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Properties

Label 32.0.72578796740...0000.3
Degree $32$
Signature $[0, 16]$
Discriminant $2^{32}\cdot 5^{24}\cdot 17^{28}$
Root discriminant $79.78$
Ramified primes $2, 5, 17$
Class number $18688$ (GRH)
Class group $[2, 4, 4, 584]$ (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![18800896, 0, -46720448, 0, 49669184, 0, -31754380, 0, 42559893, 0, -48774875, 0, 26657386, 0, -6766397, 0, 763268, 0, -141927, 0, 62416, 0, 2385, 0, -5812, 0, 675, 0, 114, 0, -23, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 23*x^30 + 114*x^28 + 675*x^26 - 5812*x^24 + 2385*x^22 + 62416*x^20 - 141927*x^18 + 763268*x^16 - 6766397*x^14 + 26657386*x^12 - 48774875*x^10 + 42559893*x^8 - 31754380*x^6 + 49669184*x^4 - 46720448*x^2 + 18800896)
 
gp: K = bnfinit(x^32 - 23*x^30 + 114*x^28 + 675*x^26 - 5812*x^24 + 2385*x^22 + 62416*x^20 - 141927*x^18 + 763268*x^16 - 6766397*x^14 + 26657386*x^12 - 48774875*x^10 + 42559893*x^8 - 31754380*x^6 + 49669184*x^4 - 46720448*x^2 + 18800896, 1)
 

Normalized defining polynomial

\( x^{32} - 23 x^{30} + 114 x^{28} + 675 x^{26} - 5812 x^{24} + 2385 x^{22} + 62416 x^{20} - 141927 x^{18} + 763268 x^{16} - 6766397 x^{14} + 26657386 x^{12} - 48774875 x^{10} + 42559893 x^{8} - 31754380 x^{6} + 49669184 x^{4} - 46720448 x^{2} + 18800896 \)

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:  \(7257879674078131427258907485712138496000000000000000000000000=2^{32}\cdot 5^{24}\cdot 17^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.78$
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:  $2, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(340=2^{2}\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{340}(1,·)$, $\chi_{340}(259,·)$, $\chi_{340}(263,·)$, $\chi_{340}(271,·)$, $\chi_{340}(149,·)$, $\chi_{340}(287,·)$, $\chi_{340}(169,·)$, $\chi_{340}(43,·)$, $\chi_{340}(257,·)$, $\chi_{340}(53,·)$, $\chi_{340}(191,·)$, $\chi_{340}(69,·)$, $\chi_{340}(247,·)$, $\chi_{340}(77,·)$, $\chi_{340}(81,·)$, $\chi_{340}(83,·)$, $\chi_{340}(213,·)$, $\chi_{340}(87,·)$, $\chi_{340}(89,·)$, $\chi_{340}(93,·)$, $\chi_{340}(223,·)$, $\chi_{340}(101,·)$, $\chi_{340}(171,·)$, $\chi_{340}(239,·)$, $\chi_{340}(319,·)$, $\chi_{340}(339,·)$, $\chi_{340}(117,·)$, $\chi_{340}(297,·)$, $\chi_{340}(251,·)$, $\chi_{340}(21,·)$, $\chi_{340}(253,·)$, $\chi_{340}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} + \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} + \frac{1}{4} a^{4}$, $\frac{1}{4} a^{11} + \frac{1}{4} a^{5}$, $\frac{1}{4} a^{12} + \frac{1}{4} a^{6}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{8} a^{9} + \frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{16} - \frac{1}{8} a^{10} - \frac{3}{16} a^{4}$, $\frac{1}{32} a^{17} - \frac{1}{16} a^{11} - \frac{3}{32} a^{5}$, $\frac{1}{32} a^{18} - \frac{1}{16} a^{12} - \frac{3}{32} a^{6}$, $\frac{1}{32} a^{19} - \frac{1}{16} a^{13} - \frac{3}{32} a^{7}$, $\frac{1}{32} a^{20} - \frac{1}{16} a^{14} - \frac{3}{32} a^{8}$, $\frac{1}{32} a^{21} - \frac{1}{16} a^{15} - \frac{3}{32} a^{9}$, $\frac{1}{128} a^{22} - \frac{1}{128} a^{20} + \frac{1}{128} a^{18} + \frac{1}{64} a^{16} + \frac{3}{64} a^{14} + \frac{5}{64} a^{12} - \frac{7}{128} a^{10} + \frac{7}{128} a^{8} + \frac{9}{128} a^{6} + \frac{7}{16} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{128} a^{23} - \frac{1}{128} a^{21} + \frac{1}{128} a^{19} - \frac{1}{64} a^{17} + \frac{3}{64} a^{15} - \frac{3}{64} a^{13} - \frac{15}{128} a^{11} - \frac{9}{128} a^{9} - \frac{7}{128} a^{7} + \frac{13}{32} a^{5} + \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{128} a^{24} - \frac{1}{128} a^{18} - \frac{5}{128} a^{12} + \frac{61}{128} a^{6} - \frac{1}{2}$, $\frac{1}{128} a^{25} - \frac{1}{128} a^{19} - \frac{5}{128} a^{13} - \frac{3}{128} a^{7}$, $\frac{1}{51712} a^{26} + \frac{67}{25856} a^{24} - \frac{53}{51712} a^{22} + \frac{49}{3232} a^{20} - \frac{3}{51712} a^{18} + \frac{287}{25856} a^{16} - \frac{795}{51712} a^{14} + \frac{167}{1616} a^{12} - \frac{6045}{51712} a^{10} + \frac{235}{25856} a^{8} + \frac{11753}{51712} a^{6} + \frac{1147}{3232} a^{4} - \frac{399}{808} a^{2} + \frac{33}{808}$, $\frac{1}{103424} a^{27} + \frac{67}{51712} a^{25} - \frac{53}{103424} a^{23} - \frac{13}{1616} a^{21} + \frac{1613}{103424} a^{19} + \frac{287}{51712} a^{17} + \frac{2437}{103424} a^{15} + \frac{33}{1616} a^{13} - \frac{6045}{103424} a^{11} + \frac{2659}{51712} a^{9} + \frac{6905}{103424} a^{7} - \frac{2085}{6464} a^{5} - \frac{399}{1616} a^{3} + \frac{33}{1616} a$, $\frac{1}{9204736} a^{28} + \frac{17}{2301184} a^{26} - \frac{13745}{9204736} a^{24} + \frac{6181}{4602368} a^{22} - \frac{4883}{9204736} a^{20} + \frac{9485}{2301184} a^{18} + \frac{274825}{9204736} a^{16} - \frac{11493}{4602368} a^{14} - \frac{696493}{9204736} a^{12} - \frac{24635}{287648} a^{10} + \frac{396045}{9204736} a^{8} + \frac{2249751}{4602368} a^{6} + \frac{73057}{287648} a^{4} + \frac{52627}{143824} a^{2} - \frac{23713}{71912}$, $\frac{1}{2494483456} a^{29} - \frac{4505}{1247241728} a^{27} + \frac{3947467}{2494483456} a^{25} - \frac{1160817}{311810432} a^{23} - \frac{28983283}{2494483456} a^{21} - \frac{17226293}{1247241728} a^{19} + \frac{29006517}{2494483456} a^{17} + \frac{15713125}{311810432} a^{15} + \frac{115900627}{2494483456} a^{13} - \frac{28471969}{1247241728} a^{11} + \frac{161814633}{2494483456} a^{9} + \frac{47344153}{311810432} a^{7} - \frac{11169061}{38976304} a^{5} + \frac{15000889}{38976304} a^{3} - \frac{2290625}{4872038} a$, $\frac{1}{1268935776296825001311684613493021811227761664} a^{30} - \frac{1329254800497669580601372934695619651}{79308486018551562581980288343313863201735104} a^{28} - \frac{10521198485574150513917766077513555282623}{1268935776296825001311684613493021811227761664} a^{26} - \frac{497756709633233045056390794959146113848711}{634467888148412500655842306746510905613880832} a^{24} + \frac{3979153693441918705861956076150742105195675}{1268935776296825001311684613493021811227761664} a^{22} - \frac{529671449994818427747489247850132661248963}{79308486018551562581980288343313863201735104} a^{20} + \frac{3274348873113634926114806427458003223804619}{1268935776296825001311684613493021811227761664} a^{18} + \frac{16396232208382676461564994514714248610021871}{634467888148412500655842306746510905613880832} a^{16} - \frac{24455367643580935552400859228570357187856219}{1268935776296825001311684613493021811227761664} a^{14} + \frac{34395324719526773973192816022221168057992271}{317233944074206250327921153373255452806940416} a^{12} - \frac{89717587112091945901439094496207569981835661}{1268935776296825001311684613493021811227761664} a^{10} + \frac{77805814365025568585243311767201384269798891}{634467888148412500655842306746510905613880832} a^{8} + \frac{207680520726360960103780548692328491163452945}{634467888148412500655842306746510905613880832} a^{6} + \frac{15026849093874532785931746614327732824510393}{39654243009275781290990144171656931600867552} a^{4} - \frac{566276880754664164392130526466485783416355}{2478390188079736330686884010728558225054222} a^{2} + \frac{1145525733883462108069147550482818134735}{36581404990106809309031498313336652768328}$, $\frac{1}{1268935776296825001311684613493021811227761664} a^{31} + \frac{48594602919341367894287750881890441}{634467888148412500655842306746510905613880832} a^{29} + \frac{694393349523961895852688089314411103073}{158616972037103125163960576686627726403470208} a^{27} + \frac{1121685688893645761220913177374914879140017}{634467888148412500655842306746510905613880832} a^{25} + \frac{675003931473109744558704495114977099626671}{317233944074206250327921153373255452806940416} a^{23} - \frac{9037692607008522903017027358236169717026087}{634467888148412500655842306746510905613880832} a^{21} + \frac{81723807573728026154254762671992305743571}{634467888148412500655842306746510905613880832} a^{19} - \frac{504949689654563257586715141066486452300559}{634467888148412500655842306746510905613880832} a^{17} + \frac{14177606214575111362383837096770172748460167}{317233944074206250327921153373255452806940416} a^{15} - \frac{14386458014037279724270520251626064263416147}{634467888148412500655842306746510905613880832} a^{13} + \frac{12444692122327090505119778154920467917472909}{158616972037103125163960576686627726403470208} a^{11} - \frac{62846164587830075387295370832886225487907175}{634467888148412500655842306746510905613880832} a^{9} + \frac{14363444190579853959756285281230108215526101}{1268935776296825001311684613493021811227761664} a^{7} + \frac{18634942744731185293599389601975059806607611}{79308486018551562581980288343313863201735104} a^{5} + \frac{108879466527114310351264118822693336907601}{222776646119526861185337888604814222476784} a^{3} + \frac{2614298523474963871778905648284958480127173}{19827121504637890645495072085828465800433776} a$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{584}$, which has order $18688$ (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:  \( -\frac{123117231451970155938895}{2678988296768260150913075097088} a^{31} + \frac{3013765236327288742224165}{2678988296768260150913075097088} a^{29} - \frac{17643082509177063029364505}{2678988296768260150913075097088} a^{27} - \frac{74470220171768369341026887}{2678988296768260150913075097088} a^{25} + \frac{878853544841045216879648875}{2678988296768260150913075097088} a^{23} - \frac{869340607843095393521312535}{2678988296768260150913075097088} a^{21} - \frac{9630437621743422773600958575}{2678988296768260150913075097088} a^{19} + \frac{24964986897666436817305443975}{2678988296768260150913075097088} a^{17} - \frac{90437044346675198796667343299}{2678988296768260150913075097088} a^{15} + \frac{953946445946078058422510195995}{2678988296768260150913075097088} a^{13} - \frac{4126389398017220820862564487935}{2678988296768260150913075097088} a^{11} + \frac{7747856241638992822657826157755}{2678988296768260150913075097088} a^{9} - \frac{2684638140625113230047366258295}{1339494148384130075456537548544} a^{7} + \frac{54007511486541511242685025739}{41859192137004064858016798392} a^{5} - \frac{67141767460283445036844171225}{20929596068502032429008399196} a^{3} + \frac{34640929156817233429116347745}{20929596068502032429008399196} a \) (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:  \( 218851674878883.47 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{-85}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-17}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(i, \sqrt{85})\), \(\Q(i, \sqrt{17})\), \(\Q(i, \sqrt{5})\), \(\Q(\sqrt{5}, \sqrt{17})\), \(\Q(\sqrt{-5}, \sqrt{-17})\), \(\Q(\sqrt{-5}, \sqrt{17})\), \(\Q(\sqrt{5}, \sqrt{-17})\), 4.4.122825.1, 4.0.1965200.1, 4.4.4913.1, 4.0.78608.1, 8.0.13363360000.1, 8.0.3862011040000.3, 8.0.6179217664.1, 8.8.15085980625.1, 8.0.3862011040000.4, 8.0.3862011040000.1, 8.0.3862011040000.2, 8.0.6411541765625.1, 8.8.1641354692000000.1, 8.0.6411541765625.2, 8.8.1641354692000000.2, 16.0.14915129273081881600000000.1, 16.0.2694045224950414864000000000000.2, 16.0.2694045224950414864000000000000.3, 16.0.41107867812353742431640625.1, 16.16.2694045224950414864000000000000.1, 16.0.2694045224950414864000000000000.1, 16.0.2694045224950414864000000000000.4

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ 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 ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
5Data not computed
17Data not computed