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Properties

Label 32.0.59816430901...0000.8
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 $16640$ (GRH)
Class group $[2, 2, 2, 2, 2, 520]$ (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![251412736, 0, -91362752, 0, 1068434336, 0, 377006912, 0, -54425047, 0, -111387544, 0, 13189776, 0, 6913746, 0, -1017614, 0, -319026, 0, 60471, 0, 5030, 0, 2375, 0, 1532, 0, 287, 0, 22, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 22*x^30 + 287*x^28 + 1532*x^26 + 2375*x^24 + 5030*x^22 + 60471*x^20 - 319026*x^18 - 1017614*x^16 + 6913746*x^14 + 13189776*x^12 - 111387544*x^10 - 54425047*x^8 + 377006912*x^6 + 1068434336*x^4 - 91362752*x^2 + 251412736)
 
gp: K = bnfinit(x^32 + 22*x^30 + 287*x^28 + 1532*x^26 + 2375*x^24 + 5030*x^22 + 60471*x^20 - 319026*x^18 - 1017614*x^16 + 6913746*x^14 + 13189776*x^12 - 111387544*x^10 - 54425047*x^8 + 377006912*x^6 + 1068434336*x^4 - 91362752*x^2 + 251412736, 1)
 

Normalized defining polynomial

\( x^{32} + 22 x^{30} + 287 x^{28} + 1532 x^{26} + 2375 x^{24} + 5030 x^{22} + 60471 x^{20} - 319026 x^{18} - 1017614 x^{16} + 6913746 x^{14} + 13189776 x^{12} - 111387544 x^{10} - 54425047 x^{8} + 377006912 x^{6} + 1068434336 x^{4} - 91362752 x^{2} + 251412736 \)

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$$\chi_{780}(1,·)$, $\chi_{780}(259,·)$, $\chi_{780}(391,·)$, $\chi_{780}(649,·)$, $\chi_{780}(151,·)$, $\chi_{780}(541,·)$, $\chi_{780}(31,·)$, $\chi_{780}(421,·)$, $\chi_{780}(47,·)$, $\chi_{780}(181,·)$, $\chi_{780}(287,·)$, $\chi_{780}(317,·)$, $\chi_{780}(53,·)$, $\chi_{780}(83,·)$, $\chi_{780}(707,·)$, $\chi_{780}(437,·)$, $\chi_{780}(203,·)$, $\chi_{780}(77,·)$, $\chi_{780}(79,·)$, $\chi_{780}(593,·)$, $\chi_{780}(467,·)$, $\chi_{780}(469,·)$, $\chi_{780}(473,·)$, $\chi_{780}(677,·)$, $\chi_{780}(571,·)$, $\chi_{780}(229,·)$, $\chi_{780}(233,·)$, $\chi_{780}(619,·)$, $\chi_{780}(109,·)$, $\chi_{780}(623,·)$, $\chi_{780}(499,·)$, $\chi_{780}(443,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{6} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{6} a^{13} + \frac{1}{6} a^{11} - \frac{1}{2} a^{9} - \frac{1}{6} a^{7} - \frac{1}{3} a^{3} + \frac{1}{6} a$, $\frac{1}{12} a^{14} - \frac{1}{12} a^{12} - \frac{5}{12} a^{10} - \frac{1}{12} a^{8} + \frac{1}{6} a^{6} + \frac{1}{3} a^{4} + \frac{5}{12} a^{2} + \frac{1}{3}$, $\frac{1}{12} a^{15} - \frac{1}{12} a^{13} - \frac{5}{12} a^{11} - \frac{1}{12} a^{9} + \frac{1}{6} a^{7} + \frac{1}{3} a^{5} + \frac{5}{12} a^{3} + \frac{1}{3} a$, $\frac{1}{12} a^{16} - \frac{1}{6} a^{12} - \frac{1}{6} a^{10} + \frac{1}{12} a^{8} + \frac{1}{6} a^{6} - \frac{1}{4} a^{4} + \frac{1}{12} a^{2} - \frac{1}{3}$, $\frac{1}{12} a^{17} - \frac{5}{12} a^{9} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{6} a$, $\frac{1}{12} a^{18} - \frac{5}{12} a^{10} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{6} a^{2}$, $\frac{1}{12} a^{19} - \frac{5}{12} a^{11} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{6} a^{3}$, $\frac{1}{348} a^{20} + \frac{1}{58} a^{18} - \frac{7}{348} a^{16} + \frac{5}{348} a^{14} + \frac{10}{87} a^{12} - \frac{17}{348} a^{10} + \frac{7}{116} a^{8} - \frac{13}{348} a^{6} + \frac{47}{116} a^{4} - \frac{21}{58} a^{2} + \frac{3}{29}$, $\frac{1}{348} a^{21} + \frac{1}{58} a^{19} - \frac{7}{348} a^{17} + \frac{5}{348} a^{15} - \frac{3}{58} a^{13} - \frac{25}{116} a^{11} - \frac{51}{116} a^{9} + \frac{15}{116} a^{7} + \frac{47}{116} a^{5} - \frac{5}{174} a^{3} - \frac{11}{174} a$, $\frac{1}{348} a^{22} - \frac{7}{174} a^{18} - \frac{11}{348} a^{16} + \frac{5}{174} a^{14} - \frac{25}{348} a^{12} - \frac{23}{58} a^{10} + \frac{151}{348} a^{8} - \frac{25}{87} a^{6} + \frac{53}{116} a^{4} + \frac{8}{29} a^{2} + \frac{11}{29}$, $\frac{1}{348} a^{23} - \frac{7}{174} a^{19} - \frac{11}{348} a^{17} + \frac{5}{174} a^{15} - \frac{25}{348} a^{13} - \frac{23}{58} a^{11} + \frac{151}{348} a^{9} - \frac{25}{87} a^{7} + \frac{53}{116} a^{5} + \frac{8}{29} a^{3} + \frac{11}{29} a$, $\frac{1}{195228} a^{24} + \frac{155}{195228} a^{22} - \frac{32}{48807} a^{20} - \frac{4373}{195228} a^{18} - \frac{188}{48807} a^{16} - \frac{2989}{195228} a^{14} - \frac{29395}{195228} a^{12} + \frac{54185}{195228} a^{10} - \frac{1058}{48807} a^{8} - \frac{37001}{195228} a^{6} - \frac{733}{4437} a^{4} - \frac{16531}{195228} a^{2} + \frac{380}{1683}$, $\frac{1}{195228} a^{25} + \frac{155}{195228} a^{23} - \frac{32}{48807} a^{21} - \frac{4373}{195228} a^{19} - \frac{188}{48807} a^{17} - \frac{2989}{195228} a^{15} + \frac{3143}{195228} a^{13} + \frac{86723}{195228} a^{11} + \frac{46691}{97614} a^{9} - \frac{69539}{195228} a^{7} - \frac{733}{4437} a^{5} - \frac{81607}{195228} a^{3} + \frac{1321}{3366} a$, $\frac{1}{585684} a^{26} + \frac{1}{585684} a^{24} + \frac{125}{585684} a^{22} - \frac{41}{65076} a^{20} - \frac{65}{11484} a^{18} + \frac{11249}{292842} a^{16} + \frac{91}{6732} a^{14} + \frac{7985}{48807} a^{12} - \frac{168781}{585684} a^{10} + \frac{2443}{16269} a^{8} + \frac{2225}{5916} a^{6} + \frac{274903}{585684} a^{4} + \frac{4838}{146421} a^{2} + \frac{5809}{13311}$, $\frac{1}{1171368} a^{27} - \frac{1}{585684} a^{25} + \frac{79}{68904} a^{23} - \frac{139}{97614} a^{21} + \frac{8317}{390456} a^{19} + \frac{9011}{585684} a^{17} + \frac{937}{43384} a^{15} + \frac{4145}{65076} a^{13} - \frac{96665}{585684} a^{11} - \frac{44375}{195228} a^{9} - \frac{22802}{48807} a^{7} - \frac{41269}{146421} a^{5} + \frac{344957}{1171368} a^{3} - \frac{37595}{146421} a$, $\frac{1}{4144299984} a^{28} - \frac{895}{2072149992} a^{26} + \frac{8627}{4144299984} a^{24} + \frac{93527}{172679166} a^{22} - \frac{659}{2802096} a^{20} - \frac{20434633}{2072149992} a^{18} - \frac{1622495}{1381433328} a^{16} + \frac{85983}{2646424} a^{14} + \frac{179630797}{2072149992} a^{12} + \frac{3284131}{40630392} a^{10} + \frac{88489871}{345358332} a^{8} + \frac{47792671}{259018749} a^{6} + \frac{952875557}{4144299984} a^{4} - \frac{26193383}{94188636} a^{2} - \frac{4023442}{86339583}$, $\frac{1}{8288599968} a^{29} - \frac{895}{4144299984} a^{27} - \frac{12601}{8288599968} a^{25} + \frac{75439}{57559722} a^{23} - \frac{116855}{95271264} a^{21} - \frac{93108691}{4144299984} a^{19} + \frac{102939557}{2762866656} a^{17} + \frac{1480715}{47635632} a^{15} - \frac{240768515}{4144299984} a^{13} - \frac{163719575}{460477776} a^{11} + \frac{14098295}{38373148} a^{9} + \frac{37649426}{259018749} a^{7} - \frac{3328493623}{8288599968} a^{5} + \frac{546886781}{2072149992} a^{3} - \frac{9493289}{57559722} a$, $\frac{1}{271746590242979994965041181722284905601651944065728} a^{30} - \frac{127553058185895739960341843060217360861}{12352117738317272498410962805558404800075088366624} a^{28} + \frac{197710728994972808438064803059940259837212023}{271746590242979994965041181722284905601651944065728} a^{26} + \frac{3690705665214054835764899987560145012871139}{5661387296728749895105024619214268866701082168036} a^{24} - \frac{263704266263606964948090839640482786475449856985}{271746590242979994965041181722284905601651944065728} a^{22} - \frac{69281534154551108775064729310705374483922226067}{135873295121489997482520590861142452800825972032864} a^{20} - \frac{62344474255708131642029020824787835966076327355}{2074401452236488511183520471162480195432457588288} a^{18} - \frac{2429103654648228060580784726184401828212576959343}{135873295121489997482520590861142452800825972032864} a^{16} + \frac{2017288830257790695258839807836575388145967363101}{135873295121489997482520590861142452800825972032864} a^{14} - \frac{21086609396403321294213982637710038934438190833907}{135873295121489997482520590861142452800825972032864} a^{12} + \frac{4133411146076910055391426624542364633558291926147}{33968323780372499370630147715285613200206493008216} a^{10} - \frac{9827664736703461491937750434275902564073577538561}{33968323780372499370630147715285613200206493008216} a^{8} - \frac{65934278805282680514807664956890707183850573729}{3123524025781379252471737720945803512662666023744} a^{6} + \frac{18091144704139038561368750782993618609275644613971}{67936647560744998741260295430571226400412986016432} a^{4} - \frac{69401535157977036043263472891214418457694975165}{292830377417004304919225411338669079312124939726} a^{2} + \frac{2117312898134147335268375961068438463056365182824}{4246040472546562421328768464410701650025811626027}$, $\frac{1}{538601741861586350020711622173568682902474153138272896} a^{31} + \frac{4818077759921165273277415917659792090766091}{269300870930793175010355811086784341451237076569136448} a^{29} + \frac{164360768786247160775183377509981104506237330735}{538601741861586350020711622173568682902474153138272896} a^{27} - \frac{2911909783330751969715233807869241286247006649}{1662351055128352932162690191893730502785414052895904} a^{25} - \frac{519607420436774230131551698640375186066559147226825}{538601741861586350020711622173568682902474153138272896} a^{23} - \frac{918929958394791634075381671323151437166367302359}{24481897357344834091850528280616758313748825142648768} a^{21} - \frac{144409460919933401764862500213483505080869644478547}{4111463678332720229165737573844035747347130939986816} a^{19} + \frac{10488413443795322177091672195877105980701125137052535}{269300870930793175010355811086784341451237076569136448} a^{17} + \frac{2779246481848347794290015105982630697402579113429041}{269300870930793175010355811086784341451237076569136448} a^{15} + \frac{182968462055845996465526090976761003526602471663005}{4414768375914642213284521493225972810676017648674368} a^{13} + \frac{14952429935098050741621886161474284764185128203514729}{33662608866349146876294476385848042681404634571142056} a^{11} + \frac{319019164126418373403088622252151293571999689732421}{3960306925452840808975820751276240315459368773075536} a^{9} + \frac{63826867763213211441235049256066192899443528028185763}{179533913953862116673570540724522894300824717712757632} a^{7} - \frac{554461504396122867946435039113811772524851206163515}{16831304433174573438147238192924021340702317285571028} a^{5} - \frac{12515248523740983875157448420391711094846014647518509}{33662608866349146876294476385848042681404634571142056} a^{3} + \frac{1884048264198092348636750839479836686814185993055203}{4207826108293643359536809548231005335175579321392757} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{520}$, which has order $16640$ (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:  \( -\frac{26305016429533230737}{17828340461005400325998976384} a^{31} - \frac{909875820357796104433}{26742510691508100488998464576} a^{29} - \frac{8160504851624452563199}{17828340461005400325998976384} a^{27} - \frac{36182109905774441993861}{13371255345754050244499232288} a^{25} - \frac{313677261134364910294997}{53485021383016200977996929152} a^{23} - \frac{5791530734009940996467}{524362954735452950764675776} a^{21} - \frac{40401021814010596623991}{408282605977222908228984192} a^{19} + \frac{9715258434821079163757611}{26742510691508100488998464576} a^{17} + \frac{17677774381381783163184815}{8914170230502700162999488192} a^{15} - \frac{233367280063697789820530459}{26742510691508100488998464576} a^{13} - \frac{3596493390457787785614133}{115269442635810777969820968} a^{11} + \frac{327561350163393007431702127}{2228542557625675040749872048} a^{9} + \frac{13899739868589003744638095381}{53485021383016200977996929152} a^{7} - \frac{466661775323909912289348935}{835703459109628140281202018} a^{5} - \frac{841121431204231629499067615}{371423759604279173458312008} a^{3} - \frac{733185175781668643598513779}{835703459109628140281202018} a \) (order $4$)
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:  \( 169883820966474.0 \) (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, \(\Q(\sqrt{5}, \sqrt{-13})\), \(\Q(\sqrt{-5}, \sqrt{13})\), 4.4.190125.1, 4.0.3042000.1, \(\Q(\zeta_{15})^+\), 4.0.18000.1, 4.4.878800.1, 4.0.54925.1, 4.4.35152.1, 4.0.2197.1, 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, 8.0.9253764000000.8, 8.0.324000000.1, 8.0.772289440000.3, 8.0.1235663104.1, 8.8.36147515625.1, 8.0.9253764000000.3, 8.8.772289440000.1, 8.0.3016755625.1, 8.0.9253764000000.1, 8.0.9253764000000.7, 8.0.772289440000.2, 8.0.772289440000.1, 16.0.2445739783817565456000000000000.9, 16.0.85632148167696000000000000.5, 16.0.596430979135513600000000.1, 16.16.2445739783817565456000000000000.1, 16.0.2445739783817565456000000000000.6, 16.0.37319027463036582275390625.2, 16.0.2445739783817565456000000000000.4

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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{32}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5Data not computed
13Data not computed