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Properties

Label 32.0.11544095681...0976.2
Degree $32$
Signature $[0, 16]$
Discriminant $2^{124}\cdot 13^{24}$
Root discriminant $100.45$
Ramified primes $2, 13$
Class number $439825$ (GRH)
Class group $[5, 87965]$ (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![58281889, -55119136, 89896776, -163307832, 350995068, -53577424, -285976060, 103143400, 173545612, -101764328, 52305080, -17522856, -15688322, 18098136, -5967804, 2415456, 2202572, -1263952, 1069852, -106160, -108044, 106392, -26884, 12896, -351, -5200, 2948, 32, -426, 104, 12, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 8*x^31 + 12*x^30 + 104*x^29 - 426*x^28 + 32*x^27 + 2948*x^26 - 5200*x^25 - 351*x^24 + 12896*x^23 - 26884*x^22 + 106392*x^21 - 108044*x^20 - 106160*x^19 + 1069852*x^18 - 1263952*x^17 + 2202572*x^16 + 2415456*x^15 - 5967804*x^14 + 18098136*x^13 - 15688322*x^12 - 17522856*x^11 + 52305080*x^10 - 101764328*x^9 + 173545612*x^8 + 103143400*x^7 - 285976060*x^6 - 53577424*x^5 + 350995068*x^4 - 163307832*x^3 + 89896776*x^2 - 55119136*x + 58281889)
 
gp: K = bnfinit(x^32 - 8*x^31 + 12*x^30 + 104*x^29 - 426*x^28 + 32*x^27 + 2948*x^26 - 5200*x^25 - 351*x^24 + 12896*x^23 - 26884*x^22 + 106392*x^21 - 108044*x^20 - 106160*x^19 + 1069852*x^18 - 1263952*x^17 + 2202572*x^16 + 2415456*x^15 - 5967804*x^14 + 18098136*x^13 - 15688322*x^12 - 17522856*x^11 + 52305080*x^10 - 101764328*x^9 + 173545612*x^8 + 103143400*x^7 - 285976060*x^6 - 53577424*x^5 + 350995068*x^4 - 163307832*x^3 + 89896776*x^2 - 55119136*x + 58281889, 1)
 

Normalized defining polynomial

\( x^{32} - 8 x^{31} + 12 x^{30} + 104 x^{29} - 426 x^{28} + 32 x^{27} + 2948 x^{26} - 5200 x^{25} - 351 x^{24} + 12896 x^{23} - 26884 x^{22} + 106392 x^{21} - 108044 x^{20} - 106160 x^{19} + 1069852 x^{18} - 1263952 x^{17} + 2202572 x^{16} + 2415456 x^{15} - 5967804 x^{14} + 18098136 x^{13} - 15688322 x^{12} - 17522856 x^{11} + 52305080 x^{10} - 101764328 x^{9} + 173545612 x^{8} + 103143400 x^{7} - 285976060 x^{6} - 53577424 x^{5} + 350995068 x^{4} - 163307832 x^{3} + 89896776 x^{2} - 55119136 x + 58281889 \)

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:  \(11544095681843725705116280971624020302917345750478466341169790976=2^{124}\cdot 13^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $100.45$
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:  $2, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(416=2^{5}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{416}(1,·)$, $\chi_{416}(261,·)$, $\chi_{416}(129,·)$, $\chi_{416}(265,·)$, $\chi_{416}(109,·)$, $\chi_{416}(21,·)$, $\chi_{416}(281,·)$, $\chi_{416}(25,·)$, $\chi_{416}(285,·)$, $\chi_{416}(5,·)$, $\chi_{416}(161,·)$, $\chi_{416}(389,·)$, $\chi_{416}(157,·)$, $\chi_{416}(385,·)$, $\chi_{416}(177,·)$, $\chi_{416}(181,·)$, $\chi_{416}(57,·)$, $\chi_{416}(317,·)$, $\chi_{416}(53,·)$, $\chi_{416}(73,·)$, $\chi_{416}(77,·)$, $\chi_{416}(333,·)$, $\chi_{416}(209,·)$, $\chi_{416}(213,·)$, $\chi_{416}(313,·)$, $\chi_{416}(229,·)$, $\chi_{416}(337,·)$, $\chi_{416}(105,·)$, $\chi_{416}(365,·)$, $\chi_{416}(369,·)$, $\chi_{416}(233,·)$, $\chi_{416}(125,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{3} a^{24} + \frac{1}{3} a^{22} + \frac{1}{3} a^{20} + \frac{1}{3} a^{16} + \frac{1}{3} a^{14} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3}$, $\frac{1}{3} a^{25} + \frac{1}{3} a^{23} + \frac{1}{3} a^{21} + \frac{1}{3} a^{17} + \frac{1}{3} a^{15} + \frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a$, $\frac{1}{3} a^{26} - \frac{1}{3} a^{20} + \frac{1}{3} a^{18} - \frac{1}{3} a^{14} + \frac{1}{3} a^{12} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{27} - \frac{1}{3} a^{21} + \frac{1}{3} a^{19} - \frac{1}{3} a^{15} + \frac{1}{3} a^{13} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{357} a^{28} - \frac{1}{51} a^{27} - \frac{13}{119} a^{26} + \frac{41}{357} a^{25} - \frac{46}{357} a^{24} - \frac{13}{51} a^{23} + \frac{67}{357} a^{22} - \frac{3}{7} a^{21} - \frac{59}{119} a^{20} - \frac{22}{51} a^{19} + \frac{53}{119} a^{18} + \frac{4}{21} a^{17} + \frac{160}{357} a^{16} + \frac{41}{119} a^{15} + \frac{19}{119} a^{14} - \frac{82}{357} a^{13} - \frac{2}{119} a^{12} - \frac{25}{357} a^{11} - \frac{23}{119} a^{10} - \frac{137}{357} a^{9} + \frac{3}{119} a^{8} - \frac{45}{119} a^{7} - \frac{95}{357} a^{6} + \frac{16}{119} a^{5} + \frac{89}{357} a^{4} + \frac{109}{357} a^{3} - \frac{59}{357} a^{2} + \frac{24}{119} a - \frac{32}{357}$, $\frac{1}{357} a^{29} + \frac{31}{357} a^{27} + \frac{2}{119} a^{26} + \frac{1}{119} a^{25} - \frac{8}{51} a^{24} - \frac{94}{357} a^{23} - \frac{41}{357} a^{22} - \frac{59}{119} a^{21} + \frac{22}{51} a^{20} - \frac{86}{357} a^{19} - \frac{3}{119} a^{18} + \frac{41}{357} a^{17} + \frac{172}{357} a^{16} - \frac{3}{7} a^{15} + \frac{79}{357} a^{14} - \frac{104}{357} a^{13} + \frac{57}{119} a^{12} - \frac{125}{357} a^{11} + \frac{94}{357} a^{10} + \frac{2}{357} a^{9} + \frac{166}{357} a^{8} + \frac{31}{357} a^{7} - \frac{22}{357} a^{6} + \frac{4}{21} a^{5} + \frac{6}{119} a^{4} - \frac{43}{119} a^{3} + \frac{45}{119} a^{2} + \frac{115}{357} a + \frac{2}{51}$, $\frac{1}{37578803258608004728723934168156054555091506526727033526579715621} a^{30} - \frac{757723004376474866220695591440502613696772451553369035712434}{1789466821838476415653520674674097835956738406034620644122843601} a^{29} - \frac{1561712528556350170255650715271743152620062392091290398333024}{37578803258608004728723934168156054555091506526727033526579715621} a^{28} + \frac{1693913928864992943533303087792554774351815291691732301252214171}{12526267752869334909574644722718684851697168842242344508859905207} a^{27} + \frac{174807551669836204558600587850125617574030802336250251115482720}{37578803258608004728723934168156054555091506526727033526579715621} a^{26} + \frac{9134511545740372511055627738182571440294246312829515240508663}{736839279580549112327920277806981461864539343661314382874112071} a^{25} + \frac{2008298564404114598645890109668109906823452936255754882114188851}{12526267752869334909574644722718684851697168842242344508859905207} a^{24} - \frac{5318713502234776943922899083936989527350596030148396066599688869}{12526267752869334909574644722718684851697168842242344508859905207} a^{23} - \frac{2977887437732975481179861765162967028751742786651409254299783813}{37578803258608004728723934168156054555091506526727033526579715621} a^{22} + \frac{5352083626539136167181405915953638729325256910721327721774937676}{12526267752869334909574644722718684851697168842242344508859905207} a^{21} + \frac{826165645665056936277923145988425075300469576440477403021250399}{2210517838741647336983760833420944385593618030983943148622336213} a^{20} - \frac{3308702170590971349337707662672766864116217640478503413513472792}{12526267752869334909574644722718684851697168842242344508859905207} a^{19} - \frac{21338862500220293712008829462197547695029701017478242845486899}{105262754225792730332560039686711637409219906237330626124873153} a^{18} - \frac{5795265160163694579746213133451156555788153033577218074228108567}{12526267752869334909574644722718684851697168842242344508859905207} a^{17} + \frac{12634987571347922872447379838261654714507892909865242890114103741}{37578803258608004728723934168156054555091506526727033526579715621} a^{16} + \frac{114174208341644291786250747461596574644219385742999314597752645}{1789466821838476415653520674674097835956738406034620644122843601} a^{15} - \frac{6153375554773562556975854584676090078684827169390046652678584761}{37578803258608004728723934168156054555091506526727033526579715621} a^{14} - \frac{2401606071783486234214029477105945953403664740160115759585844335}{12526267752869334909574644722718684851697168842242344508859905207} a^{13} - \frac{5322874174662382201204018930056270540624553325788302380312698219}{37578803258608004728723934168156054555091506526727033526579715621} a^{12} - \frac{3020193667788369348779482134288894360128709550172349172256016711}{12526267752869334909574644722718684851697168842242344508859905207} a^{11} - \frac{4420058464139963342182855545761436953246892546890620646050379260}{12526267752869334909574644722718684851697168842242344508859905207} a^{10} + \frac{604903798305237070919995272416630250049273064333070892691359151}{1789466821838476415653520674674097835956738406034620644122843601} a^{9} + \frac{3859354142085398259793066407472479164865557254714547722937180096}{12526267752869334909574644722718684851697168842242344508859905207} a^{8} + \frac{759818790304212644787094567236132576419086213516796996823135643}{1789466821838476415653520674674097835956738406034620644122843601} a^{7} + \frac{1257921380680626670262508667718568597639732815110776682141133201}{5368400465515429246960562024022293507870215218103861932368530803} a^{6} + \frac{470103817910499867263854299576396618411269284798689655434990333}{12526267752869334909574644722718684851697168842242344508859905207} a^{5} + \frac{5332900997941148843064730228892893505199609752673761561713784302}{12526267752869334909574644722718684851697168842242344508859905207} a^{4} - \frac{208081873684948933021693067262864489769402433015688426034511345}{1789466821838476415653520674674097835956738406034620644122843601} a^{3} - \frac{316421027666747303427019033517798790917024153343806298503266463}{736839279580549112327920277806981461864539343661314382874112071} a^{2} - \frac{6174898955836541565342216363450774361994802622872145080214740728}{12526267752869334909574644722718684851697168842242344508859905207} a - \frac{5782631890291195564372565358794974576855509802014612467661751959}{12526267752869334909574644722718684851697168842242344508859905207}$, $\frac{1}{53556002571030569809283654218149845838587464518619137717419653956165932065180502517} a^{31} + \frac{156337867636899456}{17852000857010189936427884739383281946195821506206379239139884652055310688393500839} a^{30} - \frac{26290394252672974004645754047699466581290068130078955963742716468107523751695641}{53556002571030569809283654218149845838587464518619137717419653956165932065180502517} a^{29} - \frac{17198115032636174799668099565512574905233075408940819506517927062541605714805798}{53556002571030569809283654218149845838587464518619137717419653956165932065180502517} a^{28} + \frac{1000156253198949851670557676401786257374292961599078517316486471528021750346289207}{53556002571030569809283654218149845838587464518619137717419653956165932065180502517} a^{27} + \frac{2537137615000604163191696127509432290212092065758333860250332881481261434518110871}{17852000857010189936427884739383281946195821506206379239139884652055310688393500839} a^{26} + \frac{117579272479555587711951467282332702573597761645929932181563324477791089666335841}{2550285836715741419489697819911897420885117358029482748448554950293615812627642977} a^{25} + \frac{2991064496933374957655014866684901851300185703337417791399281913056157024682410734}{53556002571030569809283654218149845838587464518619137717419653956165932065180502517} a^{24} + \frac{4406302671359823276310505952059902201863762296151428242092733847254151599879624064}{17852000857010189936427884739383281946195821506206379239139884652055310688393500839} a^{23} - \frac{597989903742512372068490043154245282667157841589699164717269191674217681724545856}{7650857510147224258469093459735692262655352074088448245345664850880847437882928931} a^{22} + \frac{2431647862617442467384810879682427733208931708525680515607779288646602887186431933}{53556002571030569809283654218149845838587464518619137717419653956165932065180502517} a^{21} - \frac{302519107684656085805869251644519891553969785970596882527063038829399336864986503}{2550285836715741419489697819911897420885117358029482748448554950293615812627642977} a^{20} + \frac{9577443591272129542076074673443618001927830946145970269337252793293829256283785328}{53556002571030569809283654218149845838587464518619137717419653956165932065180502517} a^{19} + \frac{2850026061839049293716542934427038718730231387271307940908027742893250861264846822}{17852000857010189936427884739383281946195821506206379239139884652055310688393500839} a^{18} + \frac{113800175564420145223436022195662894481108576626291997286967383886935151522639719}{1050117697471187643319287337610781290952695382718022308184699097179724158140794167} a^{17} + \frac{7911196297233754675592143593984628483959538916627425243651054687522219035689212105}{53556002571030569809283654218149845838587464518619137717419653956165932065180502517} a^{16} - \frac{965860769978701289594436552598874438061929745849984516520978485619323904085545424}{7650857510147224258469093459735692262655352074088448245345664850880847437882928931} a^{15} + \frac{1152554924628301600323479267540956012461994754529298771306547813950915734474242291}{2550285836715741419489697819911897420885117358029482748448554950293615812627642977} a^{14} - \frac{7462406716582897848944186262525455876694292428256067317272591869593315792586105087}{17852000857010189936427884739383281946195821506206379239139884652055310688393500839} a^{13} - \frac{8645568846175024098825234901415066806668979518717045959384422127250268236227847727}{17852000857010189936427884739383281946195821506206379239139884652055310688393500839} a^{12} + \frac{10799422367904055831422644105337373819974309445239273592087236058580618971810466089}{53556002571030569809283654218149845838587464518619137717419653956165932065180502517} a^{11} - \frac{693209542019150314146773570278771841348600330916976938707379958018564619082985568}{17852000857010189936427884739383281946195821506206379239139884652055310688393500839} a^{10} - \frac{2762948953693180019354146264504714651792463306639553102443456236518695210092880719}{53556002571030569809283654218149845838587464518619137717419653956165932065180502517} a^{9} + \frac{697017917733227574940856402902172245837202878601947557833010844665015387887582527}{17852000857010189936427884739383281946195821506206379239139884652055310688393500839} a^{8} + \frac{7786919043585506193048441796059130230341896172680261866033635672675875981410657647}{17852000857010189936427884739383281946195821506206379239139884652055310688393500839} a^{7} + \frac{19548088791445948072527140120860452767889826017456246902515752230444955914137523719}{53556002571030569809283654218149845838587464518619137717419653956165932065180502517} a^{6} - \frac{25140419044326352064244831780957679538436935074900533997598699302496410498526004208}{53556002571030569809283654218149845838587464518619137717419653956165932065180502517} a^{5} - \frac{14338866861862493954770461945672590594989975170726099762183373506194624403506477326}{53556002571030569809283654218149845838587464518619137717419653956165932065180502517} a^{4} - \frac{1544166758900578281269463438556414157352340759363480117571434034165272522385120591}{17852000857010189936427884739383281946195821506206379239139884652055310688393500839} a^{3} - \frac{7001969812082283211405782582994008787194264206293650185400371278173051191517140261}{53556002571030569809283654218149845838587464518619137717419653956165932065180502517} a^{2} + \frac{22157303925352862005201929188053429926517306039475351779913364432450003641089685077}{53556002571030569809283654218149845838587464518619137717419653956165932065180502517} a - \frac{23340293931980430921101122115842799994860867967095987424371305018233691963782998945}{53556002571030569809283654218149845838587464518619137717419653956165932065180502517}$

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

Class group and class number

$C_{5}\times C_{87965}$, which has order $439825$ (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:  \( 9664872226004.74 \) (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

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{26}) \), 4.0.140608.2, \(\Q(\sqrt{2}, \sqrt{13})\), 4.0.2197.1, 4.4.346112.1, \(\Q(\zeta_{16})^+\), 4.0.4499456.1, 4.0.4499456.2, 8.0.19770609664.2, 8.8.119793516544.1, 8.0.20245104295936.1, 8.0.10365493399519232.3, 8.0.10365493399519232.5, \(\Q(\zeta_{32})^+\), 8.8.61334280470528.1, 16.0.409864247953326282266116096.2, 16.0.107443453415476764938368737869824.1, 16.16.3761893960837392421076598784.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
13Data not computed