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Properties

Label 32.0.54568201713...0000.4
Degree $32$
Signature $[0, 16]$
Discriminant $2^{124}\cdot 3^{16}\cdot 5^{24}$
Root discriminant $84.97$
Ramified primes $2, 3, 5$
Class number $296548$ (GRH)
Class group $[14, 21182]$ (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![295089121, -206749664, 970581784, -2995482536, 4642269708, -5279972480, 4930199252, -3467919192, 1611662172, -339377432, -141389160, 176356600, -76009730, -3197336, 22324164, -10742192, 717500, 1259168, -382084, -227840, 294260, -153000, 37260, -2080, 3985, -5296, 2516, -512, 54, -40, 28, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 8*x^31 + 28*x^30 - 40*x^29 + 54*x^28 - 512*x^27 + 2516*x^26 - 5296*x^25 + 3985*x^24 - 2080*x^23 + 37260*x^22 - 153000*x^21 + 294260*x^20 - 227840*x^19 - 382084*x^18 + 1259168*x^17 + 717500*x^16 - 10742192*x^15 + 22324164*x^14 - 3197336*x^13 - 76009730*x^12 + 176356600*x^11 - 141389160*x^10 - 339377432*x^9 + 1611662172*x^8 - 3467919192*x^7 + 4930199252*x^6 - 5279972480*x^5 + 4642269708*x^4 - 2995482536*x^3 + 970581784*x^2 - 206749664*x + 295089121)
 
gp: K = bnfinit(x^32 - 8*x^31 + 28*x^30 - 40*x^29 + 54*x^28 - 512*x^27 + 2516*x^26 - 5296*x^25 + 3985*x^24 - 2080*x^23 + 37260*x^22 - 153000*x^21 + 294260*x^20 - 227840*x^19 - 382084*x^18 + 1259168*x^17 + 717500*x^16 - 10742192*x^15 + 22324164*x^14 - 3197336*x^13 - 76009730*x^12 + 176356600*x^11 - 141389160*x^10 - 339377432*x^9 + 1611662172*x^8 - 3467919192*x^7 + 4930199252*x^6 - 5279972480*x^5 + 4642269708*x^4 - 2995482536*x^3 + 970581784*x^2 - 206749664*x + 295089121, 1)
 

Normalized defining polynomial

\( x^{32} - 8 x^{31} + 28 x^{30} - 40 x^{29} + 54 x^{28} - 512 x^{27} + 2516 x^{26} - 5296 x^{25} + 3985 x^{24} - 2080 x^{23} + 37260 x^{22} - 153000 x^{21} + 294260 x^{20} - 227840 x^{19} - 382084 x^{18} + 1259168 x^{17} + 717500 x^{16} - 10742192 x^{15} + 22324164 x^{14} - 3197336 x^{13} - 76009730 x^{12} + 176356600 x^{11} - 141389160 x^{10} - 339377432 x^{9} + 1611662172 x^{8} - 3467919192 x^{7} + 4930199252 x^{6} - 5279972480 x^{5} + 4642269708 x^{4} - 2995482536 x^{3} + 970581784 x^{2} - 206749664 x + 295089121 \)

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:  \(54568201713507127370225565301626372096000000000000000000000000=2^{124}\cdot 3^{16}\cdot 5^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $84.97$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(480=2^{5}\cdot 3\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{480}(1,·)$, $\chi_{480}(259,·)$, $\chi_{480}(257,·)$, $\chi_{480}(137,·)$, $\chi_{480}(139,·)$, $\chi_{480}(17,·)$, $\chi_{480}(19,·)$, $\chi_{480}(409,·)$, $\chi_{480}(289,·)$, $\chi_{480}(347,·)$, $\chi_{480}(113,·)$, $\chi_{480}(169,·)$, $\chi_{480}(49,·)$, $\chi_{480}(443,·)$, $\chi_{480}(451,·)$, $\chi_{480}(211,·)$, $\chi_{480}(331,·)$, $\chi_{480}(467,·)$, $\chi_{480}(203,·)$, $\chi_{480}(377,·)$, $\chi_{480}(473,·)$, $\chi_{480}(91,·)$, $\chi_{480}(353,·)$, $\chi_{480}(227,·)$, $\chi_{480}(361,·)$, $\chi_{480}(323,·)$, $\chi_{480}(107,·)$, $\chi_{480}(241,·)$, $\chi_{480}(83,·)$, $\chi_{480}(233,·)$, $\chi_{480}(121,·)$, $\chi_{480}(379,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{7} a^{28} + \frac{3}{7} a^{26} + \frac{3}{7} a^{25} - \frac{2}{7} a^{24} - \frac{3}{7} a^{22} + \frac{3}{7} a^{21} - \frac{1}{7} a^{20} - \frac{1}{7} a^{19} - \frac{3}{7} a^{18} + \frac{1}{7} a^{17} + \frac{3}{7} a^{15} - \frac{1}{7} a^{14} - \frac{3}{7} a^{13} + \frac{2}{7} a^{12} + \frac{2}{7} a^{11} + \frac{1}{7} a^{9} + \frac{2}{7} a^{8} + \frac{2}{7} a^{7} + \frac{3}{7} a^{6} - \frac{2}{7} a^{5} + \frac{1}{7} a^{4} - \frac{3}{7} a^{3} + \frac{2}{7} a^{2} + \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{29} + \frac{3}{7} a^{27} + \frac{3}{7} a^{26} - \frac{2}{7} a^{25} - \frac{3}{7} a^{23} + \frac{3}{7} a^{22} - \frac{1}{7} a^{21} - \frac{1}{7} a^{20} - \frac{3}{7} a^{19} + \frac{1}{7} a^{18} + \frac{3}{7} a^{16} - \frac{1}{7} a^{15} - \frac{3}{7} a^{14} + \frac{2}{7} a^{13} + \frac{2}{7} a^{12} + \frac{1}{7} a^{10} + \frac{2}{7} a^{9} + \frac{2}{7} a^{8} + \frac{3}{7} a^{7} - \frac{2}{7} a^{6} + \frac{1}{7} a^{5} - \frac{3}{7} a^{4} + \frac{2}{7} a^{3} + \frac{2}{7} a^{2} + \frac{1}{7} a$, $\frac{1}{17637500939412847579815845212005923874063545314488527855064487} a^{30} - \frac{131734504668720129791777176227937510840124052279857659355415}{17637500939412847579815845212005923874063545314488527855064487} a^{29} + \frac{298561679461061559008560329039854552083783271829385972831487}{17637500939412847579815845212005923874063545314488527855064487} a^{28} - \frac{7582909634164899136570096368539895680614677033800292681076}{17637500939412847579815845212005923874063545314488527855064487} a^{27} - \frac{1248312509422604620206299870908750281978846890779587937533772}{17637500939412847579815845212005923874063545314488527855064487} a^{26} - \frac{6322339860492614054264809983964317487830116124588510009147727}{17637500939412847579815845212005923874063545314488527855064487} a^{25} - \frac{3301176333627832002147991087380446225488630316200204723151819}{17637500939412847579815845212005923874063545314488527855064487} a^{24} + \frac{389633814447635006605624255073060340804586026049590901478218}{2519642991344692511402263601715131982009077902069789693580641} a^{23} - \frac{2187606625353932457994884058630342461216584648422903886704228}{17637500939412847579815845212005923874063545314488527855064487} a^{22} + \frac{5109341251207540970255848678352300112466520416398647792698965}{17637500939412847579815845212005923874063545314488527855064487} a^{21} + \frac{6287561439180605690432373359376590904689840861357760135977559}{17637500939412847579815845212005923874063545314488527855064487} a^{20} + \frac{3713104924853527773091538404787602285658570380363437696223343}{17637500939412847579815845212005923874063545314488527855064487} a^{19} + \frac{282726507384432061755271680176952638384777167673543707651209}{17637500939412847579815845212005923874063545314488527855064487} a^{18} - \frac{1582918243395551449856699773516593359187682642069868624120596}{17637500939412847579815845212005923874063545314488527855064487} a^{17} - \frac{3611656334241818839366053304938707721175848216279830939596579}{17637500939412847579815845212005923874063545314488527855064487} a^{16} + \frac{1043464003762002935117523085160681729206581445627633131451559}{2519642991344692511402263601715131982009077902069789693580641} a^{15} + \frac{745678794488152006935580865229612215979693718241248986427867}{2519642991344692511402263601715131982009077902069789693580641} a^{14} - \frac{682226635719470962335624244876076335314867242808680487529823}{2519642991344692511402263601715131982009077902069789693580641} a^{13} - \frac{903367332651631228534455975263747025387139858711865660325647}{2519642991344692511402263601715131982009077902069789693580641} a^{12} + \frac{434530831372635491462664860210363905600706549167697616069607}{17637500939412847579815845212005923874063545314488527855064487} a^{11} + \frac{4776876194454908736403218486582347676054253708860211690216196}{17637500939412847579815845212005923874063545314488527855064487} a^{10} - \frac{31286622726813332452037005806684801311120599580937189114798}{92342936855564647014742645089036250649547357667479203429657} a^{9} + \frac{4430350674397312062605006235388782184996230423006984103009146}{17637500939412847579815845212005923874063545314488527855064487} a^{8} - \frac{3037498088242422726331070370640782718779486214134371654761258}{17637500939412847579815845212005923874063545314488527855064487} a^{7} - \frac{4185050590297097461456043976945442137709248911528279674898361}{17637500939412847579815845212005923874063545314488527855064487} a^{6} - \frac{81279987399599315117568942488738575098987157029413943419018}{2519642991344692511402263601715131982009077902069789693580641} a^{5} + \frac{1250219666810059973154274890229005910857420510993592873394745}{17637500939412847579815845212005923874063545314488527855064487} a^{4} + \frac{1028040487168065246156965634320142903716441112272052454070190}{2519642991344692511402263601715131982009077902069789693580641} a^{3} + \frac{8550381614369898922281691583073656391504249216434474250915496}{17637500939412847579815845212005923874063545314488527855064487} a^{2} + \frac{2310647271824226410834048527694499914004836262135001895420825}{17637500939412847579815845212005923874063545314488527855064487} a - \frac{8145067769569923450670819503274536448600993735831577634741113}{17637500939412847579815845212005923874063545314488527855064487}$, $\frac{1}{158887956301850277393729782933253675292652782000856060996856533954585406477881705015863} a^{31} + \frac{1406151162773859022505723}{158887956301850277393729782933253675292652782000856060996856533954585406477881705015863} a^{30} - \frac{1703169737037060523093221291019764398866782442373138009548406268876800225348485742769}{158887956301850277393729782933253675292652782000856060996856533954585406477881705015863} a^{29} + \frac{6684274786182188460666316467414936368073178817501699378274009862606456968854837561249}{158887956301850277393729782933253675292652782000856060996856533954585406477881705015863} a^{28} + \frac{139420959659674630993568592604114297832393563151107913482600657827000660929016773799}{22698279471692896770532826133321953613236111714408008713836647707797915211125957859409} a^{27} - \frac{60083572702588942398347612484066288281654225388257738496532712059815391119755260287772}{158887956301850277393729782933253675292652782000856060996856533954585406477881705015863} a^{26} - \frac{1060124080018247745771573600804261719259145549763432773087686446948946942456987682048}{22698279471692896770532826133321953613236111714408008713836647707797915211125957859409} a^{25} - \frac{19397588216709950844652925455651804720107731507605964785345386198203277315126794716831}{158887956301850277393729782933253675292652782000856060996856533954585406477881705015863} a^{24} - \frac{10004655099637748404452094263260352572727880399969386296771817271684278085778337022750}{22698279471692896770532826133321953613236111714408008713836647707797915211125957859409} a^{23} + \frac{74141828699290743545540173717657828412736518815144430983246901118791033114302561057280}{158887956301850277393729782933253675292652782000856060996856533954585406477881705015863} a^{22} + \frac{75917529431436231102603891800747559857970005898864446016855846629525674501437578537659}{158887956301850277393729782933253675292652782000856060996856533954585406477881705015863} a^{21} + \frac{18280270169744932493048848814641721731457074552196820208554995025014607085451691287099}{158887956301850277393729782933253675292652782000856060996856533954585406477881705015863} a^{20} + \frac{2715745182006999301798682810150299402564297434378607821739244948027124438819797531271}{158887956301850277393729782933253675292652782000856060996856533954585406477881705015863} a^{19} - \frac{78710028673131571094785353887988178093085369202852414744148129050675848329805265444852}{158887956301850277393729782933253675292652782000856060996856533954585406477881705015863} a^{18} - \frac{43011960444816751714899602005237871121593161498209666408262155267554654961599401758563}{158887956301850277393729782933253675292652782000856060996856533954585406477881705015863} a^{17} + \frac{36176477422693553706870928965485366619380533948212307377928006169203783595708288089971}{158887956301850277393729782933253675292652782000856060996856533954585406477881705015863} a^{16} + \frac{68603110482427533505925918092300718620560611586444679197567290815227004602002207790751}{158887956301850277393729782933253675292652782000856060996856533954585406477881705015863} a^{15} + \frac{42963287134851363006281169034586722678316371297492010405951588183615622249179452446068}{158887956301850277393729782933253675292652782000856060996856533954585406477881705015863} a^{14} - \frac{50911702291895141369145908187595124713949012737595447593545893909377126609143416498101}{158887956301850277393729782933253675292652782000856060996856533954585406477881705015863} a^{13} - \frac{75324895154966725643330406793803612812310402534816254323751364973634472890094164447398}{158887956301850277393729782933253675292652782000856060996856533954585406477881705015863} a^{12} + \frac{18764414648441070031792260537872133110307939116030796863955482039494953184092861045593}{158887956301850277393729782933253675292652782000856060996856533954585406477881705015863} a^{11} + \frac{44423553947417811100381316822931432616742373225490318646432449633175373176377428873194}{158887956301850277393729782933253675292652782000856060996856533954585406477881705015863} a^{10} + \frac{6291566644184766464249662875566152051802612179964580933732456863917806521028197543097}{22698279471692896770532826133321953613236111714408008713836647707797915211125957859409} a^{9} + \frac{39773432328313391421331740548717585764878822583960972117510478723618707170521707106774}{158887956301850277393729782933253675292652782000856060996856533954585406477881705015863} a^{8} + \frac{7110122194298809300536550981610053271792649852318806605690129068622939012700188377395}{22698279471692896770532826133321953613236111714408008713836647707797915211125957859409} a^{7} - \frac{6278697355232570423325630559590467119308055863469062112495969537370850262499072094475}{158887956301850277393729782933253675292652782000856060996856533954585406477881705015863} a^{6} - \frac{2622415796501924230433688289962569261211071441687995007909430050261230352828346090518}{22698279471692896770532826133321953613236111714408008713836647707797915211125957859409} a^{5} + \frac{9589817905107261234183032803963099597370958361831156468370198469135562037540370988736}{158887956301850277393729782933253675292652782000856060996856533954585406477881705015863} a^{4} - \frac{1551002355060195836022410404332312696466413616661701951514127054825345767705681985892}{158887956301850277393729782933253675292652782000856060996856533954585406477881705015863} a^{3} - \frac{10527987582098485246281867483939681687306326085847313345328929431543300684312432496689}{158887956301850277393729782933253675292652782000856060996856533954585406477881705015863} a^{2} - \frac{51742453115695828507888010594400967990393859070487034265131269659522138796239432614444}{158887956301850277393729782933253675292652782000856060996856533954585406477881705015863} a - \frac{63432532023077693071924489231323382574642557435008398844881044671672214480095654982504}{158887956301850277393729782933253675292652782000856060996856533954585406477881705015863}$

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

Class group and class number

$C_{14}\times C_{21182}$, which has order $296548$ (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:  \( -1 \) (order $2$)
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:  \( 1099105261005.4811 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.4.72000.1, \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{16})^+\), 4.4.51200.1, 4.4.2304000.1, 4.4.2304000.2, 8.8.5184000000.1, 8.8.2621440000.1, 8.8.5308416000000.1, 8.0.2147483648.1, 8.0.1342177280000.1, 8.0.2717908992000000.10, 8.0.2717908992000000.4, 16.16.28179280429056000000000000.2, 16.0.1801439850948198400000000.1, 16.0.7387029288794456064000000000000.3

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
2Data not computed
3Data not computed
5Data not computed