, $\chi_{135}(14,·)$, $\chi_{135}(16,·)$, $\chi_{135}(19,·)$, $\chi_{135}(26,·)$, $\chi_{135}(29,·)$, $\chi_{135}(31,·)$, $\chi_{135}(34,·)$, $\chi_{135}(41,·)$, $\chi_{135}(44,·)$, $\chi_{135}(46,·)$, $\chi_{135}(49,·)$, $\chi_{135}(56,·)$, $\chi_{135}(59,·)$, $\chi_{135}(61,·)$, $\chi_{135}(64,·)$, $\chi_{135}(71,·)$, $\chi_{135}(74,·)$, $\chi_{135}(76,·)$, $\chi_{135}(79,·)$, $\chi_{135}(86,·)$, $\chi_{135}(89,·)$, $\chi_{135}(91,·)$, $\chi_{135}(94,·)$, $\chi_{135}(101,·)$, $\chi_{135}(104,·)$, $\chi_{135}(106,·)$, $\chi_{135}(109,·)$, $\chi_{135}(116,·)$, $\chi_{135}(119,·)$, $\chi_{135}(121,·)$, $\chi_{135}(124,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{34} a^{18} + \frac{13}{34} a^{9} - \frac{1}{34}$, $\frac{1}{34} a^{19} + \frac{13}{34} a^{10} - \frac{1}{34} a$, $\frac{1}{34} a^{20} + \frac{13}{34} a^{11} - \frac{1}{34} a^{2}$, $\frac{1}{34} a^{21} + \frac{13}{34} a^{12} - \frac{1}{34} a^{3}$, $\frac{1}{34} a^{22} + \frac{13}{34} a^{13} - \frac{1}{34} a^{4}$, $\frac{1}{34} a^{23} + \frac{13}{34} a^{14} - \frac{1}{34} a^{5}$, $\frac{1}{34} a^{24} + \frac{13}{34} a^{15} - \frac{1}{34} a^{6}$, $\frac{1}{34} a^{25} + \frac{13}{34} a^{16} - \frac{1}{34} a^{7}$, $\frac{1}{34} a^{26} + \frac{13}{34} a^{17} - \frac{1}{34} a^{8}$, $\frac{1}{196418} a^{27} - \frac{75025}{196418}$, $\frac{1}{196418} a^{28} - \frac{75025}{196418} a$, $\frac{1}{196418} a^{29} - \frac{75025}{196418} a^{2}$, $\frac{1}{196418} a^{30} - \frac{75025}{196418} a^{3}$, $\frac{1}{196418} a^{31} - \frac{75025}{196418} a^{4}$, $\frac{1}{196418} a^{32} - \frac{75025}{196418} a^{5}$, $\frac{1}{196418} a^{33} - \frac{75025}{196418} a^{6}$, $\frac{1}{196418} a^{34} - \frac{75025}{196418} a^{7}$, $\frac{1}{196418} a^{35} - \frac{75025}{196418} a^{8}$

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

Class group and class number

$C_{37}$, which has order $37$ (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:  \( -\frac{21}{196418} a^{35} - \frac{9227465}{196418} a^{8} \) (order $54$)
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:  \( 6550249244897.024 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{18}$ (as 36T2):

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 $C_2\times C_{18}$
Character table for $C_2\times C_{18}$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\zeta_{9})\), 6.6.820125.1, 6.0.2460375.1, \(\Q(\zeta_{27})^+\), 12.0.6053445140625.1, \(\Q(\zeta_{27})\), 18.18.1923380668327365689220703125.1, 18.0.5770142004982097067662109375.1

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 $18^{2}$ R R $18^{2}$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{12}$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ $18^{2}$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{18}$ $18^{2}$

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
5Data not computed