↑

Properties

Label 36.0.21690424625...6624.1
Degree $36$
Signature $[0, 18]$
Discriminant $2^{36}\cdot 3^{54}\cdot 13^{24}$
Root discriminant $57.46$
Ramified primes $2, 3, 13$
Class number $2268$ (GRH)
Class group $[18, 126]$ (GRH)
Galois group $C_6^2$ (as 36T4)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, 0, 0, -2369, 0, 0, 0, 0, 0, 5611807, 0, 0, 0, 0, 0, -838624, 0, 0, 0, 0, 0, 122947, 0, 0, 0, 0, 0, -354, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 354*x^30 + 122947*x^24 - 838624*x^18 + 5611807*x^12 - 2369*x^6 + 1)
 
gp: K = bnfinit(x^36 - 354*x^30 + 122947*x^24 - 838624*x^18 + 5611807*x^12 - 2369*x^6 + 1, 1)
 

Normalized defining polynomial

\( x^{36} - 354 x^{30} + 122947 x^{24} - 838624 x^{18} + 5611807 x^{12} - 2369 x^{6} + 1 \)

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:  \(2169042462588586105943635605392908833154745008913475914510106624=2^{36}\cdot 3^{54}\cdot 13^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.46$
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:  $2, 3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(468=2^{2}\cdot 3^{2}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{468}(1,·)$, $\chi_{468}(131,·)$, $\chi_{468}(133,·)$, $\chi_{468}(391,·)$, $\chi_{468}(139,·)$, $\chi_{468}(269,·)$, $\chi_{468}(107,·)$, $\chi_{468}(367,·)$, $\chi_{468}(157,·)$, $\chi_{468}(287,·)$, $\chi_{468}(289,·)$, $\chi_{468}(35,·)$, $\chi_{468}(295,·)$, $\chi_{468}(425,·)$, $\chi_{468}(263,·)$, $\chi_{468}(29,·)$, $\chi_{468}(53,·)$, $\chi_{468}(55,·)$, $\chi_{468}(185,·)$, $\chi_{468}(443,·)$, $\chi_{468}(61,·)$, $\chi_{468}(191,·)$, $\chi_{468}(451,·)$, $\chi_{468}(79,·)$, $\chi_{468}(209,·)$, $\chi_{468}(419,·)$, $\chi_{468}(341,·)$, $\chi_{468}(313,·)$, $\chi_{468}(217,·)$, $\chi_{468}(347,·)$, $\chi_{468}(235,·)$, $\chi_{468}(365,·)$, $\chi_{468}(445,·)$, $\chi_{468}(113,·)$, $\chi_{468}(211,·)$, $\chi_{468}(373,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{62015} a^{24} + \frac{4335}{12403} a^{18} + \frac{14541}{62015} a^{12} + \frac{3174}{12403} a^{6} + \frac{18311}{62015}$, $\frac{1}{62015} a^{25} + \frac{4335}{12403} a^{19} + \frac{14541}{62015} a^{13} + \frac{3174}{12403} a^{7} + \frac{18311}{62015} a$, $\frac{1}{62015} a^{26} + \frac{4335}{12403} a^{20} + \frac{14541}{62015} a^{14} + \frac{3174}{12403} a^{8} + \frac{18311}{62015} a^{2}$, $\frac{1}{62015} a^{27} + \frac{4335}{12403} a^{21} + \frac{14541}{62015} a^{15} + \frac{3174}{12403} a^{9} + \frac{18311}{62015} a^{3}$, $\frac{1}{62015} a^{28} + \frac{4335}{12403} a^{22} + \frac{14541}{62015} a^{16} + \frac{3174}{12403} a^{10} + \frac{18311}{62015} a^{4}$, $\frac{1}{62015} a^{29} + \frac{4335}{12403} a^{23} + \frac{14541}{62015} a^{17} + \frac{3174}{12403} a^{11} + \frac{18311}{62015} a^{5}$, $\frac{1}{42787497099335045} a^{30} + \frac{65876963448}{8557499419867009} a^{24} + \frac{12738486085268876}{42787497099335045} a^{18} - \frac{734243975729351}{8557499419867009} a^{12} + \frac{7282103864965656}{42787497099335045} a^{6} + \frac{3195150047441441}{8557499419867009}$, $\frac{1}{42787497099335045} a^{31} + \frac{65876963448}{8557499419867009} a^{25} + \frac{12738486085268876}{42787497099335045} a^{19} - \frac{734243975729351}{8557499419867009} a^{13} + \frac{7282103864965656}{42787497099335045} a^{7} + \frac{3195150047441441}{8557499419867009} a$, $\frac{1}{42787497099335045} a^{32} + \frac{65876963448}{8557499419867009} a^{26} + \frac{12738486085268876}{42787497099335045} a^{20} - \frac{734243975729351}{8557499419867009} a^{14} + \frac{7282103864965656}{42787497099335045} a^{8} + \frac{3195150047441441}{8557499419867009} a^{2}$, $\frac{1}{42787497099335045} a^{33} + \frac{65876963448}{8557499419867009} a^{27} + \frac{12738486085268876}{42787497099335045} a^{21} - \frac{734243975729351}{8557499419867009} a^{15} + \frac{7282103864965656}{42787497099335045} a^{9} + \frac{3195150047441441}{8557499419867009} a^{3}$, $\frac{1}{42787497099335045} a^{34} + \frac{65876963448}{8557499419867009} a^{28} + \frac{12738486085268876}{42787497099335045} a^{22} - \frac{734243975729351}{8557499419867009} a^{16} + \frac{7282103864965656}{42787497099335045} a^{10} + \frac{3195150047441441}{8557499419867009} a^{4}$, $\frac{1}{42787497099335045} a^{35} + \frac{65876963448}{8557499419867009} a^{29} + \frac{12738486085268876}{42787497099335045} a^{23} - \frac{734243975729351}{8557499419867009} a^{17} + \frac{7282103864965656}{42787497099335045} a^{11} + \frac{3195150047441441}{8557499419867009} a^{5}$

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

Class group and class number

$C_{18}\times C_{126}$, which has order $2268$ (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:  \( \frac{11719091329607067}{42787497099335045} a^{35} - \frac{4148558329423663284}{42787497099335045} a^{29} + \frac{1440827121264551214062}{42787497099335045} a^{23} - \frac{9827911095609216502334}{42787497099335045} a^{17} + \frac{65765278737197795431622}{42787497099335045} a^{11} - \frac{27762527351425588474}{42787497099335045} a^{5} \) (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:  \( 860633953627025.8 \) (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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\zeta_{9})^+\), 3.3.13689.2, 3.3.13689.1, 3.3.169.1, \(\Q(\zeta_{12})\), \(\Q(\zeta_{9})\), 6.0.562166163.1, 6.0.562166163.2, 6.0.771147.1, 6.0.419904.1, \(\Q(\zeta_{36})^+\), 6.0.11992878144.4, 6.6.35978634432.1, 6.0.11992878144.3, 6.6.35978634432.2, 6.0.1827904.1, 6.6.49353408.1, 9.9.2565164201769.1, \(\Q(\zeta_{36})\), 12.0.1294462135591495962624.2, 12.0.1294462135591495962624.1, 12.0.2435758881214464.1, 18.0.177661819315004155453692747.1, 18.0.1724925183796757382490845609984.3, 18.18.46572979962512449327252831469568.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{18}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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$$[2]^{6}$
3Data not computed
$13$13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$