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Properties

Label 36.0.20068757308...7696.2
Degree $36$
Signature $[0, 18]$
Discriminant $2^{54}\cdot 3^{54}\cdot 7^{24}$
Root discriminant $53.78$
Ramified primes $2, 3, 7$
Class number $972$ (GRH)
Class group $[18, 54]$ (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![262144, 0, 0, 0, 0, 0, 4227072, 0, 0, 0, 0, 0, 68005888, 0, 0, 0, 0, 0, 2508800, 0, 0, 0, 0, 0, 84160, 0, 0, 0, 0, 0, 304, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 304*x^30 + 84160*x^24 + 2508800*x^18 + 68005888*x^12 + 4227072*x^6 + 262144)
 
gp: K = bnfinit(x^36 + 304*x^30 + 84160*x^24 + 2508800*x^18 + 68005888*x^12 + 4227072*x^6 + 262144, 1)
 

Normalized defining polynomial

\( x^{36} + 304 x^{30} + 84160 x^{24} + 2508800 x^{18} + 68005888 x^{12} + 4227072 x^{6} + 262144 \)

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:  \(200687573080369568029416132506181048520658333428355416190877696=2^{54}\cdot 3^{54}\cdot 7^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.78$
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:  \(504=2^{3}\cdot 3^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{504}(1,·)$, $\chi_{504}(389,·)$, $\chi_{504}(193,·)$, $\chi_{504}(137,·)$, $\chi_{504}(109,·)$, $\chi_{504}(401,·)$, $\chi_{504}(277,·)$, $\chi_{504}(25,·)$, $\chi_{504}(281,·)$, $\chi_{504}(29,·)$, $\chi_{504}(289,·)$, $\chi_{504}(37,·)$, $\chi_{504}(65,·)$, $\chi_{504}(169,·)$, $\chi_{504}(305,·)$, $\chi_{504}(53,·)$, $\chi_{504}(317,·)$, $\chi_{504}(449,·)$, $\chi_{504}(197,·)$, $\chi_{504}(457,·)$, $\chi_{504}(205,·)$, $\chi_{504}(337,·)$, $\chi_{504}(85,·)$, $\chi_{504}(473,·)$, $\chi_{504}(221,·)$, $\chi_{504}(421,·)$, $\chi_{504}(485,·)$, $\chi_{504}(361,·)$, $\chi_{504}(365,·)$, $\chi_{504}(445,·)$, $\chi_{504}(113,·)$, $\chi_{504}(373,·)$, $\chi_{504}(233,·)$, $\chi_{504}(121,·)$, $\chi_{504}(253,·)$, $\chi_{504}(149,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{512} a^{18}$, $\frac{1}{512} a^{19}$, $\frac{1}{1024} a^{20}$, $\frac{1}{1024} a^{21}$, $\frac{1}{2048} a^{22}$, $\frac{1}{2048} a^{23}$, $\frac{1}{2289664} a^{24} + \frac{3}{5504} a^{18} + \frac{209}{35776} a^{12} + \frac{5}{344} a^{6} + \frac{274}{559}$, $\frac{1}{2289664} a^{25} + \frac{3}{5504} a^{19} + \frac{209}{35776} a^{13} + \frac{5}{344} a^{7} + \frac{274}{559} a$, $\frac{1}{4579328} a^{26} + \frac{3}{11008} a^{20} + \frac{209}{71552} a^{14} + \frac{5}{688} a^{8} + \frac{137}{559} a^{2}$, $\frac{1}{4579328} a^{27} + \frac{3}{11008} a^{21} + \frac{209}{71552} a^{15} + \frac{5}{688} a^{9} + \frac{137}{559} a^{3}$, $\frac{1}{9158656} a^{28} + \frac{3}{22016} a^{22} + \frac{209}{143104} a^{16} + \frac{5}{1376} a^{10} + \frac{137}{1118} a^{4}$, $\frac{1}{9158656} a^{29} + \frac{3}{22016} a^{23} + \frac{209}{143104} a^{17} + \frac{5}{1376} a^{11} + \frac{137}{1118} a^{5}$, $\frac{1}{399831073980416} a^{30} + \frac{7467523}{49978884247552} a^{24} - \frac{114384399}{6247360530944} a^{18} - \frac{2751769255}{390460033184} a^{12} + \frac{1127590751}{97615008296} a^{6} - \frac{5582733509}{12201876037}$, $\frac{1}{399831073980416} a^{31} + \frac{7467523}{49978884247552} a^{25} - \frac{114384399}{6247360530944} a^{19} - \frac{2751769255}{390460033184} a^{13} + \frac{1127590751}{97615008296} a^{7} - \frac{5582733509}{12201876037} a$, $\frac{1}{799662147960832} a^{32} + \frac{7467523}{99957768495104} a^{26} - \frac{114384399}{12494721061888} a^{20} - \frac{2751769255}{780920066368} a^{14} + \frac{1127590751}{195230016592} a^{8} - \frac{5582733509}{24403752074} a^{2}$, $\frac{1}{799662147960832} a^{33} + \frac{7467523}{99957768495104} a^{27} - \frac{114384399}{12494721061888} a^{21} - \frac{2751769255}{780920066368} a^{15} + \frac{1127590751}{195230016592} a^{9} - \frac{5582733509}{24403752074} a^{3}$, $\frac{1}{1599324295921664} a^{34} + \frac{7467523}{199915536990208} a^{28} - \frac{114384399}{24989442123776} a^{22} - \frac{2751769255}{1561840132736} a^{16} + \frac{1127590751}{390460033184} a^{10} - \frac{5582733509}{48807504148} a^{4}$, $\frac{1}{1599324295921664} a^{35} + \frac{7467523}{199915536990208} a^{29} - \frac{114384399}{24989442123776} a^{23} - \frac{2751769255}{1561840132736} a^{17} + \frac{1127590751}{390460033184} a^{11} - \frac{5582733509}{48807504148} a^{5}$

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

Class group and class number

$C_{18}\times C_{54}$, which has order $972$ (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:  \( -\frac{3520629}{2324599267328} a^{34} - \frac{92024515237}{199915536990208} a^{28} - \frac{199029372171}{1561840132736} a^{22} - \frac{740253321255}{195230016592} a^{16} - \frac{5015625515159}{48807504148} a^{10} - \frac{178378536}{12201876037} a^{4} \) (order $18$)
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:  \( 160258501280890.78 \) (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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-6}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.1, 3.3.3969.2, \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\zeta_{9})\), 6.0.47258883.2, 6.0.47258883.1, 6.0.64827.1, 6.6.3359232.1, 6.0.10077696.1, 6.6.8065516032.1, 6.0.24196548096.1, 6.6.8065516032.2, 6.0.24196548096.2, 6.6.1229312.1, 6.0.33191424.1, 9.9.62523502209.1, 12.0.101559956668416.2, 12.0.585472939762041225216.2, 12.0.585472939762041225216.1, 12.0.1101670627147776.2, 18.0.105548084868928352751387.1, 18.18.524682375772545974113841184768.1, 18.0.14166424145858741301073711988736.4

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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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$2.12.18.23$x^{12} + 52 x^{10} - 28 x^{8} + 8 x^{6} + 64 x^{4} - 32 x^{2} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
2.12.18.23$x^{12} + 52 x^{10} - 28 x^{8} + 8 x^{6} + 64 x^{4} - 32 x^{2} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
2.12.18.23$x^{12} + 52 x^{10} - 28 x^{8} + 8 x^{6} + 64 x^{4} - 32 x^{2} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
3Data not computed
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$