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Properties

Label 36.0.11307847636...1152.1
Degree $36$
Signature $[0, 18]$
Discriminant $2^{36}\cdot 73^{35}$
Root discriminant $129.60$
Ramified primes $2, 73$
Class number Not computed
Class group Not computed
Galois group $C_{36}$ (as 36T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![73, 0, 7592, 0, 284846, 0, 4809386, 0, 39695721, 0, 174817626, 0, 449489251, 0, 721589086, 0, 760556194, 0, 545083627, 0, 271775788, 0, 95548021, 0, 23818951, 0, 4199763, 0, 517205, 0, 43289, 0, 2336, 0, 73, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 73*x^34 + 2336*x^32 + 43289*x^30 + 517205*x^28 + 4199763*x^26 + 23818951*x^24 + 95548021*x^22 + 271775788*x^20 + 545083627*x^18 + 760556194*x^16 + 721589086*x^14 + 449489251*x^12 + 174817626*x^10 + 39695721*x^8 + 4809386*x^6 + 284846*x^4 + 7592*x^2 + 73)
 
gp: K = bnfinit(x^36 + 73*x^34 + 2336*x^32 + 43289*x^30 + 517205*x^28 + 4199763*x^26 + 23818951*x^24 + 95548021*x^22 + 271775788*x^20 + 545083627*x^18 + 760556194*x^16 + 721589086*x^14 + 449489251*x^12 + 174817626*x^10 + 39695721*x^8 + 4809386*x^6 + 284846*x^4 + 7592*x^2 + 73, 1)
 

Normalized defining polynomial

\( x^{36} + 73 x^{34} + 2336 x^{32} + 43289 x^{30} + 517205 x^{28} + 4199763 x^{26} + 23818951 x^{24} + 95548021 x^{22} + 271775788 x^{20} + 545083627 x^{18} + 760556194 x^{16} + 721589086 x^{14} + 449489251 x^{12} + 174817626 x^{10} + 39695721 x^{8} + 4809386 x^{6} + 284846 x^{4} + 7592 x^{2} + 73 \)

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:  \(11307847636660752290696972110442359259116305023758972734266350296999135281152=2^{36}\cdot 73^{35}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $129.60$
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:  $2, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(292=2^{2}\cdot 73\)
Dirichlet character group:    $\lbrace$$\chi_{292}(1,·)$, $\chi_{292}(3,·)$, $\chi_{292}(9,·)$, $\chi_{292}(267,·)$, $\chi_{292}(109,·)$, $\chi_{292}(143,·)$, $\chi_{292}(145,·)$, $\chi_{292}(19,·)$, $\chi_{292}(23,·)$, $\chi_{292}(27,·)$, $\chi_{292}(35,·)$, $\chi_{292}(37,·)$, $\chi_{292}(41,·)$, $\chi_{292}(171,·)$, $\chi_{292}(137,·)$, $\chi_{292}(57,·)$, $\chi_{292}(65,·)$, $\chi_{292}(67,·)$, $\chi_{292}(69,·)$, $\chi_{292}(201,·)$, $\chi_{292}(77,·)$, $\chi_{292}(79,·)$, $\chi_{292}(81,·)$, $\chi_{292}(89,·)$, $\chi_{292}(207,·)$, $\chi_{292}(221,·)$, $\chi_{292}(195,·)$, $\chi_{292}(231,·)$, $\chi_{292}(105,·)$, $\chi_{292}(237,·)$, $\chi_{292}(111,·)$, $\chi_{292}(243,·)$, $\chi_{292}(119,·)$, $\chi_{292}(217,·)$, $\chi_{292}(123,·)$, $\chi_{292}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{3} a^{24} + \frac{1}{3} a^{22} + \frac{1}{3} a^{20} + \frac{1}{3} a^{18} + \frac{1}{3} a^{16} + \frac{1}{3} a^{14} + \frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{25} + \frac{1}{3} a^{23} + \frac{1}{3} a^{21} + \frac{1}{3} a^{19} + \frac{1}{3} a^{17} + \frac{1}{3} a^{15} + \frac{1}{3} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{26} - \frac{1}{3}$, $\frac{1}{3} a^{27} - \frac{1}{3} a$, $\frac{1}{3} a^{28} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{29} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{30} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{31} - \frac{1}{3} a^{5}$, $\frac{1}{1155009} a^{32} - \frac{72782}{1155009} a^{30} - \frac{12223}{385003} a^{28} - \frac{66695}{1155009} a^{26} + \frac{16311}{385003} a^{24} + \frac{6446}{385003} a^{22} + \frac{154035}{385003} a^{20} - \frac{98490}{385003} a^{18} - \frac{148098}{385003} a^{16} + \frac{10880}{385003} a^{14} + \frac{50416}{385003} a^{12} + \frac{68616}{385003} a^{10} - \frac{26961}{385003} a^{8} - \frac{219523}{1155009} a^{6} - \frac{452599}{1155009} a^{4} + \frac{93050}{385003} a^{2} + \frac{217832}{1155009}$, $\frac{1}{1155009} a^{33} - \frac{72782}{1155009} a^{31} - \frac{12223}{385003} a^{29} - \frac{66695}{1155009} a^{27} + \frac{16311}{385003} a^{25} + \frac{6446}{385003} a^{23} + \frac{154035}{385003} a^{21} - \frac{98490}{385003} a^{19} - \frac{148098}{385003} a^{17} + \frac{10880}{385003} a^{15} + \frac{50416}{385003} a^{13} + \frac{68616}{385003} a^{11} - \frac{26961}{385003} a^{9} - \frac{219523}{1155009} a^{7} - \frac{452599}{1155009} a^{5} + \frac{93050}{385003} a^{3} + \frac{217832}{1155009} a$, $\frac{1}{5285805574248283688434996707459805960035567} a^{34} + \frac{54113107022089485460648710453674276}{1761935191416094562811665569153268653345189} a^{32} + \frac{207005013985922032814688224021542520605439}{5285805574248283688434996707459805960035567} a^{30} + \frac{230906093438007590891715387286689650646386}{5285805574248283688434996707459805960035567} a^{28} + \frac{145004252862232535565585081713837323003559}{1761935191416094562811665569153268653345189} a^{26} - \frac{356272769992596750000481339649346530248574}{5285805574248283688434996707459805960035567} a^{24} + \frac{2306186055588669552523257477382577070669661}{5285805574248283688434996707459805960035567} a^{22} - \frac{3477550585566347196040419576050568237742}{9489776614449342349075398038527479281931} a^{20} + \frac{1111872361858871550825261315462574367707942}{5285805574248283688434996707459805960035567} a^{18} - \frac{336275223504254314259047868635644150615464}{5285805574248283688434996707459805960035567} a^{16} - \frac{904909256442899980090251502204856026857173}{5285805574248283688434996707459805960035567} a^{14} - \frac{693671550456153145716393379544449067439833}{5285805574248283688434996707459805960035567} a^{12} + \frac{1233029331512940235053811978746088905897277}{5285805574248283688434996707459805960035567} a^{10} - \frac{142877566700816300816145611112270684486288}{1761935191416094562811665569153268653345189} a^{8} - \frac{2062459144759175773553941408817031814259210}{5285805574248283688434996707459805960035567} a^{6} + \frac{1953574209379796406949471729468340302180442}{5285805574248283688434996707459805960035567} a^{4} - \frac{593137114670017440915981418356283434951484}{5285805574248283688434996707459805960035567} a^{2} + \frac{1218387261729163490110092456326504517884374}{5285805574248283688434996707459805960035567}$, $\frac{1}{5285805574248283688434996707459805960035567} a^{35} + \frac{54113107022089485460648710453674276}{1761935191416094562811665569153268653345189} a^{33} + \frac{207005013985922032814688224021542520605439}{5285805574248283688434996707459805960035567} a^{31} + \frac{230906093438007590891715387286689650646386}{5285805574248283688434996707459805960035567} a^{29} + \frac{145004252862232535565585081713837323003559}{1761935191416094562811665569153268653345189} a^{27} - \frac{356272769992596750000481339649346530248574}{5285805574248283688434996707459805960035567} a^{25} + \frac{2306186055588669552523257477382577070669661}{5285805574248283688434996707459805960035567} a^{23} - \frac{3477550585566347196040419576050568237742}{9489776614449342349075398038527479281931} a^{21} + \frac{1111872361858871550825261315462574367707942}{5285805574248283688434996707459805960035567} a^{19} - \frac{336275223504254314259047868635644150615464}{5285805574248283688434996707459805960035567} a^{17} - \frac{904909256442899980090251502204856026857173}{5285805574248283688434996707459805960035567} a^{15} - \frac{693671550456153145716393379544449067439833}{5285805574248283688434996707459805960035567} a^{13} + \frac{1233029331512940235053811978746088905897277}{5285805574248283688434996707459805960035567} a^{11} - \frac{142877566700816300816145611112270684486288}{1761935191416094562811665569153268653345189} a^{9} - \frac{2062459144759175773553941408817031814259210}{5285805574248283688434996707459805960035567} a^{7} + \frac{1953574209379796406949471729468340302180442}{5285805574248283688434996707459805960035567} a^{5} - \frac{593137114670017440915981418356283434951484}{5285805574248283688434996707459805960035567} a^{3} + \frac{1218387261729163490110092456326504517884374}{5285805574248283688434996707459805960035567} a$

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:  \( -1 \) (order $2$)
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

$C_{36}$ (as 36T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 36
The 36 conjugacy class representatives for $C_{36}$
Character table for $C_{36}$ is not computed

Intermediate fields

\(\Q(\sqrt{73}) \), 3.3.5329.1, 4.0.6224272.1, 6.6.2073071593.1, 9.9.806460091894081.1, 12.0.1285024504088001365512192.1, 18.18.47477585226700098686074966922953.1

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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{12}$ $36$ ${\href{/LocalNumberField/7.12.0.1}{12} }^{3}$ $36$ $36$ ${\href{/LocalNumberField/17.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{4}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{4}$ $36$ $36$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{4}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{3}$ $36$ $36$ $36$

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
2Data not computed
73Data not computed