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Properties

Label 36.0.90067643300...9616.3
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 Not computed
Class group Not computed
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![117649, 0, 0, 0, 0, 0, 285719, 0, 0, 0, 0, 0, 669879, 0, 0, 0, 0, 0, 57624, 0, 0, 0, 0, 0, 4067, 0, 0, 0, 0, 0, 70, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 70*x^30 + 4067*x^24 + 57624*x^18 + 669879*x^12 + 285719*x^6 + 117649)
 
gp: K = bnfinit(x^36 + 70*x^30 + 4067*x^24 + 57624*x^18 + 669879*x^12 + 285719*x^6 + 117649, 1)
 

Normalized defining polynomial

\( x^{36} + 70 x^{30} + 4067 x^{24} + 57624 x^{18} + 669879 x^{12} + 285719 x^{6} + 117649 \)

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$$\chi_{252}(1,·)$, $\chi_{252}(131,·)$, $\chi_{252}(65,·)$, $\chi_{252}(137,·)$, $\chi_{252}(139,·)$, $\chi_{252}(143,·)$, $\chi_{252}(19,·)$, $\chi_{252}(149,·)$, $\chi_{252}(25,·)$, $\chi_{252}(29,·)$, $\chi_{252}(31,·)$, $\chi_{252}(37,·)$, $\chi_{252}(167,·)$, $\chi_{252}(169,·)$, $\chi_{252}(47,·)$, $\chi_{252}(53,·)$, $\chi_{252}(55,·)$, $\chi_{252}(59,·)$, $\chi_{252}(193,·)$, $\chi_{252}(197,·)$, $\chi_{252}(199,·)$, $\chi_{252}(205,·)$, $\chi_{252}(83,·)$, $\chi_{252}(85,·)$, $\chi_{252}(215,·)$, $\chi_{252}(221,·)$, $\chi_{252}(223,·)$, $\chi_{252}(227,·)$, $\chi_{252}(103,·)$, $\chi_{252}(233,·)$, $\chi_{252}(109,·)$, $\chi_{252}(113,·)$, $\chi_{252}(115,·)$, $\chi_{252}(187,·)$, $\chi_{252}(121,·)$, $\chi_{252}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6}$, $\frac{1}{7} a^{7}$, $\frac{1}{7} a^{8}$, $\frac{1}{7} a^{9}$, $\frac{1}{7} a^{10}$, $\frac{1}{7} a^{11}$, $\frac{1}{49} a^{12}$, $\frac{1}{49} a^{13}$, $\frac{1}{49} a^{14}$, $\frac{1}{49} a^{15}$, $\frac{1}{343} a^{16} + \frac{1}{49} a^{10} + \frac{1}{7} a^{4}$, $\frac{1}{343} a^{17} + \frac{1}{49} a^{11} + \frac{1}{7} a^{5}$, $\frac{1}{343} a^{18}$, $\frac{1}{343} a^{19}$, $\frac{1}{343} a^{20}$, $\frac{1}{343} a^{21}$, $\frac{1}{2401} a^{22} - \frac{1}{7} a^{4}$, $\frac{1}{2401} a^{23} - \frac{1}{7} a^{5}$, $\frac{1}{31213} a^{24} - \frac{4}{637} a^{12} + \frac{3}{13}$, $\frac{1}{31213} a^{25} - \frac{4}{637} a^{13} + \frac{3}{13} a$, $\frac{1}{218491} a^{26} - \frac{2}{2401} a^{20} - \frac{4}{4459} a^{14} - \frac{2}{49} a^{8} + \frac{29}{91} a^{2}$, $\frac{1}{218491} a^{27} - \frac{2}{2401} a^{21} - \frac{4}{4459} a^{15} - \frac{2}{49} a^{9} + \frac{29}{91} a^{3}$, $\frac{1}{218491} a^{28} - \frac{4}{4459} a^{16} - \frac{2}{49} a^{10} + \frac{3}{91} a^{4}$, $\frac{1}{218491} a^{29} - \frac{4}{4459} a^{17} - \frac{2}{49} a^{11} + \frac{3}{91} a^{5}$, $\frac{1}{5022452617} a^{30} + \frac{2307}{717493231} a^{24} + \frac{35239}{102499033} a^{18} - \frac{39050}{14642719} a^{12} + \frac{89651}{2091817} a^{6} + \frac{33805}{298831}$, $\frac{1}{5022452617} a^{31} + \frac{2307}{717493231} a^{25} + \frac{35239}{102499033} a^{19} - \frac{39050}{14642719} a^{13} + \frac{89651}{2091817} a^{7} + \frac{33805}{298831} a$, $\frac{1}{35157168319} a^{32} + \frac{2307}{5022452617} a^{26} - \frac{37656}{102499033} a^{20} - \frac{636712}{102499033} a^{14} + \frac{986144}{14642719} a^{8} + \frac{33805}{2091817} a^{2}$, $\frac{1}{35157168319} a^{33} + \frac{2307}{5022452617} a^{27} - \frac{37656}{102499033} a^{21} - \frac{636712}{102499033} a^{15} + \frac{986144}{14642719} a^{9} + \frac{33805}{2091817} a^{3}$, $\frac{1}{35157168319} a^{34} + \frac{2307}{5022452617} a^{28} + \frac{35239}{717493231} a^{22} - \frac{39050}{102499033} a^{16} - \frac{72573}{2091817} a^{10} + \frac{332636}{2091817} a^{4}$, $\frac{1}{35157168319} a^{35} + \frac{2307}{5022452617} a^{29} + \frac{35239}{717493231} a^{23} - \frac{39050}{102499033} a^{17} - \frac{72573}{2091817} a^{11} + \frac{332636}{2091817} a^{5}$

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

Class group and class number

Not computed

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:  \( \frac{24252}{5022452617} a^{32} + \frac{1694080}{5022452617} a^{26} + \frac{14060864}{717493231} a^{20} + \frac{28279701}{102499033} a^{14} + \frac{47264832}{14642719} a^{8} + \frac{2879936}{2091817} a^{2} \) (order $18$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{7}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{7})^+\), 3.3.3969.1, 3.3.3969.2, \(\Q(\sqrt{-3}, \sqrt{7})\), \(\Q(\zeta_{9})\), 6.0.64827.1, 6.0.47258883.2, 6.0.47258883.1, 6.0.432081216.1, 6.6.144027072.1, 6.0.29042496.1, \(\Q(\zeta_{28})^+\), 6.0.21171979584.2, 6.6.7057326528.2, 6.0.21171979584.1, 6.6.7057326528.1, 9.9.62523502209.1, 12.0.186694177220038656.3, 12.0.843466573910016.2, 12.0.448252719505312813056.6, 12.0.448252719505312813056.5, 18.0.105548084868928352751387.1, 18.0.9490397425838961457555240648704.1, 18.18.351496200956998572502045949952.1

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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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