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Properties

Label 32.32.7274279403...0625.1
Degree $32$
Signature $[32, 0]$
Discriminant $3^{16}\cdot 5^{24}\cdot 17^{28}$
Root discriminant $69.09$
Ramified primes $3, 5, 17$
Class number $1$ (GRH)
Class group Trivial (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, 46, 67, -6488, -16252, 256798, 624842, -2410058, -5039008, 11166940, 17702718, -30305480, -32389886, 50625606, 32826914, -52954720, -18558598, 35220860, 5324168, -15130884, -283286, 4237982, -299890, -771912, 102376, 89642, -15812, -6334, 1318, 246, -57, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 4*x^31 - 57*x^30 + 246*x^29 + 1318*x^28 - 6334*x^27 - 15812*x^26 + 89642*x^25 + 102376*x^24 - 771912*x^23 - 299890*x^22 + 4237982*x^21 - 283286*x^20 - 15130884*x^19 + 5324168*x^18 + 35220860*x^17 - 18558598*x^16 - 52954720*x^15 + 32826914*x^14 + 50625606*x^13 - 32389886*x^12 - 30305480*x^11 + 17702718*x^10 + 11166940*x^9 - 5039008*x^8 - 2410058*x^7 + 624842*x^6 + 256798*x^5 - 16252*x^4 - 6488*x^3 + 67*x^2 + 46*x + 1)
 
gp: K = bnfinit(x^32 - 4*x^31 - 57*x^30 + 246*x^29 + 1318*x^28 - 6334*x^27 - 15812*x^26 + 89642*x^25 + 102376*x^24 - 771912*x^23 - 299890*x^22 + 4237982*x^21 - 283286*x^20 - 15130884*x^19 + 5324168*x^18 + 35220860*x^17 - 18558598*x^16 - 52954720*x^15 + 32826914*x^14 + 50625606*x^13 - 32389886*x^12 - 30305480*x^11 + 17702718*x^10 + 11166940*x^9 - 5039008*x^8 - 2410058*x^7 + 624842*x^6 + 256798*x^5 - 16252*x^4 - 6488*x^3 + 67*x^2 + 46*x + 1, 1)
 

Normalized defining polynomial

\( x^{32} - 4 x^{31} - 57 x^{30} + 246 x^{29} + 1318 x^{28} - 6334 x^{27} - 15812 x^{26} + 89642 x^{25} + 102376 x^{24} - 771912 x^{23} - 299890 x^{22} + 4237982 x^{21} - 283286 x^{20} - 15130884 x^{19} + 5324168 x^{18} + 35220860 x^{17} - 18558598 x^{16} - 52954720 x^{15} + 32826914 x^{14} + 50625606 x^{13} - 32389886 x^{12} - 30305480 x^{11} + 17702718 x^{10} + 11166940 x^{9} - 5039008 x^{8} - 2410058 x^{7} + 624842 x^{6} + 256798 x^{5} - 16252 x^{4} - 6488 x^{3} + 67 x^{2} + 46 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[32, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(72742794030721358896825924818930661340866148471832275390625=3^{16}\cdot 5^{24}\cdot 17^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.09$
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:  $3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(255=3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{255}(128,·)$, $\chi_{255}(1,·)$, $\chi_{255}(2,·)$, $\chi_{255}(4,·)$, $\chi_{255}(8,·)$, $\chi_{255}(137,·)$, $\chi_{255}(16,·)$, $\chi_{255}(19,·)$, $\chi_{255}(151,·)$, $\chi_{255}(152,·)$, $\chi_{255}(154,·)$, $\chi_{255}(32,·)$, $\chi_{255}(38,·)$, $\chi_{255}(169,·)$, $\chi_{255}(47,·)$, $\chi_{255}(49,·)$, $\chi_{255}(53,·)$, $\chi_{255}(188,·)$, $\chi_{255}(64,·)$, $\chi_{255}(196,·)$, $\chi_{255}(203,·)$, $\chi_{255}(76,·)$, $\chi_{255}(77,·)$, $\chi_{255}(83,·)$, $\chi_{255}(212,·)$, $\chi_{255}(94,·)$, $\chi_{255}(98,·)$, $\chi_{255}(229,·)$, $\chi_{255}(106,·)$, $\chi_{255}(242,·)$, $\chi_{255}(121,·)$, $\chi_{255}(166,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{24} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{25} - \frac{1}{2} a^{10}$, $\frac{1}{2} a^{26} - \frac{1}{2} a^{11}$, $\frac{1}{2} a^{27} - \frac{1}{2} a^{12}$, $\frac{1}{2} a^{28} - \frac{1}{2} a^{13}$, $\frac{1}{2} a^{29} - \frac{1}{2} a^{14}$, $\frac{1}{1463636} a^{30} + \frac{34283}{731818} a^{29} - \frac{116421}{731818} a^{28} + \frac{22043}{731818} a^{27} + \frac{120719}{731818} a^{26} - \frac{45797}{365909} a^{25} - \frac{117075}{731818} a^{24} - \frac{27720}{365909} a^{23} + \frac{3731}{731818} a^{22} - \frac{58957}{731818} a^{21} - \frac{101681}{731818} a^{20} - \frac{63785}{731818} a^{19} - \frac{25421}{731818} a^{18} + \frac{96357}{731818} a^{17} + \frac{21813}{731818} a^{16} - \frac{4091}{731818} a^{15} + \frac{176179}{731818} a^{14} + \frac{105695}{731818} a^{13} - \frac{355709}{731818} a^{12} - \frac{329855}{731818} a^{11} + \frac{33203}{365909} a^{10} - \frac{138459}{731818} a^{9} - \frac{147120}{365909} a^{8} + \frac{217281}{731818} a^{7} - \frac{110735}{731818} a^{6} - \frac{25879}{731818} a^{5} + \frac{30829}{731818} a^{4} - \frac{307537}{731818} a^{3} - \frac{192921}{731818} a^{2} - \frac{295697}{731818} a - \frac{646803}{1463636}$, $\frac{1}{589436544203686818071634020717366890195190208305904636584656151921406808} a^{31} + \frac{137108018950716215463299377457110321525051661564392340451712471709}{589436544203686818071634020717366890195190208305904636584656151921406808} a^{30} - \frac{3007253465736140475693847138201268820751971193146566272125426549985157}{73679568025460852258954252589670861274398776038238079573082018990175851} a^{29} + \frac{662059070987482850172182721966646053060472965935483166906185943943103}{294718272101843409035817010358683445097595104152952318292328075960703404} a^{28} - \frac{12656583695454744763399965688432463575795457175815528525264104033348823}{73679568025460852258954252589670861274398776038238079573082018990175851} a^{27} - \frac{4023496977384103373773512116057425815826692686194614377787855371966081}{294718272101843409035817010358683445097595104152952318292328075960703404} a^{26} + \frac{32706161431847590148445989698821835548446276495241226423997948240154233}{294718272101843409035817010358683445097595104152952318292328075960703404} a^{25} - \frac{14401466697670927245477859541063159597567556377387159552536746831656971}{73679568025460852258954252589670861274398776038238079573082018990175851} a^{24} - \frac{8798477690005140861530585701081401049401757785242511991609116403890485}{147359136050921704517908505179341722548797552076476159146164037980351702} a^{23} + \frac{29594343515580980341439188732786297630318736622135230673081746297452943}{147359136050921704517908505179341722548797552076476159146164037980351702} a^{22} - \frac{26843763880129897370329924750388652995924270504528651986128909536064899}{294718272101843409035817010358683445097595104152952318292328075960703404} a^{21} - \frac{11986725657551875525452289399319577977058723693396973717279359058441165}{147359136050921704517908505179341722548797552076476159146164037980351702} a^{20} - \frac{43836233809931323431917707651580564544841597333937399016516322652508293}{294718272101843409035817010358683445097595104152952318292328075960703404} a^{19} + \frac{1126539971600841402729470032881482433531584335995912159975566617355419}{294718272101843409035817010358683445097595104152952318292328075960703404} a^{18} + \frac{38284625072689738796034168669780128810955701939252060638786077872278217}{294718272101843409035817010358683445097595104152952318292328075960703404} a^{17} - \frac{66154279284327533744843874255672310190188533229010728985983454071983391}{294718272101843409035817010358683445097595104152952318292328075960703404} a^{16} + \frac{11831608298564774905467879023468394071505033352621598225709156951113191}{73679568025460852258954252589670861274398776038238079573082018990175851} a^{15} - \frac{38108142063993727262753726115073749636854036803469236007484074579108565}{147359136050921704517908505179341722548797552076476159146164037980351702} a^{14} - \frac{86511914301110987004447884350931581202146453100475040989982524709417469}{294718272101843409035817010358683445097595104152952318292328075960703404} a^{13} + \frac{29764443325039427338548042415776223331849359541510345129217243920497805}{73679568025460852258954252589670861274398776038238079573082018990175851} a^{12} + \frac{46669235135084179576941264104352259744298306869088074722502122406876355}{294718272101843409035817010358683445097595104152952318292328075960703404} a^{11} + \frac{6071295417537594320118832789689118930046563306214592850662852568517955}{294718272101843409035817010358683445097595104152952318292328075960703404} a^{10} - \frac{20875858740046527948416546022818984129359607677172876796044583399605456}{73679568025460852258954252589670861274398776038238079573082018990175851} a^{9} - \frac{7388669400608680495859845255693564085316002151277356485236663627738645}{73679568025460852258954252589670861274398776038238079573082018990175851} a^{8} + \frac{23311377612277260093463567515782599757041653071380337367532258448560395}{73679568025460852258954252589670861274398776038238079573082018990175851} a^{7} - \frac{10264062514178843079782926269080925416192972083961111564143391632679985}{294718272101843409035817010358683445097595104152952318292328075960703404} a^{6} + \frac{4348236286624636618779741098097866790008907981505778569196540622169241}{147359136050921704517908505179341722548797552076476159146164037980351702} a^{5} + \frac{102839458012496896830584065444093997050393740491133699248813265573965933}{294718272101843409035817010358683445097595104152952318292328075960703404} a^{4} + \frac{121478679092692200592140672701110220544194502842216640949637474742432701}{294718272101843409035817010358683445097595104152952318292328075960703404} a^{3} - \frac{91299461310095561988576559533400511503280519558686491847456034464474161}{294718272101843409035817010358683445097595104152952318292328075960703404} a^{2} + \frac{21895604946529282410199593229738192596979234730234901602147976580991949}{589436544203686818071634020717366890195190208305904636584656151921406808} a - \frac{49625363938326468335506249271874713446700325382515400951583797018055681}{589436544203686818071634020717366890195190208305904636584656151921406808}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $31$
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:  \( 24920675451007074000 \) (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

\(\Q(\sqrt{85}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{5}) \), 4.4.5527125.2, \(\Q(\sqrt{5}, \sqrt{17})\), 4.4.5527125.1, 4.4.122825.1, 4.4.4913.1, 4.4.325125.1, \(\Q(\zeta_{15})^+\), 8.8.30549110765625.1, 8.8.15085980625.1, 8.8.105706265625.1, 8.8.519334883015625.2, 8.8.519334883015625.1, 8.8.256461670625.1, \(\Q(\zeta_{17})^+\), 16.16.933248168570425273681640625.1, 16.16.269708720716852904093994140625.1, 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{8}$ R 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
3Data not computed
5Data not computed
17Data not computed