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Properties

Label 32.0.59816430901...0000.5
Degree $32$
Signature $[0, 16]$
Discriminant $2^{32}\cdot 3^{16}\cdot 5^{24}\cdot 13^{24}$
Root discriminant $79.30$
Ramified primes $2, 3, 5, 13$
Class number $532480$ (GRH)
Class group $[2, 2, 2, 8, 8, 1040]$ (GRH)
Galois group $C_2\times C_4^2$ (as 32T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![65536, 0, 872448, 0, 8588288, 0, 31866640, 0, 81082665, 0, 125633292, 0, 139391684, 0, 95642940, 0, 47307204, 0, 16368060, 0, 4239806, 0, 792804, 0, 111660, 0, 11180, 0, 812, 0, 36, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 36*x^30 + 812*x^28 + 11180*x^26 + 111660*x^24 + 792804*x^22 + 4239806*x^20 + 16368060*x^18 + 47307204*x^16 + 95642940*x^14 + 139391684*x^12 + 125633292*x^10 + 81082665*x^8 + 31866640*x^6 + 8588288*x^4 + 872448*x^2 + 65536)
 
gp: K = bnfinit(x^32 + 36*x^30 + 812*x^28 + 11180*x^26 + 111660*x^24 + 792804*x^22 + 4239806*x^20 + 16368060*x^18 + 47307204*x^16 + 95642940*x^14 + 139391684*x^12 + 125633292*x^10 + 81082665*x^8 + 31866640*x^6 + 8588288*x^4 + 872448*x^2 + 65536, 1)
 

Normalized defining polynomial

\( x^{32} + 36 x^{30} + 812 x^{28} + 11180 x^{26} + 111660 x^{24} + 792804 x^{22} + 4239806 x^{20} + 16368060 x^{18} + 47307204 x^{16} + 95642940 x^{14} + 139391684 x^{12} + 125633292 x^{10} + 81082665 x^{8} + 31866640 x^{6} + 8588288 x^{4} + 872448 x^{2} + 65536 \)

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:  \(5981643090147991811559885370844487936000000000000000000000000=2^{32}\cdot 3^{16}\cdot 5^{24}\cdot 13^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.30$
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:  $2, 3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(780=2^{2}\cdot 3\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{780}(1,·)$, $\chi_{780}(389,·)$, $\chi_{780}(577,·)$, $\chi_{780}(649,·)$, $\chi_{780}(239,·)$, $\chi_{780}(151,·)$, $\chi_{780}(103,·)$, $\chi_{780}(31,·)$, $\chi_{780}(547,·)$, $\chi_{780}(551,·)$, $\chi_{780}(359,·)$, $\chi_{780}(181,·)$, $\chi_{780}(521,·)$, $\chi_{780}(697,·)$, $\chi_{780}(287,·)$, $\chi_{780}(671,·)$, $\chi_{780}(317,·)$, $\chi_{780}(437,·)$, $\chi_{780}(701,·)$, $\chi_{780}(73,·)$, $\chi_{780}(209,·)$, $\chi_{780}(467,·)$, $\chi_{780}(469,·)$, $\chi_{780}(727,·)$, $\chi_{780}(473,·)$, $\chi_{780}(733,·)$, $\chi_{780}(443,·)$, $\chi_{780}(593,·)$, $\chi_{780}(619,·)$, $\chi_{780}(623,·)$, $\chi_{780}(499,·)$, $\chi_{780}(703,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{4} a^{7} + \frac{1}{4} a$, $\frac{1}{4} a^{8} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} + \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} + \frac{1}{4} a^{4}$, $\frac{1}{4} a^{11} + \frac{1}{4} a^{5}$, $\frac{1}{4} a^{12} + \frac{1}{4} a^{6}$, $\frac{1}{4} a^{13} - \frac{1}{4} a$, $\frac{1}{16} a^{14} - \frac{1}{8} a^{8} - \frac{3}{16} a^{2}$, $\frac{1}{32} a^{15} - \frac{1}{16} a^{9} + \frac{13}{32} a^{3}$, $\frac{1}{32} a^{16} - \frac{1}{16} a^{10} + \frac{13}{32} a^{4}$, $\frac{1}{32} a^{17} - \frac{1}{16} a^{11} + \frac{13}{32} a^{5}$, $\frac{1}{32} a^{18} - \frac{1}{16} a^{12} + \frac{13}{32} a^{6}$, $\frac{1}{32} a^{19} - \frac{1}{16} a^{13} - \frac{3}{32} a^{7} - \frac{1}{2} a$, $\frac{1}{32} a^{20} + \frac{1}{32} a^{8} - \frac{7}{16} a^{2}$, $\frac{1}{64} a^{21} - \frac{1}{64} a^{15} - \frac{5}{64} a^{9} - \frac{3}{64} a^{3}$, $\frac{1}{128} a^{22} - \frac{1}{128} a^{16} + \frac{11}{128} a^{10} + \frac{13}{128} a^{4}$, $\frac{1}{128} a^{23} - \frac{1}{128} a^{17} + \frac{11}{128} a^{11} + \frac{13}{128} a^{5}$, $\frac{1}{5376} a^{24} + \frac{5}{5376} a^{22} + \frac{13}{1344} a^{20} - \frac{41}{5376} a^{18} - \frac{65}{5376} a^{16} + \frac{19}{672} a^{14} - \frac{5}{5376} a^{12} - \frac{593}{5376} a^{10} - \frac{95}{1344} a^{8} + \frac{1829}{5376} a^{6} + \frac{1717}{5376} a^{4} - \frac{73}{168} a^{2} + \frac{4}{21}$, $\frac{1}{10752} a^{25} + \frac{5}{10752} a^{23} + \frac{13}{2688} a^{21} - \frac{41}{10752} a^{19} - \frac{65}{10752} a^{17} + \frac{19}{1344} a^{15} - \frac{5}{10752} a^{13} - \frac{593}{10752} a^{11} - \frac{95}{2688} a^{9} - \frac{859}{10752} a^{7} - \frac{3659}{10752} a^{5} - \frac{73}{336} a^{3} - \frac{13}{84} a$, $\frac{1}{655872} a^{26} + \frac{17}{218624} a^{24} - \frac{233}{109312} a^{22} - \frac{3025}{655872} a^{20} - \frac{10015}{655872} a^{18} - \frac{1649}{109312} a^{16} + \frac{1209}{218624} a^{14} + \frac{52937}{655872} a^{12} + \frac{31363}{327936} a^{10} - \frac{4769}{218624} a^{8} - \frac{52919}{218624} a^{6} - \frac{155717}{327936} a^{4} - \frac{5699}{20496} a^{2} - \frac{580}{1281}$, $\frac{1}{655872} a^{27} - \frac{5}{327936} a^{25} - \frac{1703}{655872} a^{23} + \frac{4051}{655872} a^{21} - \frac{3757}{327936} a^{19} - \frac{847}{93696} a^{17} + \frac{4603}{655872} a^{15} + \frac{3803}{46848} a^{13} - \frac{65069}{655872} a^{11} + \frac{80609}{655872} a^{9} + \frac{4115}{46848} a^{7} - \frac{36029}{93696} a^{5} - \frac{5287}{11712} a^{3} - \frac{41}{854} a$, $\frac{1}{54386873856} a^{28} - \frac{27815}{54386873856} a^{26} + \frac{1941059}{27193436928} a^{24} - \frac{167378003}{54386873856} a^{22} + \frac{2687425}{415166976} a^{20} - \frac{75834175}{9064478976} a^{18} - \frac{130971923}{54386873856} a^{16} - \frac{1296262757}{54386873856} a^{14} - \frac{2989784603}{27193436928} a^{12} - \frac{2147286931}{18128957952} a^{10} + \frac{5658124081}{54386873856} a^{8} + \frac{7130205125}{27193436928} a^{6} - \frac{6532685501}{27193436928} a^{4} + \frac{29917429}{121399272} a^{2} + \frac{48202774}{106224363}$, $\frac{1}{108773747712} a^{29} + \frac{13777}{27193436928} a^{27} + \frac{381611}{13596718464} a^{25} - \frac{9643621}{3399179616} a^{23} - \frac{44117}{29654784} a^{21} + \frac{8631963}{1510746496} a^{19} + \frac{538483709}{54386873856} a^{17} - \frac{16193321}{27193436928} a^{15} + \frac{1079119501}{13596718464} a^{13} - \frac{68334943}{4532239488} a^{11} + \frac{748680061}{27193436928} a^{9} + \frac{42764549}{13596718464} a^{7} - \frac{52674288259}{108773747712} a^{5} - \frac{198174143}{971194176} a^{3} + \frac{86206019}{424897452} a$, $\frac{1}{6506126820124963695224072309939355648} a^{30} + \frac{9753137216880705999201005}{1626531705031240923806018077484838912} a^{28} + \frac{471336360198536808891399048851}{1626531705031240923806018077484838912} a^{26} + \frac{2065824279102734356068885106859}{1626531705031240923806018077484838912} a^{24} - \frac{1721485554153032878473090117404879}{542177235010413641268672692494946304} a^{22} + \frac{8404948550603261771155539789996305}{1626531705031240923806018077484838912} a^{20} + \frac{47592018071937345873390794091094559}{3253063410062481847612036154969677824} a^{18} + \frac{260160762501221583911954531351899}{56087300173491066338138554396028928} a^{16} - \frac{9712949063032292187494306635974589}{542177235010413641268672692494946304} a^{14} + \frac{104509647667071480967715924913617615}{1626531705031240923806018077484838912} a^{12} - \frac{166100020962798208782521291838054935}{1626531705031240923806018077484838912} a^{10} + \frac{69273028797567199611825203231099915}{1626531705031240923806018077484838912} a^{8} - \frac{266490862344460036585498976281631071}{722902980013884855024896923326595072} a^{6} - \frac{137602375738027351272530036077497}{2823839765679237714941003606744512} a^{4} + \frac{3627092064082103917066496962885}{34719341381302103052553323033744} a^{2} - \frac{1324851456588462007590819665515}{26039506035976577289414992275308}$, $\frac{1}{26024507280499854780896289239757422592} a^{31} - \frac{20153558824112662248237947}{6506126820124963695224072309939355648} a^{29} - \frac{3656714801035818803670846934661}{6506126820124963695224072309939355648} a^{27} - \frac{64436439626020496894294720019557}{6506126820124963695224072309939355648} a^{25} + \frac{20995265110246587409386374546375099}{6506126820124963695224072309939355648} a^{23} - \frac{7405454212997017224247683225490429}{2168708940041654565074690769979785216} a^{21} - \frac{155629815126742347534367230190882561}{13012253640249927390448144619878711296} a^{19} - \frac{362052358406135132840084141309467}{32049885813423466478936316797730816} a^{17} + \frac{37626757944826510799467626847284737}{6506126820124963695224072309939355648} a^{15} + \frac{81207901863553516415303600506965543}{722902980013884855024896923326595072} a^{13} + \frac{285754474746911948535578831200386529}{6506126820124963695224072309939355648} a^{11} + \frac{211851797066702988740070037466818099}{6506126820124963695224072309939355648} a^{9} + \frac{1462137784814798253989066519419196393}{26024507280499854780896289239757422592} a^{7} + \frac{12704899848361236116070521397128065}{25414557891113139434469032460700608} a^{5} + \frac{5296124878385051093317643607025807}{12707278945556569717234516230350304} a^{3} - \frac{745833851813222635186225362029941}{2117879824259428286205752705058384} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{8}\times C_{8}\times C_{1040}$, which has order $532480$ (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:  \( -\frac{4791620338837015389013750471}{2365004296664835948827361799323648} a^{30} - \frac{14317883363616494339768282825}{197083691388736329068946816610304} a^{28} - \frac{322203471702339594203946772735}{197083691388736329068946816610304} a^{26} - \frac{4418697129918190957162194838535}{197083691388736329068946816610304} a^{24} - \frac{131898747342179263825974121757845}{591251074166208987206840449830912} a^{22} - \frac{931319638685040475484019827139199}{591251074166208987206840449830912} a^{20} - \frac{9900585199215569113964693362487945}{1182502148332417974413680899661824} a^{18} - \frac{217596585270549252908162499718255}{6795989358232287209274028158976} a^{16} - \frac{54124578016579579601763780342681895}{591251074166208987206840449830912} a^{14} - \frac{107430885942588686297196512144059105}{591251074166208987206840449830912} a^{12} - \frac{153272390491201448601163863163834831}{591251074166208987206840449830912} a^{10} - \frac{43901839177011233642394919422124535}{197083691388736329068946816610304} a^{8} - \frac{330267147442790631823960090469689855}{2365004296664835948827361799323648} a^{6} - \frac{1854036779779793105866025637538685}{36953192135388061700427528114432} a^{4} - \frac{32905218490535578561718943088885}{2309574508461753856276720507152} a^{2} - \frac{255340307060119734811613829103}{577393627115438464069180126788} \) (order $6$)
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:  \( 9245193308827.074 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_4^2$ (as 32T36):

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 $C_2\times C_4^2$
Character table for $C_2\times C_4^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-195}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{-3}, \sqrt{65})\), 4.0.2471625.2, 4.4.274625.1, \(\Q(\sqrt{13}, \sqrt{-15})\), \(\Q(\sqrt{5}, \sqrt{-39})\), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{-15}, \sqrt{-39})\), 4.0.2471625.1, 4.4.274625.2, \(\Q(\sqrt{-3}, \sqrt{13})\), \(\Q(\sqrt{-3}, \sqrt{5})\), 4.0.316368.2, 4.4.878800.1, 4.0.7909200.1, 4.4.35152.1, 4.4.338000.1, 4.0.18000.1, \(\Q(\zeta_{20})^+\), 4.0.3042000.1, 8.0.6108930140625.1, 8.0.1445900625.1, 8.0.6108930140625.3, 8.0.6108930140625.4, 8.0.6108930140625.5, 8.8.75418890625.1, 8.0.6108930140625.2, 8.0.62555444640000.22, 8.0.62555444640000.40, 8.0.9253764000000.2, 8.0.9253764000000.5, 8.0.62555444640000.63, 8.8.772289440000.1, 8.8.114244000000.1, 8.0.9253764000000.3, 8.0.100088711424.2, 8.0.62555444640000.70, 8.0.9253764000000.9, 8.0.324000000.2, 16.0.37319027463036582275390625.1, 16.0.3913183654108104729600000000.6, 16.0.85632148167696000000000000.7, 16.0.2445739783817565456000000000000.5, 16.0.2445739783817565456000000000000.4, 16.0.2445739783817565456000000000000.13, 16.16.372769361959696000000000000.2

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
$2$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
13Data not computed