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Properties

Label 16.0.17828665137...0000.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 19^{8}$
Root discriminant $50.49$
Ramified primes $2, 3, 5, 19$
Class number $2048$ (GRH)
Class group $[2, 4, 16, 16]$ (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![3167221, -2467436, 3402871, -2095374, 1646891, -829962, 475958, -201062, 91167, -32324, 11912, -3486, 1036, -238, 54, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 54*x^14 - 238*x^13 + 1036*x^12 - 3486*x^11 + 11912*x^10 - 32324*x^9 + 91167*x^8 - 201062*x^7 + 475958*x^6 - 829962*x^5 + 1646891*x^4 - 2095374*x^3 + 3402871*x^2 - 2467436*x + 3167221)
 
gp: K = bnfinit(x^16 - 8*x^15 + 54*x^14 - 238*x^13 + 1036*x^12 - 3486*x^11 + 11912*x^10 - 32324*x^9 + 91167*x^8 - 201062*x^7 + 475958*x^6 - 829962*x^5 + 1646891*x^4 - 2095374*x^3 + 3402871*x^2 - 2467436*x + 3167221, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 54 x^{14} - 238 x^{13} + 1036 x^{12} - 3486 x^{11} + 11912 x^{10} - 32324 x^{9} + 91167 x^{8} - 201062 x^{7} + 475958 x^{6} - 829962 x^{5} + 1646891 x^{4} - 2095374 x^{3} + 3402871 x^{2} - 2467436 x + 3167221 \)

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:  \(1782866513792016000000000000=2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.49$
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:  $2, 3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1140=2^{2}\cdot 3\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{1140}(1,·)$, $\chi_{1140}(1027,·)$, $\chi_{1140}(77,·)$, $\chi_{1140}(911,·)$, $\chi_{1140}(721,·)$, $\chi_{1140}(533,·)$, $\chi_{1140}(343,·)$, $\chi_{1140}(797,·)$, $\chi_{1140}(607,·)$, $\chi_{1140}(419,·)$, $\chi_{1140}(229,·)$, $\chi_{1140}(1063,·)$, $\chi_{1140}(113,·)$, $\chi_{1140}(1139,·)$, $\chi_{1140}(949,·)$, $\chi_{1140}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{29} a^{13} + \frac{8}{29} a^{12} + \frac{8}{29} a^{11} + \frac{13}{29} a^{10} - \frac{5}{29} a^{9} - \frac{1}{29} a^{8} + \frac{10}{29} a^{7} - \frac{9}{29} a^{6} + \frac{14}{29} a^{5} - \frac{1}{29} a^{4} - \frac{11}{29} a^{3} - \frac{2}{29} a^{2} + \frac{2}{29}$, $\frac{1}{14745812611919141} a^{14} - \frac{7}{14745812611919141} a^{13} + \frac{6454258592663777}{14745812611919141} a^{12} + \frac{5511886279774852}{14745812611919141} a^{11} + \frac{2737134566943659}{14745812611919141} a^{10} + \frac{2144859687648196}{14745812611919141} a^{9} - \frac{1333945444023242}{14745812611919141} a^{8} - \frac{5885877719949114}{14745812611919141} a^{7} - \frac{4594343200086883}{14745812611919141} a^{6} - \frac{1018342896207646}{14745812611919141} a^{5} - \frac{1422392925042460}{14745812611919141} a^{4} - \frac{875532519242759}{14745812611919141} a^{3} - \frac{2177700863097423}{14745812611919141} a^{2} + \frac{459996440619049}{14745812611919141} a + \frac{3273618762534704}{14745812611919141}$, $\frac{1}{210113083907235840109} a^{15} + \frac{7117}{210113083907235840109} a^{14} + \frac{121502924299441738}{7245278755421925521} a^{13} - \frac{81976477840629932482}{210113083907235840109} a^{12} - \frac{34044731409591128829}{210113083907235840109} a^{11} + \frac{96853956666346386766}{210113083907235840109} a^{10} + \frac{54405389044346758883}{210113083907235840109} a^{9} - \frac{93051568811917899822}{210113083907235840109} a^{8} - \frac{36270652795655811230}{210113083907235840109} a^{7} + \frac{34455887722558071548}{210113083907235840109} a^{6} + \frac{95673234179766030719}{210113083907235840109} a^{5} + \frac{40731423636144867716}{210113083907235840109} a^{4} + \frac{55417347449158080543}{210113083907235840109} a^{3} - \frac{84155747137045986487}{210113083907235840109} a^{2} + \frac{86520400456016190725}{210113083907235840109} a - \frac{45718126108411224137}{210113083907235840109}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{16}\times C_{16}$, which has order $2048$ (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:  \( 3121.7160224989234 \) (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

\(\Q(\sqrt{-285}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-95}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-57}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{3}, \sqrt{-95})\), \(\Q(\sqrt{5}, \sqrt{-57})\), \(\Q(\sqrt{15}, \sqrt{-19})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{-19})\), \(\Q(\sqrt{5}, \sqrt{-19})\), \(\Q(\sqrt{15}, \sqrt{-57})\), 4.0.722000.3, \(\Q(\zeta_{15})^+\), 4.0.406125.2, \(\Q(\zeta_{20})^+\), 8.0.1688960160000.4, 8.0.42224004000000.63, 8.0.42224004000000.59, 8.0.42224004000000.54, \(\Q(\zeta_{60})^+\), 8.0.521284000000.1, 8.0.164937515625.1

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$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}$
$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}$
$19$19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$