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Properties

Label 32.0.59816430901...0000.2
Degree $32$
Signature $[0, 16]$
Discriminant $2^{32}\cdot 3^{16}\cdot 5^{24}\cdot 13^{24}$
Root discriminant $79.30$
Ramified primes $2, 3, 5, 13$
Class number $26624$ (GRH)
Class group $[2, 8, 8, 208]$ (GRH)
Galois group $C_2\times C_4^2$ (as 32T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6561, 0, -107163, 0, 1670868, 0, -26009748, 0, 404862948, 0, -306502677, 0, 169153947, 0, -82519608, 0, 37477864, 0, -10960760, 0, 2866035, 0, -672841, 0, 125492, 0, -7300, 0, 420, 0, -23, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 23*x^30 + 420*x^28 - 7300*x^26 + 125492*x^24 - 672841*x^22 + 2866035*x^20 - 10960760*x^18 + 37477864*x^16 - 82519608*x^14 + 169153947*x^12 - 306502677*x^10 + 404862948*x^8 - 26009748*x^6 + 1670868*x^4 - 107163*x^2 + 6561)
 
gp: K = bnfinit(x^32 - 23*x^30 + 420*x^28 - 7300*x^26 + 125492*x^24 - 672841*x^22 + 2866035*x^20 - 10960760*x^18 + 37477864*x^16 - 82519608*x^14 + 169153947*x^12 - 306502677*x^10 + 404862948*x^8 - 26009748*x^6 + 1670868*x^4 - 107163*x^2 + 6561, 1)
 

Normalized defining polynomial

\( x^{32} - 23 x^{30} + 420 x^{28} - 7300 x^{26} + 125492 x^{24} - 672841 x^{22} + 2866035 x^{20} - 10960760 x^{18} + 37477864 x^{16} - 82519608 x^{14} + 169153947 x^{12} - 306502677 x^{10} + 404862948 x^{8} - 26009748 x^{6} + 1670868 x^{4} - 107163 x^{2} + 6561 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 16]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5981643090147991811559885370844487936000000000000000000000000=2^{32}\cdot 3^{16}\cdot 5^{24}\cdot 13^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.30$
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:  $2, 3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(780=2^{2}\cdot 3\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{780}(1,·)$, $\chi_{780}(259,·)$, $\chi_{780}(391,·)$, $\chi_{780}(649,·)$, $\chi_{780}(493,·)$, $\chi_{780}(281,·)$, $\chi_{780}(103,·)$, $\chi_{780}(157,·)$, $\chi_{780}(671,·)$, $\chi_{780}(161,·)$, $\chi_{780}(547,·)$, $\chi_{780}(551,·)$, $\chi_{780}(47,·)$, $\chi_{780}(437,·)$, $\chi_{780}(313,·)$, $\chi_{780}(571,·)$, $\chi_{780}(317,·)$, $\chi_{780}(181,·)$, $\chi_{780}(707,·)$, $\chi_{780}(203,·)$, $\chi_{780}(79,·)$, $\chi_{780}(337,·)$, $\chi_{780}(83,·)$, $\chi_{780}(469,·)$, $\chi_{780}(727,·)$, $\chi_{780}(473,·)$, $\chi_{780}(593,·)$, $\chi_{780}(359,·)$, $\chi_{780}(749,·)$, $\chi_{780}(239,·)$, $\chi_{780}(629,·)$, $\chi_{780}(703,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{4} a^{10} + \frac{1}{4}$, $\frac{1}{4} a^{11} + \frac{1}{4} a$, $\frac{1}{4} a^{12} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{13} + \frac{1}{4} a^{3}$, $\frac{1}{4} a^{14} + \frac{1}{4} a^{4}$, $\frac{1}{4} a^{15} + \frac{1}{4} a^{5}$, $\frac{1}{4} a^{16} + \frac{1}{4} a^{6}$, $\frac{1}{4} a^{17} + \frac{1}{4} a^{7}$, $\frac{1}{12} a^{18} + \frac{1}{12} a^{16} - \frac{1}{12} a^{12} - \frac{1}{12} a^{10} + \frac{5}{12} a^{8} - \frac{1}{4} a^{6} + \frac{1}{3} a^{4} - \frac{5}{12} a^{2} + \frac{1}{4}$, $\frac{1}{12} a^{19} + \frac{1}{12} a^{17} - \frac{1}{12} a^{13} - \frac{1}{12} a^{11} + \frac{5}{12} a^{9} - \frac{1}{4} a^{7} + \frac{1}{3} a^{5} - \frac{5}{12} a^{3} + \frac{1}{4} a$, $\frac{1}{144} a^{20} + \frac{1}{36} a^{18} - \frac{1}{12} a^{16} + \frac{1}{18} a^{14} - \frac{1}{36} a^{12} - \frac{5}{72} a^{10} - \frac{5}{12} a^{8} + \frac{13}{36} a^{6} - \frac{1}{18} a^{4} + \frac{5}{12} a^{2} - \frac{3}{16}$, $\frac{1}{144} a^{21} + \frac{1}{36} a^{19} - \frac{1}{12} a^{17} + \frac{1}{18} a^{15} - \frac{1}{36} a^{13} - \frac{5}{72} a^{11} - \frac{5}{12} a^{9} + \frac{13}{36} a^{7} - \frac{1}{18} a^{5} + \frac{5}{12} a^{3} - \frac{3}{16} a$, $\frac{1}{432} a^{22} + \frac{1}{432} a^{20} + \frac{1}{36} a^{18} + \frac{1}{54} a^{16} - \frac{7}{108} a^{14} + \frac{1}{216} a^{12} + \frac{1}{72} a^{10} - \frac{5}{108} a^{8} + \frac{11}{54} a^{6} - \frac{17}{36} a^{4} - \frac{7}{48} a^{2} - \frac{1}{16}$, $\frac{1}{432} a^{23} + \frac{1}{432} a^{21} + \frac{1}{36} a^{19} + \frac{1}{54} a^{17} - \frac{7}{108} a^{15} + \frac{1}{216} a^{13} + \frac{1}{72} a^{11} - \frac{5}{108} a^{9} + \frac{11}{54} a^{7} - \frac{17}{36} a^{5} - \frac{7}{48} a^{3} - \frac{1}{16} a$, $\frac{1}{79056} a^{24} - \frac{2}{4941} a^{22} + \frac{13}{13176} a^{20} - \frac{25}{19764} a^{18} - \frac{403}{4941} a^{16} + \frac{751}{39528} a^{14} + \frac{403}{6588} a^{12} - \frac{1463}{19764} a^{10} + \frac{3643}{19764} a^{8} + \frac{479}{1647} a^{6} - \frac{1435}{8784} a^{4} - \frac{329}{732} a^{2} - \frac{93}{488}$, $\frac{1}{237168} a^{25} - \frac{215}{237168} a^{23} - \frac{35}{79056} a^{21} - \frac{2221}{59292} a^{19} + \frac{6257}{59292} a^{17} + \frac{3313}{118584} a^{15} + \frac{1843}{39528} a^{13} - \frac{10063}{118584} a^{11} + \frac{16087}{59292} a^{9} + \frac{5515}{19764} a^{7} - \frac{215}{26352} a^{5} - \frac{2597}{8784} a^{3} + \frac{1339}{2928} a$, $\frac{1}{1161968663849456701872} a^{26} + \frac{92743903282721}{145246082981182087734} a^{24} - \frac{15586835315202901}{24207680496863681289} a^{22} + \frac{988539839745682319}{1161968663849456701872} a^{20} + \frac{2572369557750760553}{290492165962364175468} a^{18} + \frac{22921593476517147973}{580984331924728350936} a^{16} - \frac{5978577496395308153}{96830721987454725156} a^{14} - \frac{57619815091020725}{1190541663780181047} a^{12} + \frac{11245837616308580657}{580984331924728350936} a^{10} - \frac{27313329213376279945}{96830721987454725156} a^{8} - \frac{8091259118777296351}{129107629316606300208} a^{6} - \frac{1598764867983112705}{3586323036572397228} a^{4} - \frac{414463055599336216}{896580759143099307} a^{2} + \frac{1807037988250598881}{4781764048763196304}$, $\frac{1}{3485905991548370105616} a^{27} + \frac{92743903282721}{435738248943546263202} a^{25} - \frac{15586835315202901}{72623041490591043867} a^{23} - \frac{1770171748135552861}{871476497887092526404} a^{21} - \frac{29704537771400814499}{871476497887092526404} a^{19} - \frac{122324489504664939761}{1742952995774185052808} a^{17} - \frac{8891435637029396321}{72623041490591043867} a^{15} - \frac{891891295797516815}{14286499965362172564} a^{13} + \frac{50003666385737706025}{871476497887092526404} a^{11} - \frac{27313329213376279945}{290492165962364175468} a^{9} - \frac{54713458594218460315}{387322887949818900624} a^{7} - \frac{5237319705927816863}{16138453664575787526} a^{5} + \frac{4618213091604350285}{10758969109717191684} a^{3} + \frac{786242722995761927}{1793161518286198614} a$, $\frac{1}{3485905991548370105616} a^{28} + \frac{1}{3485905991548370105616} a^{26} - \frac{1203242655334945}{580984331924728350936} a^{24} + \frac{2168773165652904251}{3485905991548370105616} a^{22} - \frac{679501119700381321}{3485905991548370105616} a^{20} + \frac{30285753547147977277}{1742952995774185052808} a^{18} + \frac{49287908876716116889}{580984331924728350936} a^{16} + \frac{76891893677574217753}{871476497887092526404} a^{14} + \frac{149474644808120018693}{1742952995774185052808} a^{12} - \frac{13036607772515479399}{580984331924728350936} a^{10} - \frac{7291598920040375663}{387322887949818900624} a^{8} + \frac{44245981633543435519}{129107629316606300208} a^{6} + \frac{329824439257926215}{7172646073144794456} a^{4} - \frac{1721505579382127291}{14345292146289588912} a^{2} + \frac{1849302919121146199}{4781764048763196304}$, $\frac{1}{10457717974645110316848} a^{29} + \frac{1}{10457717974645110316848} a^{27} - \frac{1203242655334945}{1742952995774185052808} a^{25} + \frac{2168773165652904251}{10457717974645110316848} a^{23} + \frac{1470511211072706248}{653607373415319394803} a^{21} + \frac{78701114540875339855}{5228858987322555158424} a^{19} + \frac{146118630864170842045}{1742952995774185052808} a^{17} - \frac{46280934900235775635}{1307214746830638789606} a^{15} - \frac{334678965129153607087}{5228858987322555158424} a^{13} - \frac{26691370966977474107}{871476497887092526404} a^{11} + \frac{218646752384020649701}{1161968663849456701872} a^{9} - \frac{5962540878470125673}{387322887949818900624} a^{7} + \frac{15932485970158767095}{64553814658303150104} a^{5} - \frac{1519546189032235339}{4781764048763196304} a^{3} + \frac{477873850728436933}{1195441012190799076} a$, $\frac{1}{41830871898580441267392} a^{30} + \frac{1}{10457717974645110316848} a^{28} - \frac{1}{3485905991548370105616} a^{26} + \frac{2401103703331429}{1307214746830638789606} a^{24} - \frac{1158713903525396167}{10457717974645110316848} a^{22} - \frac{57233623161081512413}{41830871898580441267392} a^{20} + \frac{45781540626187594787}{1742952995774185052808} a^{18} + \frac{569286111768105935945}{5228858987322555158424} a^{16} + \frac{58318589582285784095}{1307214746830638789606} a^{14} + \frac{103836777462534818419}{1742952995774185052808} a^{12} - \frac{114936835224297127415}{1549291551799275602496} a^{10} - \frac{15781139678915014709}{43035876438868766736} a^{8} + \frac{52324478796279651349}{129107629316606300208} a^{6} - \frac{36241552349556679}{92749733704458549} a^{4} + \frac{6320154977465760689}{14345292146289588912} a^{2} + \frac{356095852479750035}{19127056195052785216}$, $\frac{1}{125492615695741323802176} a^{31} + \frac{1}{31373153923935330950544} a^{29} - \frac{1}{10457717974645110316848} a^{27} + \frac{2401103703331429}{3921644240491916368818} a^{25} - \frac{1585399650024317341}{1960822120245958184409} a^{23} + \frac{426919986776192113367}{125492615695741323802176} a^{21} + \frac{94196901619914957365}{5228858987322555158424} a^{19} - \frac{399021108106441315615}{15686576961967665475272} a^{17} - \frac{730631638225657276925}{7843288480983832737636} a^{15} + \frac{217337538793169231351}{2614429493661277579212} a^{13} + \frac{423011620261562456785}{4647874655397826807488} a^{11} - \frac{19848275756356665379}{387322887949818900624} a^{9} + \frac{39756391826321466535}{129107629316606300208} a^{7} + \frac{174505095583797175}{1112996804453502588} a^{5} - \frac{1184418596324782691}{10758969109717191684} a^{3} + \frac{7689825028034977493}{19127056195052785216} a$

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

Class group and class number

$C_{2}\times C_{8}\times C_{8}\times C_{208}$, which has order $26624$ (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:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{4847999581601825359}{125492615695741323802176} a^{31} - \frac{13937999304009886927}{15686576961967665475272} a^{29} + \frac{169679985356063887565}{10457717974645110316848} a^{27} - \frac{8847599236423331280175}{31373153923935330950544} a^{25} + \frac{152096290873594066987907}{31373153923935330950544} a^{23} - \frac{3261932886484553776374919}{125492615695741323802176} a^{21} + \frac{144734692519207470147863}{1307214746830638789606} a^{19} - \frac{6642219986754752915239105}{15686576961967665475272} a^{17} + \frac{22711583623916264119544147}{15686576961967665475272} a^{15} - \frac{16668959377414443362880803}{5228858987322555158424} a^{13} + \frac{30372528306751753402945999}{4647874655397826807488} a^{11} - \frac{6879296644852159814050031}{580984331924728350936} a^{9} + \frac{6057948773179264126623143}{387322887949818900624} a^{7} - \frac{129727620804083244781481}{129107629316606300208} a^{5} + \frac{925967920085948643569}{14345292146289588912} a^{3} - \frac{237551979498489442591}{57381168585158355648} a \) (order $20$)
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:  \( 95287215485273.48 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_4^2$ (as 32T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2\times C_4^2$
Character table for $C_2\times C_4^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{-65}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-13}) \), \(\Q(i, \sqrt{65})\), 4.4.39546000.2, 4.0.2471625.2, \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{13})\), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{-5}, \sqrt{-13})\), 4.4.39546000.1, 4.0.2471625.1, \(\Q(\sqrt{5}, \sqrt{-13})\), \(\Q(\sqrt{-5}, \sqrt{13})\), \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{5})\), 4.4.338000.1, 4.0.21125.1, 4.4.19773.1, 4.0.316368.2, 4.4.494325.1, 4.0.7909200.1, 8.0.1563886116000000.46, 8.0.4569760000.1, 8.0.1563886116000000.44, 8.8.1563886116000000.7, 8.0.1563886116000000.65, 8.0.6108930140625.4, 8.0.1563886116000000.37, \(\Q(\zeta_{20})\), 8.0.114244000000.2, 8.0.100088711424.1, 8.0.62555444640000.62, 8.8.114244000000.1, 8.0.446265625.1, 8.8.244357205625.1, 8.0.62555444640000.63, 8.0.114244000000.1, 8.0.114244000000.3, 8.0.62555444640000.43, 8.0.62555444640000.68, 16.0.2445739783817565456000000000000.9, 16.0.13051691536000000000000.1, 16.0.3913183654108104729600000000.8, 16.16.2445739783817565456000000000000.3, 16.0.2445739783817565456000000000000.14, 16.0.2445739783817565456000000000000.5, 16.0.37319027463036582275390625.3

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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
13Data not computed