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Properties

Label 32.0.40466747549...2736.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{32}\cdot 7^{16}\cdot 17^{28}$
Root discriminant $63.13$
Ramified primes $2, 7, 17$
Class number $360$ (GRH)
Class group $[3, 120]$ (GRH)
Galois group $C_2^2\times C_8$ (as 32T37)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![65536, 0, -1769472, 0, 22093824, 0, -88639488, 0, 193736448, 0, -255944832, 0, 224682160, 0, -138056868, 0, 61603217, 0, -20412324, 0, 5086474, 0, -955614, 0, 134311, 0, -13806, 0, 991, 0, -45, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 45*x^30 + 991*x^28 - 13806*x^26 + 134311*x^24 - 955614*x^22 + 5086474*x^20 - 20412324*x^18 + 61603217*x^16 - 138056868*x^14 + 224682160*x^12 - 255944832*x^10 + 193736448*x^8 - 88639488*x^6 + 22093824*x^4 - 1769472*x^2 + 65536)
 
gp: K = bnfinit(x^32 - 45*x^30 + 991*x^28 - 13806*x^26 + 134311*x^24 - 955614*x^22 + 5086474*x^20 - 20412324*x^18 + 61603217*x^16 - 138056868*x^14 + 224682160*x^12 - 255944832*x^10 + 193736448*x^8 - 88639488*x^6 + 22093824*x^4 - 1769472*x^2 + 65536, 1)
 

Normalized defining polynomial

\( x^{32} - 45 x^{30} + 991 x^{28} - 13806 x^{26} + 134311 x^{24} - 955614 x^{22} + 5086474 x^{20} - 20412324 x^{18} + 61603217 x^{16} - 138056868 x^{14} + 224682160 x^{12} - 255944832 x^{10} + 193736448 x^{8} - 88639488 x^{6} + 22093824 x^{4} - 1769472 x^{2} + 65536 \)

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:  \(4046674754963777806715657550724832045209369204607193972736=2^{32}\cdot 7^{16}\cdot 17^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.13$
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:  $2, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(476=2^{2}\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{476}(1,·)$, $\chi_{476}(393,·)$, $\chi_{476}(13,·)$, $\chi_{476}(15,·)$, $\chi_{476}(407,·)$, $\chi_{476}(281,·)$, $\chi_{476}(155,·)$, $\chi_{476}(237,·)$, $\chi_{476}(293,·)$, $\chi_{476}(169,·)$, $\chi_{476}(43,·)$, $\chi_{476}(433,·)$, $\chi_{476}(307,·)$, $\chi_{476}(55,·)$, $\chi_{476}(321,·)$, $\chi_{476}(195,·)$, $\chi_{476}(69,·)$, $\chi_{476}(183,·)$, $\chi_{476}(239,·)$, $\chi_{476}(461,·)$, $\chi_{476}(463,·)$, $\chi_{476}(83,·)$, $\chi_{476}(475,·)$, $\chi_{476}(349,·)$, $\chi_{476}(223,·)$, $\chi_{476}(225,·)$, $\chi_{476}(421,·)$, $\chi_{476}(365,·)$, $\chi_{476}(111,·)$, $\chi_{476}(251,·)$, $\chi_{476}(253,·)$, $\chi_{476}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{16} - \frac{1}{4} a^{14} - \frac{1}{2} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{19} - \frac{1}{8} a^{17} + \frac{3}{8} a^{15} - \frac{1}{4} a^{13} - \frac{1}{8} a^{11} - \frac{1}{4} a^{9} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{20} - \frac{1}{16} a^{18} + \frac{3}{16} a^{16} + \frac{3}{8} a^{14} - \frac{1}{16} a^{12} + \frac{3}{8} a^{10} + \frac{1}{8} a^{8} + \frac{1}{4} a^{6} + \frac{1}{16} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{32} a^{21} - \frac{1}{32} a^{19} + \frac{3}{32} a^{17} + \frac{3}{16} a^{15} - \frac{1}{32} a^{13} - \frac{5}{16} a^{11} + \frac{1}{16} a^{9} - \frac{3}{8} a^{7} + \frac{1}{32} a^{5} + \frac{1}{4} a^{3}$, $\frac{1}{64} a^{22} - \frac{1}{64} a^{20} + \frac{3}{64} a^{18} + \frac{3}{32} a^{16} + \frac{31}{64} a^{14} + \frac{11}{32} a^{12} + \frac{1}{32} a^{10} + \frac{5}{16} a^{8} - \frac{31}{64} a^{6} + \frac{1}{8} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{128} a^{23} - \frac{1}{128} a^{21} + \frac{3}{128} a^{19} + \frac{3}{64} a^{17} - \frac{33}{128} a^{15} + \frac{11}{64} a^{13} + \frac{1}{64} a^{11} - \frac{11}{32} a^{9} + \frac{33}{128} a^{7} - \frac{7}{16} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{256} a^{24} - \frac{1}{256} a^{22} + \frac{3}{256} a^{20} + \frac{3}{128} a^{18} - \frac{33}{256} a^{16} - \frac{53}{128} a^{14} + \frac{1}{128} a^{12} - \frac{11}{64} a^{10} + \frac{33}{256} a^{8} - \frac{7}{32} a^{6} + \frac{3}{8} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{512} a^{25} - \frac{1}{512} a^{23} + \frac{3}{512} a^{21} + \frac{3}{256} a^{19} - \frac{33}{512} a^{17} - \frac{53}{256} a^{15} + \frac{1}{256} a^{13} - \frac{11}{128} a^{11} + \frac{33}{512} a^{9} + \frac{25}{64} a^{7} + \frac{3}{16} a^{5} + \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{1024} a^{26} - \frac{1}{1024} a^{24} + \frac{3}{1024} a^{22} + \frac{3}{512} a^{20} - \frac{33}{1024} a^{18} + \frac{203}{512} a^{16} + \frac{1}{512} a^{14} - \frac{11}{256} a^{12} - \frac{479}{1024} a^{10} + \frac{25}{128} a^{8} - \frac{13}{32} a^{6} + \frac{1}{16} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{2048} a^{27} - \frac{1}{2048} a^{25} + \frac{3}{2048} a^{23} + \frac{3}{1024} a^{21} - \frac{33}{2048} a^{19} + \frac{203}{1024} a^{17} + \frac{1}{1024} a^{15} + \frac{245}{512} a^{13} - \frac{479}{2048} a^{11} - \frac{103}{256} a^{9} + \frac{19}{64} a^{7} - \frac{15}{32} a^{5} - \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{274432} a^{28} + \frac{83}{274432} a^{26} - \frac{193}{274432} a^{24} + \frac{985}{137216} a^{22} - \frac{1721}{274432} a^{20} + \frac{14833}{137216} a^{18} - \frac{36587}{137216} a^{16} + \frac{22695}{68608} a^{14} + \frac{24913}{274432} a^{12} - \frac{169}{68608} a^{10} - \frac{2679}{17152} a^{8} + \frac{381}{2144} a^{6} - \frac{255}{1072} a^{4} + \frac{29}{134} a^{2} + \frac{1}{67}$, $\frac{1}{548864} a^{29} + \frac{83}{548864} a^{27} - \frac{193}{548864} a^{25} + \frac{985}{274432} a^{23} - \frac{1721}{548864} a^{21} + \frac{14833}{274432} a^{19} - \frac{36587}{274432} a^{17} - \frac{45913}{137216} a^{15} + \frac{24913}{548864} a^{13} - \frac{169}{137216} a^{11} - \frac{2679}{34304} a^{9} - \frac{1763}{4288} a^{7} + \frac{817}{2144} a^{5} - \frac{105}{268} a^{3} + \frac{1}{134} a$, $\frac{1}{773002429543109060063313563592094401347584} a^{30} - \frac{47716590753688898974014283547237349}{59461725349469927697177966430161107795968} a^{28} - \frac{10868583391807147834022774959642068557}{773002429543109060063313563592094401347584} a^{26} + \frac{424418181703347258171071858047093611979}{386501214771554530031656781796047200673792} a^{24} + \frac{5590340536574526115222377736841456197071}{773002429543109060063313563592094401347584} a^{22} - \frac{12035866073063120407701290681210551978101}{386501214771554530031656781796047200673792} a^{20} + \frac{17023783142503132028502907859222341744281}{386501214771554530031656781796047200673792} a^{18} - \frac{79129992037207870863672558708396057983395}{193250607385777265015828390898023600336896} a^{16} + \frac{223523103979098207368737366129652330195905}{773002429543109060063313563592094401347584} a^{14} - \frac{3629880557189399743193226912930012441919}{7432715668683740962147245803770138474496} a^{12} - \frac{18150831793596387355832204274514341439769}{48312651846444316253957097724505900084224} a^{10} - \frac{165747709174926898005274801766371837907}{12078162961611079063489274431126475021056} a^{8} + \frac{156062894099062465317359658339889143801}{1509770370201384882936159303890809377632} a^{6} + \frac{55204587116124774666686840927004017131}{188721296275173110367019912986351172204} a^{4} - \frac{52576656292083148392521027483997592839}{188721296275173110367019912986351172204} a^{2} - \frac{5705969223469131917267849311120117497}{47180324068793277591754978246587793051}$, $\frac{1}{1546004859086218120126627127184188802695168} a^{31} - \frac{47716590753688898974014283547237349}{118923450698939855394355932860322215591936} a^{29} - \frac{10868583391807147834022774959642068557}{1546004859086218120126627127184188802695168} a^{27} + \frac{424418181703347258171071858047093611979}{773002429543109060063313563592094401347584} a^{25} + \frac{5590340536574526115222377736841456197071}{1546004859086218120126627127184188802695168} a^{23} - \frac{12035866073063120407701290681210551978101}{773002429543109060063313563592094401347584} a^{21} + \frac{17023783142503132028502907859222341744281}{773002429543109060063313563592094401347584} a^{19} - \frac{79129992037207870863672558708396057983395}{386501214771554530031656781796047200673792} a^{17} + \frac{223523103979098207368737366129652330195905}{1546004859086218120126627127184188802695168} a^{15} + \frac{3802835111494341218954018890840126032577}{14865431337367481924294491607540276948992} a^{13} + \frac{30161820052847928898124893449991558644455}{96625303692888632507914195449011800168448} a^{11} + \frac{11912415252436152165483999629360103183149}{24156325923222158126978548862252950042112} a^{9} + \frac{156062894099062465317359658339889143801}{3019540740402769765872318607781618755264} a^{7} - \frac{133516709159048335700333072059347155073}{377442592550346220734039825972702344408} a^{5} + \frac{136144639983089961974498885502353579365}{377442592550346220734039825972702344408} a^{3} + \frac{20737177422662072837243564467733837777}{47180324068793277591754978246587793051} a$

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

Class group and class number

$C_{3}\times C_{120}$, which has order $360$ (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:  \( \frac{407994202068554377171}{56862968055878267369455616} a^{31} - \frac{1412217611216386191567}{4374074465836789797650432} a^{29} + \frac{404278775057077237277737}{56862968055878267369455616} a^{27} - \frac{2815872336681880618842735}{28431484027939133684727808} a^{25} + \frac{54782667711934794516342381}{56862968055878267369455616} a^{23} - \frac{194859562130025958110199415}{28431484027939133684727808} a^{21} + \frac{1036957412031248295976424323}{28431484027939133684727808} a^{19} - \frac{2079942311201895363050381945}{14215742013969566842363904} a^{17} + \frac{25092854252642908498656206579}{56862968055878267369455616} a^{15} - \frac{540077568429486069053622551}{546759308229598724706304} a^{13} + \frac{2849667876515826623737028837}{1776967751746195855295488} a^{11} - \frac{403571521856565707558667655}{222120968968274481911936} a^{9} + \frac{150721644874025912329968511}{111060484484137240955968} a^{7} - \frac{8306666533486751680405267}{13882560560517155119496} a^{5} + \frac{461808600036415282777527}{3470640140129288779874} a^{3} - \frac{8485425913117011379564}{1735320070064644389937} a \) (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:  \( 20780022383218.207 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_8$ (as 32T37):

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 $C_2^2\times C_8$
Character table for $C_2^2\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{119}) \), \(\Q(\sqrt{-119}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-17}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-7}) \), \(\Q(i, \sqrt{119})\), \(\Q(i, \sqrt{17})\), \(\Q(i, \sqrt{7})\), \(\Q(\sqrt{7}, \sqrt{17})\), \(\Q(\sqrt{-7}, \sqrt{-17})\), \(\Q(\sqrt{-7}, \sqrt{17})\), \(\Q(\sqrt{7}, \sqrt{-17})\), 4.4.4913.1, 4.0.78608.1, 4.4.3851792.1, 4.0.240737.1, 8.0.51336683776.1, 8.0.6179217664.1, 8.0.14836301611264.3, 8.8.14836301611264.1, 8.0.14836301611264.1, 8.0.57954303169.1, 8.0.14836301611264.2, 8.0.105046700288.1, \(\Q(\zeta_{17})^+\), 8.0.985223153873.1, 8.8.252217127391488.1, 16.0.220115845500394762571677696.1, 16.0.11034809241396899282944.1, 16.0.63613479349614086383214854144.2, 16.0.63613479349614086383214854144.1, 16.16.63613479349614086383214854144.1, 16.0.63613479349614086383214854144.3, 16.0.970664662927461034900129.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{4}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{16}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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}$
7Data not computed
17Data not computed