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Properties

Label 32.0.22300745198...0000.2
Degree $32$
Signature $[0, 16]$
Discriminant $2^{160}\cdot 5^{16}$
Root discriminant $71.55$
Ramified primes $2, 5$
Class number $578$ (GRH)
Class group $[17, 34]$ (GRH)
Galois group $C_2\times C_{16}$ (as 32T32)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4870849, 0, -256, 0, 5440, 0, -45696, 0, 201552, 0, -537472, 0, 940576, 0, -1136960, 0, 980628, 0, -615296, 0, 283360, 0, -95680, 0, 23400, 0, -4032, 0, 464, 0, -32, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 32*x^30 + 464*x^28 - 4032*x^26 + 23400*x^24 - 95680*x^22 + 283360*x^20 - 615296*x^18 + 980628*x^16 - 1136960*x^14 + 940576*x^12 - 537472*x^10 + 201552*x^8 - 45696*x^6 + 5440*x^4 - 256*x^2 + 4870849)
 
gp: K = bnfinit(x^32 - 32*x^30 + 464*x^28 - 4032*x^26 + 23400*x^24 - 95680*x^22 + 283360*x^20 - 615296*x^18 + 980628*x^16 - 1136960*x^14 + 940576*x^12 - 537472*x^10 + 201552*x^8 - 45696*x^6 + 5440*x^4 - 256*x^2 + 4870849, 1)
 

Normalized defining polynomial

\( x^{32} - 32 x^{30} + 464 x^{28} - 4032 x^{26} + 23400 x^{24} - 95680 x^{22} + 283360 x^{20} - 615296 x^{18} + 980628 x^{16} - 1136960 x^{14} + 940576 x^{12} - 537472 x^{10} + 201552 x^{8} - 45696 x^{6} + 5440 x^{4} - 256 x^{2} + 4870849 \)

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:  \(223007451985306231415357182726483615059804160000000000000000=2^{160}\cdot 5^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.55$
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:  $2, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(320=2^{6}\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{320}(1,·)$, $\chi_{320}(131,·)$, $\chi_{320}(11,·)$, $\chi_{320}(269,·)$, $\chi_{320}(149,·)$, $\chi_{320}(279,·)$, $\chi_{320}(281,·)$, $\chi_{320}(29,·)$, $\chi_{320}(159,·)$, $\chi_{320}(161,·)$, $\chi_{320}(291,·)$, $\chi_{320}(39,·)$, $\chi_{320}(41,·)$, $\chi_{320}(171,·)$, $\chi_{320}(51,·)$, $\chi_{320}(309,·)$, $\chi_{320}(189,·)$, $\chi_{320}(319,·)$, $\chi_{320}(69,·)$, $\chi_{320}(199,·)$, $\chi_{320}(201,·)$, $\chi_{320}(79,·)$, $\chi_{320}(81,·)$, $\chi_{320}(211,·)$, $\chi_{320}(91,·)$, $\chi_{320}(229,·)$, $\chi_{320}(109,·)$, $\chi_{320}(239,·)$, $\chi_{320}(241,·)$, $\chi_{320}(119,·)$, $\chi_{320}(121,·)$, $\chi_{320}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{987} a^{16} - \frac{16}{987} a^{14} + \frac{104}{987} a^{12} - \frac{352}{987} a^{10} - \frac{109}{329} a^{8} + \frac{15}{47} a^{6} + \frac{16}{47} a^{4} - \frac{64}{987} a^{2} + \frac{2}{987}$, $\frac{1}{2178309} a^{17} - \frac{832057}{2178309} a^{15} - \frac{589135}{2178309} a^{13} - \frac{821536}{2178309} a^{11} + \frac{29830}{726103} a^{9} + \frac{11906}{103729} a^{7} + \frac{39778}{103729} a^{5} - \frac{1035427}{2178309} a^{3} - \frac{589237}{2178309} a$, $\frac{1}{2178309} a^{18} - \frac{6}{726103} a^{16} - \frac{831905}{2178309} a^{14} + \frac{108067}{311187} a^{12} - \frac{894832}{2178309} a^{10} + \frac{153966}{726103} a^{8} - \frac{30846}{103729} a^{6} - \frac{293875}{2178309} a^{4} + \frac{617992}{2178309} a^{2} - \frac{233}{987}$, $\frac{1}{2178309} a^{19} - \frac{560768}{2178309} a^{15} + \frac{1043584}{2178309} a^{13} - \frac{434317}{2178309} a^{11} - \frac{35197}{726103} a^{9} - \frac{23996}{103729} a^{7} - \frac{505954}{2178309} a^{5} - \frac{84746}{311187} a^{3} - \frac{228952}{2178309} a$, $\frac{1}{2178309} a^{20} - \frac{190}{2178309} a^{16} + \frac{262524}{726103} a^{14} - \frac{948548}{2178309} a^{12} + \frac{797072}{2178309} a^{10} - \frac{278322}{726103} a^{8} - \frac{366913}{2178309} a^{6} + \frac{60916}{311187} a^{4} + \frac{132187}{311187} a^{2} - \frac{479}{987}$, $\frac{1}{2178309} a^{21} - \frac{66430}{311187} a^{15} + \frac{18470}{103729} a^{13} - \frac{13507}{46347} a^{11} + \frac{306657}{726103} a^{9} - \frac{784771}{2178309} a^{7} + \frac{17725}{311187} a^{5} + \frac{80663}{726103} a^{3} + \frac{259885}{2178309} a$, $\frac{1}{2178309} a^{22} + \frac{667}{2178309} a^{16} - \frac{528035}{2178309} a^{14} - \frac{127219}{2178309} a^{12} + \frac{374842}{2178309} a^{10} - \frac{579520}{2178309} a^{8} + \frac{123661}{311187} a^{6} - \frac{42929}{726103} a^{4} + \frac{952883}{2178309} a^{2} + \frac{422}{987}$, $\frac{1}{2178309} a^{23} - \frac{144973}{311187} a^{15} + \frac{243402}{726103} a^{13} - \frac{594514}{2178309} a^{11} + \frac{723302}{2178309} a^{9} - \frac{50033}{311187} a^{7} + \frac{115957}{726103} a^{5} + \frac{352913}{726103} a^{3} - \frac{45928}{311187} a$, $\frac{1}{2178309} a^{24} + \frac{409}{2178309} a^{16} - \frac{265151}{2178309} a^{14} + \frac{20454}{103729} a^{12} + \frac{28978}{103729} a^{10} + \frac{954106}{2178309} a^{8} - \frac{23084}{726103} a^{6} + \frac{59382}{726103} a^{4} + \frac{17898}{726103} a^{2} - \frac{67}{987}$, $\frac{1}{2178309} a^{25} + \frac{229958}{2178309} a^{15} - \frac{8650}{46347} a^{13} - \frac{1021133}{2178309} a^{11} - \frac{113480}{311187} a^{9} + \frac{16879}{726103} a^{7} + \frac{173139}{726103} a^{5} + \frac{135913}{311187} a^{3} - \frac{134605}{311187} a$, $\frac{1}{2178309} a^{26} + \frac{430}{2178309} a^{16} + \frac{1087589}{2178309} a^{14} - \frac{930646}{2178309} a^{12} - \frac{597937}{2178309} a^{10} + \frac{347929}{726103} a^{8} + \frac{34098}{726103} a^{6} + \frac{10114}{311187} a^{4} + \frac{225901}{726103} a^{2} - \frac{208}{987}$, $\frac{1}{2178309} a^{27} - \frac{182962}{726103} a^{15} - \frac{13640}{103729} a^{13} - \frac{74505}{726103} a^{11} - \frac{135220}{726103} a^{9} - \frac{223915}{726103} a^{7} + \frac{42349}{311187} a^{5} - \frac{642032}{2178309} a^{3} + \frac{229010}{2178309} a$, $\frac{1}{2178309} a^{28} + \frac{219}{726103} a^{16} - \frac{121964}{726103} a^{14} + \frac{97641}{726103} a^{12} + \frac{8235}{726103} a^{10} + \frac{142447}{726103} a^{8} - \frac{123176}{311187} a^{6} + \frac{1026460}{2178309} a^{4} - \frac{88798}{2178309} a^{2} - \frac{163}{329}$, $\frac{1}{2178309} a^{29} - \frac{153334}{726103} a^{15} - \frac{18304}{103729} a^{13} - \frac{21275}{103729} a^{11} + \frac{21274}{103729} a^{9} + \frac{60310}{311187} a^{7} - \frac{1035047}{2178309} a^{5} + \frac{554333}{2178309} a^{3} + \frac{162931}{726103} a$, $\frac{1}{2178309} a^{30} - \frac{946}{2178309} a^{16} + \frac{983956}{2178309} a^{14} - \frac{627749}{2178309} a^{12} + \frac{53908}{2178309} a^{10} + \frac{614179}{2178309} a^{8} - \frac{200801}{2178309} a^{6} + \frac{137210}{2178309} a^{4} - \frac{572774}{2178309} a^{2} + \frac{416}{987}$, $\frac{1}{2178309} a^{31} + \frac{75861}{726103} a^{15} - \frac{100785}{726103} a^{13} + \frac{179055}{726103} a^{11} + \frac{317668}{2178309} a^{9} + \frac{1066423}{2178309} a^{7} - \frac{359209}{2178309} a^{5} + \frac{7254}{103729} a^{3} - \frac{343765}{726103} a$

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

Class group and class number

$C_{17}\times C_{34}$, which has order $578$ (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:  \( 67435414427975.664 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{16}$ (as 32T32):

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 $C_2\times C_{16}$
Character table for $C_2\times C_{16}$ is not computed

Intermediate fields

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\zeta_{16})^+\), 4.0.51200.2, 8.0.10485760000.2, \(\Q(\zeta_{32})^+\), 8.0.1342177280000.1, 16.0.7205759403792793600000000.1, 16.0.604462909807314587353088.1, 16.16.236118324143482260684800000000.1

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 $16^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{4}$ $16^{2}$ $16^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ $16^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{4}$ $16^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ $16^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{4}$ $16^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ $16^{2}$ $16^{2}$

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