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Properties

Label 16.0.47991168245...000.39
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{8}\cdot 5^{12}\cdot 17^{8}$
Root discriminant $95.52$
Ramified primes $2, 3, 5, 17$
Class number $130560$ (GRH)
Class group $[2, 2, 8, 4080]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![210560401, -76098832, 115796640, -33425040, 28648659, -6891612, 4275808, -854320, 426051, -70272, 30078, -4056, 1529, -164, 54, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 54*x^14 - 164*x^13 + 1529*x^12 - 4056*x^11 + 30078*x^10 - 70272*x^9 + 426051*x^8 - 854320*x^7 + 4275808*x^6 - 6891612*x^5 + 28648659*x^4 - 33425040*x^3 + 115796640*x^2 - 76098832*x + 210560401)
 
gp: K = bnfinit(x^16 - 4*x^15 + 54*x^14 - 164*x^13 + 1529*x^12 - 4056*x^11 + 30078*x^10 - 70272*x^9 + 426051*x^8 - 854320*x^7 + 4275808*x^6 - 6891612*x^5 + 28648659*x^4 - 33425040*x^3 + 115796640*x^2 - 76098832*x + 210560401, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 54 x^{14} - 164 x^{13} + 1529 x^{12} - 4056 x^{11} + 30078 x^{10} - 70272 x^{9} + 426051 x^{8} - 854320 x^{7} + 4275808 x^{6} - 6891612 x^{5} + 28648659 x^{4} - 33425040 x^{3} + 115796640 x^{2} - 76098832 x + 210560401 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(47991168245852798976000000000000=2^{32}\cdot 3^{8}\cdot 5^{12}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $95.52$
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:  $2, 3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2040=2^{3}\cdot 3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{2040}(1,·)$, $\chi_{2040}(203,·)$, $\chi_{2040}(1157,·)$, $\chi_{2040}(1223,·)$, $\chi_{2040}(137,·)$, $\chi_{2040}(1291,·)$, $\chi_{2040}(271,·)$, $\chi_{2040}(1427,·)$, $\chi_{2040}(1429,·)$, $\chi_{2040}(407,·)$, $\chi_{2040}(409,·)$, $\chi_{2040}(1699,·)$, $\chi_{2040}(679,·)$, $\chi_{2040}(1973,·)$, $\chi_{2040}(953,·)$, $\chi_{2040}(1021,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{1593} a^{12} + \frac{77}{531} a^{11} + \frac{109}{1593} a^{10} - \frac{107}{1593} a^{9} + \frac{8}{531} a^{8} - \frac{41}{531} a^{7} + \frac{103}{1593} a^{6} - \frac{8}{59} a^{5} + \frac{196}{1593} a^{4} - \frac{763}{1593} a^{3} + \frac{98}{531} a^{2} + \frac{34}{1593} a + \frac{289}{1593}$, $\frac{1}{1593} a^{13} - \frac{152}{1593} a^{11} + \frac{202}{1593} a^{10} - \frac{8}{59} a^{9} + \frac{58}{531} a^{8} - \frac{158}{1593} a^{7} - \frac{38}{531} a^{6} + \frac{178}{1593} a^{5} + \frac{158}{1593} a^{4} - \frac{92}{531} a^{3} + \frac{88}{1593} a^{2} - \frac{662}{1593} a + \frac{49}{531}$, $\frac{1}{11062195029} a^{14} - \frac{1800068}{11062195029} a^{13} - \frac{207940}{3687398343} a^{12} - \frac{31098076}{11062195029} a^{11} + \frac{484696927}{3687398343} a^{10} - \frac{370362628}{11062195029} a^{9} - \frac{177573023}{11062195029} a^{8} + \frac{1555676917}{11062195029} a^{7} + \frac{58984978}{3687398343} a^{6} - \frac{1081717069}{3687398343} a^{5} + \frac{26866205}{187494831} a^{4} - \frac{78890621}{187494831} a^{3} - \frac{169054786}{582220791} a^{2} + \frac{88422787}{1229132781} a - \frac{3731162320}{11062195029}$, $\frac{1}{3145683089218357346577275709} a^{15} - \frac{57196735782604}{285971189928941576961570519} a^{14} + \frac{20401159216555069066054}{349520343246484149619697301} a^{13} + \frac{2580148384525443035003}{101473648039301849889589539} a^{12} - \frac{143813100685662707693699485}{1048561029739452448859091903} a^{11} - \frac{182066492821625127352305934}{3145683089218357346577275709} a^{10} + \frac{490172586647415219150755}{285971189928941576961570519} a^{9} - \frac{181304384786819045280504155}{3145683089218357346577275709} a^{8} - \frac{9335353169047791157181329}{1048561029739452448859091903} a^{7} - \frac{108586888410854919699157882}{1048561029739452448859091903} a^{6} + \frac{6924611087009232608969045}{108471830662701977468181921} a^{5} - \frac{965706506462949195854858059}{3145683089218357346577275709} a^{4} - \frac{1311973801835740589079641710}{3145683089218357346577275709} a^{3} + \frac{358052383542341528673080821}{1048561029739452448859091903} a^{2} - \frac{307666045914712185348524248}{3145683089218357346577275709} a - \frac{433713814077858755031389}{1252761086904961109748019}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{8}\times C_{4080}$, which has order $130560$ (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:  $7$
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:  \( 16694.393243512957 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{-170}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-85}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-34}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-17}) \), \(\Q(\sqrt{2}, \sqrt{-85})\), \(\Q(\sqrt{5}, \sqrt{-34})\), \(\Q(\sqrt{10}, \sqrt{-17})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-17})\), \(\Q(\sqrt{5}, \sqrt{-17})\), \(\Q(\sqrt{10}, \sqrt{-34})\), \(\Q(\zeta_{15})^+\), 4.0.20808000.2, 4.4.72000.1, 4.0.5202000.2, 8.0.3421020160000.8, 8.0.432972864000000.134, 8.0.6927565824000000.128, 8.8.5184000000.1, 8.0.6927565824000000.151, 8.0.27060804000000.43, 8.0.6927565824000000.89

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$17$17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$