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Properties

Label 32.0.42127744383...0000.2
Degree $32$
Signature $[0, 16]$
Discriminant $2^{124}\cdot 5^{24}\cdot 7^{16}$
Root discriminant $129.80$
Ramified primes $2, 5, 7$
Class number Not computed
Class group Not computed
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![1117847129071, 858240692296, -954628846756, -1277065516216, 603312953478, 456373127360, 223883790332, -483652890552, 54615249427, 52356953808, 38583306500, -20749097200, -14042867340, 6386853784, 2360618384, -813999752, -478813865, 38552968, 120802336, -4950760, -22883300, 2536640, 2902260, -567280, -240585, 71064, 12196, -5712, -206, 280, -12, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 8*x^31 - 12*x^30 + 280*x^29 - 206*x^28 - 5712*x^27 + 12196*x^26 + 71064*x^25 - 240585*x^24 - 567280*x^23 + 2902260*x^22 + 2536640*x^21 - 22883300*x^20 - 4950760*x^19 + 120802336*x^18 + 38552968*x^17 - 478813865*x^16 - 813999752*x^15 + 2360618384*x^14 + 6386853784*x^13 - 14042867340*x^12 - 20749097200*x^11 + 38583306500*x^10 + 52356953808*x^9 + 54615249427*x^8 - 483652890552*x^7 + 223883790332*x^6 + 456373127360*x^5 + 603312953478*x^4 - 1277065516216*x^3 - 954628846756*x^2 + 858240692296*x + 1117847129071)
 
gp: K = bnfinit(x^32 - 8*x^31 - 12*x^30 + 280*x^29 - 206*x^28 - 5712*x^27 + 12196*x^26 + 71064*x^25 - 240585*x^24 - 567280*x^23 + 2902260*x^22 + 2536640*x^21 - 22883300*x^20 - 4950760*x^19 + 120802336*x^18 + 38552968*x^17 - 478813865*x^16 - 813999752*x^15 + 2360618384*x^14 + 6386853784*x^13 - 14042867340*x^12 - 20749097200*x^11 + 38583306500*x^10 + 52356953808*x^9 + 54615249427*x^8 - 483652890552*x^7 + 223883790332*x^6 + 456373127360*x^5 + 603312953478*x^4 - 1277065516216*x^3 - 954628846756*x^2 + 858240692296*x + 1117847129071, 1)
 

Normalized defining polynomial

\( x^{32} - 8 x^{31} - 12 x^{30} + 280 x^{29} - 206 x^{28} - 5712 x^{27} + 12196 x^{26} + 71064 x^{25} - 240585 x^{24} - 567280 x^{23} + 2902260 x^{22} + 2536640 x^{21} - 22883300 x^{20} - 4950760 x^{19} + 120802336 x^{18} + 38552968 x^{17} - 478813865 x^{16} - 813999752 x^{15} + 2360618384 x^{14} + 6386853784 x^{13} - 14042867340 x^{12} - 20749097200 x^{11} + 38583306500 x^{10} + 52356953808 x^{9} + 54615249427 x^{8} - 483652890552 x^{7} + 223883790332 x^{6} + 456373127360 x^{5} + 603312953478 x^{4} - 1277065516216 x^{3} - 954628846756 x^{2} + 858240692296 x + 1117847129071 \)

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:  \(42127744383897781264481528176959874165374976000000000000000000000000=2^{124}\cdot 5^{24}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $129.80$
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:  $2, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1120=2^{5}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{1120}(1,·)$, $\chi_{1120}(771,·)$, $\chi_{1120}(363,·)$, $\chi_{1120}(657,·)$, $\chi_{1120}(659,·)$, $\chi_{1120}(281,·)$, $\chi_{1120}(153,·)$, $\chi_{1120}(923,·)$, $\chi_{1120}(1051,·)$, $\chi_{1120}(561,·)$, $\chi_{1120}(169,·)$, $\chi_{1120}(939,·)$, $\chi_{1120}(27,·)$, $\chi_{1120}(433,·)$, $\chi_{1120}(307,·)$, $\chi_{1120}(841,·)$, $\chi_{1120}(867,·)$, $\chi_{1120}(449,·)$, $\chi_{1120}(97,·)$, $\chi_{1120}(713,·)$, $\chi_{1120}(587,·)$, $\chi_{1120}(83,·)$, $\chi_{1120}(729,·)$, $\chi_{1120}(993,·)$, $\chi_{1120}(99,·)$, $\chi_{1120}(491,·)$, $\chi_{1120}(1009,·)$, $\chi_{1120}(211,·)$, $\chi_{1120}(643,·)$, $\chi_{1120}(937,·)$, $\chi_{1120}(377,·)$, $\chi_{1120}(379,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $\frac{1}{31} a^{29} - \frac{1}{31} a^{28} - \frac{11}{31} a^{27} - \frac{8}{31} a^{26} + \frac{8}{31} a^{25} - \frac{15}{31} a^{24} + \frac{15}{31} a^{23} - \frac{15}{31} a^{22} - \frac{3}{31} a^{21} - \frac{4}{31} a^{20} - \frac{5}{31} a^{19} - \frac{13}{31} a^{18} + \frac{14}{31} a^{16} - \frac{5}{31} a^{15} - \frac{12}{31} a^{14} - \frac{7}{31} a^{13} - \frac{14}{31} a^{12} - \frac{8}{31} a^{11} - \frac{5}{31} a^{10} - \frac{5}{31} a^{9} - \frac{5}{31} a^{8} - \frac{11}{31} a^{7} - \frac{7}{31} a^{6} - \frac{3}{31} a^{5} - \frac{9}{31} a^{4} - \frac{10}{31} a^{3} - \frac{4}{31} a^{2} + \frac{5}{31} a + \frac{5}{31}$, $\frac{1}{7576337217254497052011232092161707258480016239563653448563550980181849229809} a^{30} - \frac{57560613629979796216998165234796704056038404618154425821032303647915936045}{7576337217254497052011232092161707258480016239563653448563550980181849229809} a^{29} + \frac{3452895108167667046007651888607882990250275237254357687405596345468825800425}{7576337217254497052011232092161707258480016239563653448563550980181849229809} a^{28} + \frac{3093197327066673532838668242359388805940369957684685459002679409051964445672}{7576337217254497052011232092161707258480016239563653448563550980181849229809} a^{27} + \frac{2759096388243550707978563186245427929105659731425869444563385092532659517436}{7576337217254497052011232092161707258480016239563653448563550980181849229809} a^{26} + \frac{273030323513022492207533467155438526956209485957871433731558460984105051820}{7576337217254497052011232092161707258480016239563653448563550980181849229809} a^{25} - \frac{1548416899930391117620124824996170277585724142904327521019045375245512625784}{7576337217254497052011232092161707258480016239563653448563550980181849229809} a^{24} - \frac{917402590705951514762029802799188717575932172743417552845499354370725390404}{7576337217254497052011232092161707258480016239563653448563550980181849229809} a^{23} + \frac{1916949042830248360751277599929460580788245478887435173020799702888304621872}{7576337217254497052011232092161707258480016239563653448563550980181849229809} a^{22} - \frac{1607585226792869954160749929366612449801033291067057719848015336297642411160}{7576337217254497052011232092161707258480016239563653448563550980181849229809} a^{21} - \frac{3163107983273950428560527730940017676721448239663996828505207178306374396172}{7576337217254497052011232092161707258480016239563653448563550980181849229809} a^{20} + \frac{2763597317965544389572748386180564905788582563582805029901253980169804457631}{7576337217254497052011232092161707258480016239563653448563550980181849229809} a^{19} - \frac{3501456459020732461715568535411756995350360913424812657721504307281736397388}{7576337217254497052011232092161707258480016239563653448563550980181849229809} a^{18} - \frac{739029283119979124417619724126411553546955038961749158680670036587541611347}{7576337217254497052011232092161707258480016239563653448563550980181849229809} a^{17} + \frac{1840737967258589447935576500990612559818352803540465265903422623759337438258}{7576337217254497052011232092161707258480016239563653448563550980181849229809} a^{16} + \frac{1094577824892614595351858589719637751241457711555497621843356744444825820520}{7576337217254497052011232092161707258480016239563653448563550980181849229809} a^{15} + \frac{2754264925314071003529128077552383986116672686737229847279835601301626714830}{7576337217254497052011232092161707258480016239563653448563550980181849229809} a^{14} - \frac{464914094911136180036981228217676363878170564023447635794404223584845581227}{7576337217254497052011232092161707258480016239563653448563550980181849229809} a^{13} + \frac{2713139606672366744137871138506517444773857410956714395544043361944639428746}{7576337217254497052011232092161707258480016239563653448563550980181849229809} a^{12} - \frac{97379423115787188681322262513368321781813758879547640845295208909137743464}{244397974750145066193910712650377653499355362566569466082695192909091910639} a^{11} - \frac{814542781868320391892513898874327691013811547852408976052572509930945320229}{7576337217254497052011232092161707258480016239563653448563550980181849229809} a^{10} + \frac{673786585244679213543412297348165630087634916166485761431899590012722375706}{7576337217254497052011232092161707258480016239563653448563550980181849229809} a^{9} + \frac{1594923129986733403879746043538230750292611627155066350046094105514125329256}{7576337217254497052011232092161707258480016239563653448563550980181849229809} a^{8} + \frac{2059596888807790322840699539817088848328209253627064344950804193073396890266}{7576337217254497052011232092161707258480016239563653448563550980181849229809} a^{7} + \frac{3060232870026675778074495552428063117925008137206275746719692811501342410594}{7576337217254497052011232092161707258480016239563653448563550980181849229809} a^{6} + \frac{936382066613864032063483192223413990985704570777731893014822941603849032372}{7576337217254497052011232092161707258480016239563653448563550980181849229809} a^{5} + \frac{299518895524706421308595294942568671557355776778635212039053554503750333796}{7576337217254497052011232092161707258480016239563653448563550980181849229809} a^{4} - \frac{1478936602230284663417618076553576883405522681577032331831383717167583348102}{7576337217254497052011232092161707258480016239563653448563550980181849229809} a^{3} + \frac{2754705717117619849469662119450315536519475680621241588590637651662111486488}{7576337217254497052011232092161707258480016239563653448563550980181849229809} a^{2} - \frac{3697492753316624329308442286504243322608191669652369773156060914653380496994}{7576337217254497052011232092161707258480016239563653448563550980181849229809} a + \frac{573130405157322281957100458043503703741836208043913330610344375790412327924}{7576337217254497052011232092161707258480016239563653448563550980181849229809}$, $\frac{1}{14179132158128207695767716105091383681070095452324750414278340089997282967068894649530365333091208852191} a^{31} + \frac{375621567408823381718694813}{14179132158128207695767716105091383681070095452324750414278340089997282967068894649530365333091208852191} a^{30} - \frac{111701153074237295088410331558892724843889461293223742306624228387959197686527104405574935481654786235}{14179132158128207695767716105091383681070095452324750414278340089997282967068894649530365333091208852191} a^{29} + \frac{5652363145413163305377439870761815534966798866685409711924521848661901633985719721889429182168795171656}{14179132158128207695767716105091383681070095452324750414278340089997282967068894649530365333091208852191} a^{28} - \frac{209178304822567563783908136477293956723480565851463956519260376676629441707202366409459858997720424465}{457391359939619603089281164680367215518390175881443561750914196451525257002222408049366623648103511361} a^{27} + \frac{5916864052568588426920067873484294359796063161682380173875220605284133893639207615118945699299412251089}{14179132158128207695767716105091383681070095452324750414278340089997282967068894649530365333091208852191} a^{26} + \frac{3856447951531617914360062389339375496356006046400219563285270873168625570064412670984609228823614185722}{14179132158128207695767716105091383681070095452324750414278340089997282967068894649530365333091208852191} a^{25} - \frac{2217452779650060771321661937731298513866786183311308493913254663150738987514132396400212100975056500578}{14179132158128207695767716105091383681070095452324750414278340089997282967068894649530365333091208852191} a^{24} + \frac{1541128628347652819528736162827454345168064525699099457168895060737473382071077629791147399270355259650}{14179132158128207695767716105091383681070095452324750414278340089997282967068894649530365333091208852191} a^{23} + \frac{219261173728976449070462646112055133259955932099440934427992847388075779203088081120065114213326917954}{14179132158128207695767716105091383681070095452324750414278340089997282967068894649530365333091208852191} a^{22} + \frac{2003020170511071688937913617567813971488366886210079158783231353559124204009804082536078288217214341984}{14179132158128207695767716105091383681070095452324750414278340089997282967068894649530365333091208852191} a^{21} + \frac{6596160549077795807797500049160402186052636559644229511549764425648493634922539771513156386109696041749}{14179132158128207695767716105091383681070095452324750414278340089997282967068894649530365333091208852191} a^{20} + \frac{825413322951440342076905675160979418022559771611912623852521559442064531126683143572116439366853762902}{14179132158128207695767716105091383681070095452324750414278340089997282967068894649530365333091208852191} a^{19} - \frac{4203327879275907614104172924189949262098417070633444956658279548339994032483185659503876564553478919993}{14179132158128207695767716105091383681070095452324750414278340089997282967068894649530365333091208852191} a^{18} + \frac{4116856076662102882229985517298454957780949476705931825240294658942338451439173284675951390176885322614}{14179132158128207695767716105091383681070095452324750414278340089997282967068894649530365333091208852191} a^{17} - \frac{42898794617782873271321379578335319782485040594876508889937287519476069765643346689380487198759852414}{457391359939619603089281164680367215518390175881443561750914196451525257002222408049366623648103511361} a^{16} + \frac{4833291351481231587937359129632176005603664844578139291159930242389260695506645477410742246304509568076}{14179132158128207695767716105091383681070095452324750414278340089997282967068894649530365333091208852191} a^{15} - \frac{3405227762624718343182185815473367855952471217602703389535747626845488897380989347743857676520975591387}{14179132158128207695767716105091383681070095452324750414278340089997282967068894649530365333091208852191} a^{14} + \frac{6984143438516476595019412688930927047660751668550005300228530908781158142342924293877305570693035728983}{14179132158128207695767716105091383681070095452324750414278340089997282967068894649530365333091208852191} a^{13} - \frac{985359264169574726250095555761671211053494197333172334040497079541417333155556511898055789379044149221}{14179132158128207695767716105091383681070095452324750414278340089997282967068894649530365333091208852191} a^{12} + \frac{5033591300644328025132324586288470038019739025145123279291690721372025495602842468068326386192889673090}{14179132158128207695767716105091383681070095452324750414278340089997282967068894649530365333091208852191} a^{11} - \frac{2995188504429485032587511255921652347329483603152076406057422297936289989977111752433117785024340607162}{14179132158128207695767716105091383681070095452324750414278340089997282967068894649530365333091208852191} a^{10} + \frac{1194825648363087318070267490181866788863231181554843018231471254922695290884274336622610300491771301228}{14179132158128207695767716105091383681070095452324750414278340089997282967068894649530365333091208852191} a^{9} + \frac{3368153510304606139036010308299659770068649389135590516781870413604462415281200293495983052327968794702}{14179132158128207695767716105091383681070095452324750414278340089997282967068894649530365333091208852191} a^{8} - \frac{3136144187536761988697792328977067541167388062238429573542267055724410423379886469853812389445675542470}{14179132158128207695767716105091383681070095452324750414278340089997282967068894649530365333091208852191} a^{7} + \frac{6906961294621908658854701125873936700764847726658057565323726126105421418638349294731495803288682631224}{14179132158128207695767716105091383681070095452324750414278340089997282967068894649530365333091208852191} a^{6} + \frac{554482040192364424790508503677670841382305314235643544878496057874138033591870777642678019706571618123}{14179132158128207695767716105091383681070095452324750414278340089997282967068894649530365333091208852191} a^{5} - \frac{2369814153744729554872914711948851854179704764241537925738114900083280615019594243885213095364323072276}{14179132158128207695767716105091383681070095452324750414278340089997282967068894649530365333091208852191} a^{4} - \frac{6702748773844446956428741532020784659835742847259816929224255328686748189794385546639492441402926999904}{14179132158128207695767716105091383681070095452324750414278340089997282967068894649530365333091208852191} a^{3} + \frac{4769534918668787081946445812553279597663119579027296694438892965498622654652216182130483255464252087716}{14179132158128207695767716105091383681070095452324750414278340089997282967068894649530365333091208852191} a^{2} + \frac{968420094520719356656189193486329821463078248460759115704213234765212581889161641985823632857266920588}{14179132158128207695767716105091383681070095452324750414278340089997282967068894649530365333091208852191} a + \frac{5043104044655971382769460216449081815301898913924702949559770006560949325612916144784423577632275598149}{14179132158128207695767716105091383681070095452324750414278340089997282967068894649530365333091208852191}$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.4.392000.1, \(\Q(\sqrt{2}, \sqrt{5})\), 4.4.6125.1, \(\Q(\zeta_{16})^+\), 4.4.51200.1, 4.4.12544000.2, 4.4.12544000.1, 8.8.153664000000.1, 8.8.2621440000.1, 8.8.157351936000000.4, 8.0.1342177280000.1, 8.0.2147483648.1, 8.0.80564191232000000.92, 8.0.80564191232000000.78, 16.16.24759631762948096000000000000.2, 16.0.1801439850948198400000000.1, 16.0.6490588908866265677824000000000000.9

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

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
5Data not computed
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$