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Properties

Label 32.0.16898567960...0625.1
Degree $32$
Signature $[0, 16]$
Discriminant $5^{24}\cdot 17^{28}$
Root discriminant $39.89$
Ramified primes $5, 17$
Class number $365$ (GRH)
Class group $[365]$ (GRH)
Galois group $C_4\times C_8$ (as 32T43)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, 26, 134, 741, -664, 3055, -3657, 12428, -2094, 19367, -90, 27196, -7798, 26197, -8919, 21029, -4801, 11628, -2513, 5640, -1493, 2257, -556, 766, -145, 192, -38, 44, -9, 8, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - x^31 + 8*x^30 - 9*x^29 + 44*x^28 - 38*x^27 + 192*x^26 - 145*x^25 + 766*x^24 - 556*x^23 + 2257*x^22 - 1493*x^21 + 5640*x^20 - 2513*x^19 + 11628*x^18 - 4801*x^17 + 21029*x^16 - 8919*x^15 + 26197*x^14 - 7798*x^13 + 27196*x^12 - 90*x^11 + 19367*x^10 - 2094*x^9 + 12428*x^8 - 3657*x^7 + 3055*x^6 - 664*x^5 + 741*x^4 + 134*x^3 + 26*x^2 + 4*x + 1)
 
gp: K = bnfinit(x^32 - x^31 + 8*x^30 - 9*x^29 + 44*x^28 - 38*x^27 + 192*x^26 - 145*x^25 + 766*x^24 - 556*x^23 + 2257*x^22 - 1493*x^21 + 5640*x^20 - 2513*x^19 + 11628*x^18 - 4801*x^17 + 21029*x^16 - 8919*x^15 + 26197*x^14 - 7798*x^13 + 27196*x^12 - 90*x^11 + 19367*x^10 - 2094*x^9 + 12428*x^8 - 3657*x^7 + 3055*x^6 - 664*x^5 + 741*x^4 + 134*x^3 + 26*x^2 + 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{32} - x^{31} + 8 x^{30} - 9 x^{29} + 44 x^{28} - 38 x^{27} + 192 x^{26} - 145 x^{25} + 766 x^{24} - 556 x^{23} + 2257 x^{22} - 1493 x^{21} + 5640 x^{20} - 2513 x^{19} + 11628 x^{18} - 4801 x^{17} + 21029 x^{16} - 8919 x^{15} + 26197 x^{14} - 7798 x^{13} + 27196 x^{12} - 90 x^{11} + 19367 x^{10} - 2094 x^{9} + 12428 x^{8} - 3657 x^{7} + 3055 x^{6} - 664 x^{5} + 741 x^{4} + 134 x^{3} + 26 x^{2} + 4 x + 1 \)

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:  \(1689856796077948861582881651286068022251129150390625=5^{24}\cdot 17^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.89$
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:  $5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(85=5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{85}(1,·)$, $\chi_{85}(2,·)$, $\chi_{85}(4,·)$, $\chi_{85}(8,·)$, $\chi_{85}(9,·)$, $\chi_{85}(13,·)$, $\chi_{85}(16,·)$, $\chi_{85}(18,·)$, $\chi_{85}(19,·)$, $\chi_{85}(21,·)$, $\chi_{85}(26,·)$, $\chi_{85}(32,·)$, $\chi_{85}(33,·)$, $\chi_{85}(36,·)$, $\chi_{85}(38,·)$, $\chi_{85}(42,·)$, $\chi_{85}(43,·)$, $\chi_{85}(47,·)$, $\chi_{85}(49,·)$, $\chi_{85}(52,·)$, $\chi_{85}(53,·)$, $\chi_{85}(59,·)$, $\chi_{85}(64,·)$, $\chi_{85}(66,·)$, $\chi_{85}(67,·)$, $\chi_{85}(69,·)$, $\chi_{85}(72,·)$, $\chi_{85}(76,·)$, $\chi_{85}(77,·)$, $\chi_{85}(81,·)$, $\chi_{85}(83,·)$, $\chi_{85}(84,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{24} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{25} - \frac{1}{2} a^{10}$, $\frac{1}{1084} a^{26} + \frac{33}{542} a^{25} - \frac{111}{542} a^{24} + \frac{239}{1084} a^{23} - \frac{165}{1084} a^{22} + \frac{115}{542} a^{21} - \frac{253}{1084} a^{20} - \frac{113}{542} a^{19} - \frac{109}{1084} a^{18} - \frac{247}{1084} a^{17} + \frac{189}{1084} a^{16} + \frac{41}{1084} a^{15} + \frac{267}{542} a^{14} + \frac{81}{542} a^{13} - \frac{181}{542} a^{12} + \frac{297}{1084} a^{11} - \frac{117}{542} a^{10} - \frac{175}{542} a^{9} - \frac{519}{1084} a^{8} + \frac{183}{1084} a^{7} + \frac{227}{542} a^{6} - \frac{339}{1084} a^{5} - \frac{110}{271} a^{4} + \frac{391}{1084} a^{3} - \frac{73}{1084} a^{2} + \frac{489}{1084} a - \frac{105}{1084}$, $\frac{1}{1084} a^{27} - \frac{121}{542} a^{25} + \frac{257}{1084} a^{24} - \frac{221}{1084} a^{23} - \frac{131}{542} a^{22} - \frac{257}{1084} a^{21} + \frac{53}{271} a^{20} + \frac{173}{1084} a^{19} - \frac{99}{1084} a^{18} + \frac{231}{1084} a^{17} + \frac{33}{1084} a^{16} - \frac{1}{271} a^{15} - \frac{197}{542} a^{14} - \frac{107}{542} a^{13} + \frac{341}{1084} a^{12} - \frac{81}{271} a^{11} - \frac{41}{542} a^{10} + \frac{359}{1084} a^{9} + \frac{291}{1084} a^{8} - \frac{121}{542} a^{7} + \frac{49}{1084} a^{6} + \frac{127}{542} a^{5} - \frac{379}{1084} a^{4} - \frac{405}{1084} a^{3} - \frac{113}{1084} a^{2} - \frac{401}{1084} a + \frac{213}{542}$, $\frac{1}{1084} a^{28} - \frac{31}{1084} a^{25} + \frac{255}{1084} a^{24} + \frac{31}{271} a^{23} - \frac{79}{1084} a^{22} + \frac{23}{542} a^{21} + \frac{193}{1084} a^{20} - \frac{49}{1084} a^{19} - \frac{131}{1084} a^{18} - \frac{121}{1084} a^{17} + \frac{103}{542} a^{16} - \frac{57}{271} a^{15} + \frac{9}{542} a^{14} + \frac{521}{1084} a^{13} - \frac{31}{271} a^{12} + \frac{62}{271} a^{11} + \frac{99}{1084} a^{10} + \frac{143}{1084} a^{9} - \frac{24}{271} a^{8} - \frac{109}{1084} a^{7} + \frac{24}{271} a^{6} + \frac{509}{1084} a^{5} - \frac{111}{1084} a^{4} + \frac{201}{1084} a^{3} + \frac{361}{1084} a^{2} - \frac{119}{271} a - \frac{239}{542}$, $\frac{1}{6994690103778888916} a^{29} + \frac{238992580538113}{6994690103778888916} a^{28} + \frac{11176854233145}{25810664589589996} a^{27} - \frac{677989716707749}{3497345051889444458} a^{26} + \frac{118919764428686289}{1748672525944722229} a^{25} - \frac{106657785389522033}{1748672525944722229} a^{24} - \frac{698773070032505533}{6994690103778888916} a^{23} - \frac{371535526848374415}{3497345051889444458} a^{22} + \frac{47718023896353045}{1748672525944722229} a^{21} + \frac{818706205806022961}{6994690103778888916} a^{20} + \frac{240757971105817625}{6994690103778888916} a^{19} - \frac{505612948403975327}{3497345051889444458} a^{18} - \frac{793790987222013933}{6994690103778888916} a^{17} - \frac{424983883583533873}{3497345051889444458} a^{16} - \frac{1746590709729208965}{6994690103778888916} a^{15} - \frac{1665521171382451361}{6994690103778888916} a^{14} - \frac{2331973421906031387}{6994690103778888916} a^{13} - \frac{2383808486743315463}{6994690103778888916} a^{12} - \frac{364541401960640012}{1748672525944722229} a^{11} + \frac{1275934640593223007}{3497345051889444458} a^{10} - \frac{1478957734038702985}{3497345051889444458} a^{9} + \frac{1151716964890354903}{6994690103778888916} a^{8} + \frac{420009084031151661}{3497345051889444458} a^{7} - \frac{692530452832543281}{1748672525944722229} a^{6} + \frac{1281886295787421019}{6994690103778888916} a^{5} - \frac{525234219424847}{78592023637965044} a^{4} - \frac{1022606909164384901}{3497345051889444458} a^{3} + \frac{174298592997779577}{6994690103778888916} a^{2} + \frac{667982046625337338}{1748672525944722229} a + \frac{2455766378331128239}{6994690103778888916}$, $\frac{1}{6994690103778888916} a^{30} - \frac{768783998027255}{1748672525944722229} a^{28} - \frac{261369195419286}{1748672525944722229} a^{27} - \frac{561238360446757}{3497345051889444458} a^{26} + \frac{341964516196992609}{3497345051889444458} a^{25} + \frac{46424210449192113}{1748672525944722229} a^{24} + \frac{401764436203986046}{1748672525944722229} a^{23} - \frac{289975995360056245}{1748672525944722229} a^{22} - \frac{54627792963365033}{1748672525944722229} a^{21} - \frac{43496485087228783}{1748672525944722229} a^{20} + \frac{134956827610176183}{3497345051889444458} a^{19} + \frac{213367843794216672}{1748672525944722229} a^{18} - \frac{422592922581644560}{1748672525944722229} a^{17} + \frac{418695181300051129}{3497345051889444458} a^{16} - \frac{266084463220423162}{1748672525944722229} a^{15} + \frac{402418041184279804}{1748672525944722229} a^{14} + \frac{833184553214866472}{1748672525944722229} a^{13} - \frac{121445817676648863}{1748672525944722229} a^{12} - \frac{1511564017088487105}{3497345051889444458} a^{11} - \frac{975255567396235899}{3497345051889444458} a^{10} - \frac{431113377550052514}{1748672525944722229} a^{9} + \frac{124598039070150487}{1748672525944722229} a^{8} - \frac{827724636837817853}{1748672525944722229} a^{7} + \frac{553695699835829497}{1748672525944722229} a^{6} + \frac{199514949659538217}{1748672525944722229} a^{5} - \frac{460990733715285161}{3497345051889444458} a^{4} + \frac{260276915848824781}{1748672525944722229} a^{3} - \frac{483587524257293127}{1748672525944722229} a^{2} - \frac{1627700963448124247}{3497345051889444458} a + \frac{356428630790383079}{6994690103778888916}$, $\frac{1}{13989380207557777832} a^{31} - \frac{4356973972692453}{13989380207557777832} a^{28} + \frac{5063314461365647}{13989380207557777832} a^{27} - \frac{4516368649464381}{13989380207557777832} a^{26} + \frac{1206343396195167469}{13989380207557777832} a^{25} + \frac{243027655217430363}{1748672525944722229} a^{24} - \frac{402468218491128641}{3497345051889444458} a^{23} - \frac{802160556079857189}{3497345051889444458} a^{22} + \frac{2898298222545634327}{13989380207557777832} a^{21} - \frac{181119462553860909}{1748672525944722229} a^{20} + \frac{1361037987664504055}{6994690103778888916} a^{19} - \frac{1017905265630036765}{13989380207557777832} a^{18} - \frac{1392772855137488721}{13989380207557777832} a^{17} - \frac{1358217831420281895}{6994690103778888916} a^{16} + \frac{1218816771079836903}{13989380207557777832} a^{15} + \frac{325003902824605323}{6994690103778888916} a^{14} + \frac{6006520249301455623}{13989380207557777832} a^{13} - \frac{4685296753849334383}{13989380207557777832} a^{12} - \frac{6737927339016199061}{13989380207557777832} a^{11} - \frac{6301593504074454125}{13989380207557777832} a^{10} + \frac{2234617461232295119}{6994690103778888916} a^{9} - \frac{3050702031215734479}{6994690103778888916} a^{8} + \frac{1290856213855959783}{6994690103778888916} a^{7} - \frac{59910188491136897}{157184047275930088} a^{6} + \frac{623756617958205042}{1748672525944722229} a^{5} + \frac{2337793944771734303}{6994690103778888916} a^{4} - \frac{2463453890936182603}{13989380207557777832} a^{3} - \frac{3083330581979061409}{13989380207557777832} a^{2} - \frac{6811727986398740463}{13989380207557777832} a - \frac{6307587425954534063}{13989380207557777832}$

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

Class group and class number

$C_{365}$, which has order $365$ (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:  \( \frac{2528535812253219747}{13989380207557777832} a^{31} - \frac{312543835478787294}{1748672525944722229} a^{30} + \frac{5054686180429912637}{3497345051889444458} a^{29} - \frac{22568921454126168423}{13989380207557777832} a^{28} + \frac{111163504319626792311}{13989380207557777832} a^{27} - \frac{95154251627665944351}{13989380207557777832} a^{26} + \frac{485341708327626640635}{13989380207557777832} a^{25} - \frac{181343349173891893349}{6994690103778888916} a^{24} + \frac{968372845836137320311}{6994690103778888916} a^{23} - \frac{173780009551890319743}{1748672525944722229} a^{22} + \frac{5706837683947301957631}{13989380207557777832} a^{21} - \frac{933720007854356298855}{3497345051889444458} a^{20} + \frac{6580160458217253382}{6452666147397499} a^{19} - \frac{6260090957345399045841}{13989380207557777832} a^{18} + \frac{29447006459793751382859}{13989380207557777832} a^{17} - \frac{1493687120986818596160}{1748672525944722229} a^{16} + \frac{53257777602069764852469}{13989380207557777832} a^{15} - \frac{11118721754867681199515}{6994690103778888916} a^{14} + \frac{66381899029907353452729}{13989380207557777832} a^{13} - \frac{19483948346692734930555}{13989380207557777832} a^{12} + \frac{69039029160797434893801}{13989380207557777832} a^{11} + \frac{6734749476955779405}{13989380207557777832} a^{10} + \frac{12350937480213690861817}{3497345051889444458} a^{9} - \frac{644220557806567902507}{1748672525944722229} a^{8} + \frac{15756081767730935265735}{6994690103778888916} a^{7} - \frac{9248046756838033425741}{13989380207557777832} a^{6} + \frac{968592671135134576767}{1748672525944722229} a^{5} - \frac{234013867799071988415}{1748672525944722229} a^{4} + \frac{1883705248718057242449}{13989380207557777832} a^{3} + \frac{340667106240790505331}{13989380207557777832} a^{2} + \frac{66085789685343354441}{13989380207557777832} a + \frac{10172392514420249871}{13989380207557777832} \) (order $10$)
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:  \( 42597284566.379654 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times C_8$ (as 32T43):

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 $C_4\times C_8$
Character table for $C_4\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{85}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{5}) \), 4.0.614125.2, \(\Q(\sqrt{5}, \sqrt{17})\), 4.0.614125.1, 4.4.122825.1, 4.4.4913.1, 4.0.36125.1, \(\Q(\zeta_{5})\), 8.0.377149515625.1, 8.8.15085980625.1, 8.0.1305015625.1, 8.0.6411541765625.1, 8.0.6411541765625.2, 8.8.256461670625.1, \(\Q(\zeta_{17})^+\), 16.0.142241757136172119140625.1, 16.0.41107867812353742431640625.1, 16.16.65772588499765987890625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
5Data not computed
17Data not computed