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Properties

Label 32.0.15693807744...0000.6
Degree $32$
Signature $[0, 16]$
Discriminant $2^{96}\cdot 5^{24}\cdot 7^{16}$
Root discriminant $70.77$
Ramified primes $2, 5, 7$
Class number $870400$ (GRH)
Class group $[2, 4, 80, 1360]$ (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, 2359296, 0, 39043072, 0, 180264960, 0, 436417280, 0, 570415104, 0, 471188096, 0, 267441600, 0, 109690049, 0, 33445200, 0, 7694204, 0, 1339752, 0, 175155, 0, 16800, 0, 1128, 0, 48, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 48*x^30 + 1128*x^28 + 16800*x^26 + 175155*x^24 + 1339752*x^22 + 7694204*x^20 + 33445200*x^18 + 109690049*x^16 + 267441600*x^14 + 471188096*x^12 + 570415104*x^10 + 436417280*x^8 + 180264960*x^6 + 39043072*x^4 + 2359296*x^2 + 65536)
 
gp: K = bnfinit(x^32 + 48*x^30 + 1128*x^28 + 16800*x^26 + 175155*x^24 + 1339752*x^22 + 7694204*x^20 + 33445200*x^18 + 109690049*x^16 + 267441600*x^14 + 471188096*x^12 + 570415104*x^10 + 436417280*x^8 + 180264960*x^6 + 39043072*x^4 + 2359296*x^2 + 65536, 1)
 

Normalized defining polynomial

\( x^{32} + 48 x^{30} + 1128 x^{28} + 16800 x^{26} + 175155 x^{24} + 1339752 x^{22} + 7694204 x^{20} + 33445200 x^{18} + 109690049 x^{16} + 267441600 x^{14} + 471188096 x^{12} + 570415104 x^{10} + 436417280 x^{8} + 180264960 x^{6} + 39043072 x^{4} + 2359296 x^{2} + 65536 \)

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:  \(156938077449417789520626992646455296000000000000000000000000=2^{96}\cdot 5^{24}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.77$
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:  \(560=2^{4}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{560}(1,·)$, $\chi_{560}(321,·)$, $\chi_{560}(267,·)$, $\chi_{560}(141,·)$, $\chi_{560}(407,·)$, $\chi_{560}(281,·)$, $\chi_{560}(27,·)$, $\chi_{560}(29,·)$, $\chi_{560}(547,·)$, $\chi_{560}(421,·)$, $\chi_{560}(167,·)$, $\chi_{560}(41,·)$, $\chi_{560}(43,·)$, $\chi_{560}(307,·)$, $\chi_{560}(181,·)$, $\chi_{560}(183,·)$, $\chi_{560}(309,·)$, $\chi_{560}(449,·)$, $\chi_{560}(323,·)$, $\chi_{560}(69,·)$, $\chi_{560}(461,·)$, $\chi_{560}(463,·)$, $\chi_{560}(209,·)$, $\chi_{560}(83,·)$, $\chi_{560}(169,·)$, $\chi_{560}(349,·)$, $\chi_{560}(223,·)$, $\chi_{560}(489,·)$, $\chi_{560}(363,·)$, $\chi_{560}(503,·)$, $\chi_{560}(447,·)$, $\chi_{560}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{10} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{19} + \frac{3}{8} a^{11} - \frac{1}{2} a^{7} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{20} - \frac{1}{2} a^{16} + \frac{3}{16} a^{12} - \frac{1}{2} a^{10} - \frac{1}{4} a^{8} + \frac{1}{16} a^{4}$, $\frac{1}{32} a^{21} - \frac{1}{4} a^{17} - \frac{13}{32} a^{13} - \frac{1}{4} a^{11} + \frac{3}{8} a^{9} - \frac{1}{2} a^{7} + \frac{1}{32} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{64} a^{22} - \frac{1}{8} a^{18} - \frac{1}{2} a^{16} - \frac{13}{64} a^{14} + \frac{3}{8} a^{12} - \frac{5}{16} a^{10} + \frac{1}{4} a^{8} + \frac{1}{64} a^{6} + \frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{128} a^{23} - \frac{1}{16} a^{19} - \frac{1}{4} a^{17} + \frac{51}{128} a^{15} - \frac{5}{16} a^{13} + \frac{11}{32} a^{11} - \frac{3}{8} a^{9} + \frac{1}{128} a^{7} + \frac{1}{8} a^{5} + \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{256} a^{24} - \frac{1}{32} a^{20} - \frac{1}{8} a^{18} - \frac{77}{256} a^{16} - \frac{5}{32} a^{14} - \frac{21}{64} a^{12} - \frac{3}{16} a^{10} - \frac{127}{256} a^{8} - \frac{7}{16} a^{6} + \frac{1}{16} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{512} a^{25} - \frac{1}{64} a^{21} - \frac{1}{16} a^{19} - \frac{77}{512} a^{17} - \frac{5}{64} a^{15} + \frac{43}{128} a^{13} + \frac{13}{32} a^{11} + \frac{129}{512} a^{9} - \frac{7}{32} a^{7} - \frac{15}{32} a^{5} + \frac{3}{8} a^{3}$, $\frac{1}{1024} a^{26} - \frac{1}{128} a^{22} - \frac{1}{32} a^{20} - \frac{77}{1024} a^{18} - \frac{5}{128} a^{16} + \frac{43}{256} a^{14} + \frac{13}{64} a^{12} - \frac{383}{1024} a^{10} - \frac{7}{64} a^{8} - \frac{15}{64} a^{6} - \frac{5}{16} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2048} a^{27} - \frac{1}{256} a^{23} - \frac{1}{64} a^{21} - \frac{77}{2048} a^{19} - \frac{5}{256} a^{17} - \frac{213}{512} a^{15} + \frac{13}{128} a^{13} + \frac{641}{2048} a^{11} + \frac{57}{128} a^{9} - \frac{15}{128} a^{7} + \frac{11}{32} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{323584} a^{28} + \frac{13}{40448} a^{24} + \frac{53}{10112} a^{22} + \frac{7475}{323584} a^{20} - \frac{3461}{40448} a^{18} - \frac{21825}{80896} a^{16} + \frac{6777}{20224} a^{14} - \frac{128319}{323584} a^{12} - \frac{9735}{20224} a^{10} - \frac{1057}{2528} a^{8} - \frac{623}{2528} a^{6} + \frac{153}{632} a^{4} + \frac{51}{158} a^{2} + \frac{21}{79}$, $\frac{1}{647168} a^{29} + \frac{13}{80896} a^{25} + \frac{53}{20224} a^{23} + \frac{7475}{647168} a^{21} - \frac{3461}{80896} a^{19} - \frac{21825}{161792} a^{17} + \frac{6777}{40448} a^{15} - \frac{128319}{647168} a^{13} + \frac{10489}{40448} a^{11} + \frac{1471}{5056} a^{9} + \frac{1905}{5056} a^{7} + \frac{153}{1264} a^{5} + \frac{51}{316} a^{3} - \frac{29}{79} a$, $\frac{1}{28609011724986074536124433741148889836404736} a^{30} + \frac{2107281496622078069378366286412976417}{3576126465623259317015554217643611229550592} a^{28} - \frac{1449712813286389198406819310051641858729}{3576126465623259317015554217643611229550592} a^{26} - \frac{1559849707053297117528632475259266320527}{894031616405814829253888554410902807387648} a^{24} + \frac{53331493514643676291335221269051723610547}{28609011724986074536124433741148889836404736} a^{22} + \frac{17425752500843862847855641775516556283527}{1788063232811629658507777108821805614775296} a^{20} - \frac{23406831460992854244307200899378428838933}{7152252931246518634031108435287222459101184} a^{18} + \frac{29377255241943984093272468329768944876539}{1788063232811629658507777108821805614775296} a^{16} + \frac{1957896690610294059322399205866484855769345}{28609011724986074536124433741148889836404736} a^{14} - \frac{1618518617910553108992009210109076739128365}{3576126465623259317015554217643611229550592} a^{12} + \frac{15126947771077145098838133098549418308261}{1788063232811629658507777108821805614775296} a^{10} + \frac{151102427614781187619811190639413716324087}{447015808202907414626944277205451403693824} a^{8} - \frac{13157541765866654141062984552126758992695}{111753952050726853656736069301362850923456} a^{6} - \frac{13945142262387176979747126323124102256627}{27938488012681713414184017325340712730864} a^{4} - \frac{2600965418867678295154107543370497779453}{6984622003170428353546004331335178182716} a^{2} - \frac{102724643999952174695437240972561465969}{1746155500792607088386501082833794545679}$, $\frac{1}{57218023449972149072248867482297779672809472} a^{31} + \frac{2107281496622078069378366286412976417}{7152252931246518634031108435287222459101184} a^{29} - \frac{1449712813286389198406819310051641858729}{7152252931246518634031108435287222459101184} a^{27} - \frac{1559849707053297117528632475259266320527}{1788063232811629658507777108821805614775296} a^{25} + \frac{53331493514643676291335221269051723610547}{57218023449972149072248867482297779672809472} a^{23} + \frac{17425752500843862847855641775516556283527}{3576126465623259317015554217643611229550592} a^{21} - \frac{23406831460992854244307200899378428838933}{14304505862493037268062216870574444918202368} a^{19} + \frac{29377255241943984093272468329768944876539}{3576126465623259317015554217643611229550592} a^{17} - \frac{26651115034375780476802034535282404980635391}{57218023449972149072248867482297779672809472} a^{15} + \frac{1957607847712706208023545007534534490422227}{7152252931246518634031108435287222459101184} a^{13} + \frac{15126947771077145098838133098549418308261}{3576126465623259317015554217643611229550592} a^{11} + \frac{151102427614781187619811190639413716324087}{894031616405814829253888554410902807387648} a^{9} - \frac{13157541765866654141062984552126758992695}{223507904101453707313472138602725701846912} a^{7} + \frac{13993345750294536434436891002216610474237}{55876976025363426828368034650681425461728} a^{5} - \frac{2600965418867678295154107543370497779453}{13969244006340856707092008662670356365432} a^{3} + \frac{821715428396327456845531920930616539855}{1746155500792607088386501082833794545679} a$

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

Class group and class number

$C_{2}\times C_{4}\times C_{80}\times C_{1360}$, which has order $870400$ (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:  \( 48613521256.81357 \) (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

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-70}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{2}, \sqrt{-35})\), 4.0.2508800.1, \(\Q(\zeta_{16})^+\), \(\Q(\sqrt{10}, \sqrt{-14})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-7})\), 4.0.100352.5, 4.4.51200.1, \(\Q(\sqrt{-7}, \sqrt{10})\), \(\Q(\sqrt{5}, \sqrt{-14})\), 4.0.12544000.2, 4.4.256000.1, 4.0.12544000.1, 4.4.256000.2, 4.4.8000.1, 4.0.392000.2, \(\Q(\zeta_{20})^+\), 4.0.98000.1, 8.0.6294077440000.6, 8.0.6146560000.2, 8.0.6294077440000.2, 8.0.6294077440000.7, 8.0.6294077440000.5, 8.8.2621440000.1, 8.0.10070523904.2, 8.0.157351936000000.68, 8.0.157351936000000.70, 8.0.153664000000.4, 8.0.9604000000.2, 8.0.157351936000000.83, 8.8.65536000000.1, \(\Q(\zeta_{40})^+\), 8.0.2458624000000.6, 8.0.157351936000000.65, 8.0.157351936000000.31, 8.0.2458624000000.2, 8.0.2458624000000.4, 16.0.39615410820716953600000000.1, 16.0.24759631762948096000000000000.1, 16.0.6044831973376000000000000.2, 16.0.396154108207169536000000000000.12, 16.0.396154108207169536000000000000.6, 16.0.396154108207169536000000000000.14, \(\Q(\zeta_{80})^+\)

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{8}$ R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
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}$