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Properties

Label 32.0.47565240232...0000.2
Degree $32$
Signature $[0, 16]$
Discriminant $2^{48}\cdot 5^{24}\cdot 17^{28}$
Root discriminant $112.83$
Ramified primes $2, 5, 17$
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![132892327936, 0, 223409602560, 0, 469219672064, 0, 269225361408, 0, 49845997568, 0, -9379649536, 0, -3522902016, 0, 322428416, 0, 132440064, 0, -20680576, 0, 3957248, 0, -701280, 0, 26112, 0, 5912, 0, -376, 0, -14, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 14*x^30 - 376*x^28 + 5912*x^26 + 26112*x^24 - 701280*x^22 + 3957248*x^20 - 20680576*x^18 + 132440064*x^16 + 322428416*x^14 - 3522902016*x^12 - 9379649536*x^10 + 49845997568*x^8 + 269225361408*x^6 + 469219672064*x^4 + 223409602560*x^2 + 132892327936)
 
gp: K = bnfinit(x^32 - 14*x^30 - 376*x^28 + 5912*x^26 + 26112*x^24 - 701280*x^22 + 3957248*x^20 - 20680576*x^18 + 132440064*x^16 + 322428416*x^14 - 3522902016*x^12 - 9379649536*x^10 + 49845997568*x^8 + 269225361408*x^6 + 469219672064*x^4 + 223409602560*x^2 + 132892327936, 1)
 

Normalized defining polynomial

\( x^{32} - 14 x^{30} - 376 x^{28} + 5912 x^{26} + 26112 x^{24} - 701280 x^{22} + 3957248 x^{20} - 20680576 x^{18} + 132440064 x^{16} + 322428416 x^{14} - 3522902016 x^{12} - 9379649536 x^{10} + 49845997568 x^{8} + 269225361408 x^{6} + 469219672064 x^{4} + 223409602560 x^{2} + 132892327936 \)

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:  \(475652402320384421216839760983630708473856000000000000000000000000=2^{48}\cdot 5^{24}\cdot 17^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $112.83$
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:  $2, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(680=2^{3}\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{680}(1,·)$, $\chi_{680}(387,·)$, $\chi_{680}(179,·)$, $\chi_{680}(257,·)$, $\chi_{680}(393,·)$, $\chi_{680}(523,·)$, $\chi_{680}(19,·)$, $\chi_{680}(409,·)$, $\chi_{680}(667,·)$, $\chi_{680}(417,·)$, $\chi_{680}(291,·)$, $\chi_{680}(297,·)$, $\chi_{680}(433,·)$, $\chi_{680}(307,·)$, $\chi_{680}(441,·)$, $\chi_{680}(59,·)$, $\chi_{680}(169,·)$, $\chi_{680}(67,·)$, $\chi_{680}(457,·)$, $\chi_{680}(331,·)$, $\chi_{680}(593,·)$, $\chi_{680}(203,·)$, $\chi_{680}(89,·)$, $\chi_{680}(219,·)$, $\chi_{680}(443,·)$, $\chi_{680}(81,·)$, $\chi_{680}(361,·)$, $\chi_{680}(491,·)$, $\chi_{680}(451,·)$, $\chi_{680}(489,·)$, $\chi_{680}(553,·)$, $\chi_{680}(123,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{16} a^{7} - \frac{1}{2} a$, $\frac{1}{64} a^{8} + \frac{1}{8} a^{2}$, $\frac{1}{64} a^{9} + \frac{1}{8} a^{3}$, $\frac{1}{128} a^{10} + \frac{1}{16} a^{4}$, $\frac{1}{128} a^{11} + \frac{1}{16} a^{5}$, $\frac{1}{256} a^{12} + \frac{1}{32} a^{6}$, $\frac{1}{256} a^{13} - \frac{1}{32} a^{7} - \frac{1}{2} a$, $\frac{1}{512} a^{14} - \frac{1}{8} a^{2}$, $\frac{1}{1024} a^{15} - \frac{1}{16} a^{3}$, $\frac{1}{8192} a^{16} - \frac{1}{1024} a^{14} - \frac{1}{512} a^{10} - \frac{1}{16} a^{6} + \frac{13}{128} a^{4} + \frac{1}{16} a^{2} - \frac{1}{2}$, $\frac{1}{8192} a^{17} - \frac{1}{512} a^{11} + \frac{13}{128} a^{5}$, $\frac{1}{16384} a^{18} - \frac{1}{1024} a^{12} + \frac{13}{256} a^{6}$, $\frac{1}{16384} a^{19} - \frac{1}{1024} a^{13} - \frac{3}{256} a^{7} - \frac{1}{2} a$, $\frac{1}{131072} a^{20} - \frac{1}{65536} a^{18} + \frac{1}{32768} a^{16} - \frac{7}{8192} a^{14} - \frac{3}{4096} a^{12} - \frac{5}{2048} a^{10} + \frac{13}{2048} a^{8} + \frac{59}{1024} a^{6} + \frac{5}{512} a^{4} - \frac{13}{64} a^{2} - \frac{3}{8}$, $\frac{1}{131072} a^{21} - \frac{1}{65536} a^{19} + \frac{1}{32768} a^{17} + \frac{1}{8192} a^{15} - \frac{3}{4096} a^{13} - \frac{5}{2048} a^{11} + \frac{13}{2048} a^{9} - \frac{5}{1024} a^{7} + \frac{5}{512} a^{5} + \frac{15}{64} a^{3} + \frac{1}{8} a$, $\frac{1}{262144} a^{22} - \frac{1}{32768} a^{16} - \frac{1}{4096} a^{14} - \frac{1}{512} a^{12} + \frac{11}{4096} a^{10} + \frac{1}{256} a^{8} + \frac{13}{512} a^{4} - \frac{13}{64} a^{2} + \frac{1}{8}$, $\frac{1}{23330816} a^{23} - \frac{31}{11665408} a^{21} + \frac{95}{5832704} a^{19} - \frac{9}{729088} a^{17} - \frac{217}{729088} a^{15} + \frac{501}{364544} a^{13} - \frac{1047}{364544} a^{11} - \frac{203}{182272} a^{9} + \frac{1499}{91136} a^{7} - \frac{1873}{22784} a^{5} - \frac{9}{2848} a^{3} + \frac{43}{356} a$, $\frac{1}{46661632} a^{24} - \frac{31}{23330816} a^{22} + \frac{3}{5832704} a^{20} + \frac{53}{5832704} a^{18} - \frac{167}{2916352} a^{16} + \frac{103}{182272} a^{14} - \frac{513}{729088} a^{12} - \frac{25}{364544} a^{10} + \frac{171}{91136} a^{8} - \frac{3301}{91136} a^{6} + \frac{4111}{45568} a^{4} - \frac{991}{5696} a^{2} - \frac{1}{8}$, $\frac{1}{93323264} a^{25} - \frac{65}{23330816} a^{21} - \frac{7}{2916352} a^{19} - \frac{37}{5832704} a^{17} + \frac{271}{1458176} a^{15} + \frac{1001}{1458176} a^{13} + \frac{313}{364544} a^{11} - \frac{77}{364544} a^{9} + \frac{2627}{91136} a^{7} + \frac{8135}{91136} a^{5} - \frac{1015}{11392} a^{3} - \frac{93}{1424} a$, $\frac{1}{101162418176} a^{26} - \frac{251}{25290604544} a^{24} - \frac{34521}{25290604544} a^{22} - \frac{3697}{3161325568} a^{20} + \frac{40641}{6322651136} a^{18} + \frac{37115}{1580662784} a^{16} - \frac{1232871}{1580662784} a^{14} - \frac{18183}{12348928} a^{12} + \frac{434155}{395165696} a^{10} - \frac{393}{1110016} a^{8} - \frac{4474695}{98791424} a^{6} - \frac{696325}{12348928} a^{4} + \frac{228897}{1543616} a^{2} - \frac{239}{542}$, $\frac{1}{202324836352} a^{27} - \frac{251}{50581209088} a^{25} + \frac{167}{50581209088} a^{23} - \frac{1805}{1580662784} a^{21} - \frac{40659}{12645302272} a^{19} - \frac{56651}{3161325568} a^{17} - \frac{272447}{3161325568} a^{15} + \frac{123097}{197582848} a^{13} - \frac{188061}{790331392} a^{11} + \frac{254993}{197582848} a^{9} - \frac{233003}{197582848} a^{7} + \frac{333163}{6174464} a^{5} - \frac{36123}{385904} a^{3} - \frac{135219}{385904} a$, $\frac{1}{40869616943104} a^{28} + \frac{39}{10217404235776} a^{26} - \frac{56849}{10217404235776} a^{24} - \frac{25823}{1277175529472} a^{22} - \frac{7019883}{2554351058944} a^{20} - \frac{2229429}{638587764736} a^{18} - \frac{14844487}{638587764736} a^{16} + \frac{29381343}{79823470592} a^{14} - \frac{293195789}{159646941184} a^{12} - \frac{11350877}{39911735296} a^{10} - \frac{36945115}{39911735296} a^{8} - \frac{151742105}{4988966912} a^{6} - \frac{15453489}{155905216} a^{4} + \frac{13897585}{77952608} a^{2} + \frac{10159}{109484}$, $\frac{1}{40869616943104} a^{29} - \frac{23}{20434808471552} a^{27} - \frac{6147}{10217404235776} a^{25} + \frac{98809}{5108702117888} a^{23} + \frac{8597141}{2554351058944} a^{21} + \frac{961509}{1277175529472} a^{19} + \frac{8204319}{638587764736} a^{17} + \frac{88986711}{319293882368} a^{15} - \frac{290181141}{159646941184} a^{13} + \frac{195772255}{79823470592} a^{11} + \frac{120441771}{39911735296} a^{9} - \frac{352642845}{19955867648} a^{7} + \frac{108484531}{1247241728} a^{5} + \frac{483861}{155905216} a^{3} - \frac{12122731}{38976304} a$, $\frac{1}{18058685267633370027771879004596913307648} a^{30} - \frac{33426120723022209934278705}{4514671316908342506942969751149228326912} a^{28} + \frac{8514009375840411336909516153}{4514671316908342506942969751149228326912} a^{26} - \frac{2064620027696687062443008191317}{282166957306771406683935609446826770432} a^{24} - \frac{1925040015305416971791718807732909}{1128667829227085626735742437787307081728} a^{22} + \frac{959002368601018176698460245834823}{282166957306771406683935609446826770432} a^{20} - \frac{5424142441525206131090966847780141}{282166957306771406683935609446826770432} a^{18} + \frac{4096988610150697986606332837283}{68888417311223487959945217150104192} a^{16} + \frac{8721893348915772610552412518070285}{70541739326692851670983902361706692608} a^{14} + \frac{17201895811751541345068693525011467}{17635434831673212917745975590426673152} a^{12} - \frac{4006046235884328251471820179030725}{17635434831673212917745975590426673152} a^{10} - \frac{7845887064152525422685028821622675}{1102214676979575807359123474401667072} a^{8} - \frac{92225121562848252117369292542405291}{2204429353959151614718246948803334144} a^{6} - \frac{7224663265676308749019289574804789}{68888417311223487959945217150104192} a^{4} - \frac{695702097730407663753245720009945}{4305526081951467997496576071881512} a^{2} + \frac{4151748860656034332676485636381}{48376697550016494353894113167208}$, $\frac{1}{18058685267633370027771879004596913307648} a^{31} - \frac{33426120723022209934278705}{4514671316908342506942969751149228326912} a^{29} + \frac{8514009375840411336909516153}{4514671316908342506942969751149228326912} a^{27} + \frac{1917847138358687669350747763267}{564333914613542813367871218893653540864} a^{25} + \frac{10027886695242802364045718955411}{1128667829227085626735742437787307081728} a^{23} + \frac{318011126063299626509363246369317}{282166957306771406683935609446826770432} a^{21} - \frac{3102060959124414402104049415754157}{282166957306771406683935609446826770432} a^{19} - \frac{472353888947468893408182349319029}{35270869663346425835491951180853346304} a^{17} - \frac{8548587676440115873787785882622971}{70541739326692851670983902361706692608} a^{15} - \frac{25938024228475667495016381891846267}{17635434831673212917745975590426673152} a^{13} + \frac{105973055867073768609179440181955}{17635434831673212917745975590426673152} a^{11} + \frac{12650923548810862784217655908592637}{2204429353959151614718246948803334144} a^{9} - \frac{63755435054663545190102606943503383}{2204429353959151614718246948803334144} a^{7} + \frac{11206005842494307141881399070437999}{551107338489787903679561737200833536} a^{5} + \frac{2009086908336707658824556968883753}{68888417311223487959945217150104192} a^{3} + \frac{878094302653071532483860018631541}{8611052163902935994993152143763024} a$

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

\(\Q(\sqrt{85}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{5}) \), 4.4.39304000.2, \(\Q(\sqrt{5}, \sqrt{17})\), 4.4.39304000.1, 4.4.4913.1, 4.4.122825.1, 4.4.2312000.1, 4.4.8000.1, 8.8.1544804416000000.3, 8.8.15085980625.1, 8.8.5345344000000.2, 8.0.1050467002880000.6, 8.0.1680747204608.1, 8.0.6411541765625.1, 8.0.6411541765625.2, 16.16.2386420683693101056000000000000.1, 16.0.1103480924139689928294400000000.2, 16.0.41107867812353742431640625.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ 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
$2$2.4.6.4$x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.4$x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.4$x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.4$x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.4$x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.4$x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.4$x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.4$x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
5Data not computed
17Data not computed