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Properties

Label 16.0.64467159966...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{20}\cdot 5^{12}\cdot 17^{8}\cdot 19^{2}$
Root discriminant $47.38$
Ramified primes $2, 5, 17, 19$
Class number $104$ (GRH)
Class group $[2, 2, 26]$ (GRH)
Galois group $C_2^6.C_2^2$ (as 16T528)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32661, -59964, 109408, -109846, 55825, -15830, 4044, -882, -219, 518, -276, 42, -27, 16, 4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 4*x^14 + 16*x^13 - 27*x^12 + 42*x^11 - 276*x^10 + 518*x^9 - 219*x^8 - 882*x^7 + 4044*x^6 - 15830*x^5 + 55825*x^4 - 109846*x^3 + 109408*x^2 - 59964*x + 32661)
 
gp: K = bnfinit(x^16 - 4*x^15 + 4*x^14 + 16*x^13 - 27*x^12 + 42*x^11 - 276*x^10 + 518*x^9 - 219*x^8 - 882*x^7 + 4044*x^6 - 15830*x^5 + 55825*x^4 - 109846*x^3 + 109408*x^2 - 59964*x + 32661, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 4 x^{14} + 16 x^{13} - 27 x^{12} + 42 x^{11} - 276 x^{10} + 518 x^{9} - 219 x^{8} - 882 x^{7} + 4044 x^{6} - 15830 x^{5} + 55825 x^{4} - 109846 x^{3} + 109408 x^{2} - 59964 x + 32661 \)

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:  \(644671599667456000000000000=2^{20}\cdot 5^{12}\cdot 17^{8}\cdot 19^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.38$
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:  $2, 5, 17, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{6} a^{8} + \frac{1}{3} a^{7} + \frac{1}{6} a^{6} + \frac{1}{3} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} + \frac{1}{6} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{6} a$, $\frac{1}{6} a^{10} + \frac{1}{3} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{11} + \frac{1}{3} a^{3} - \frac{1}{2} a$, $\frac{1}{6} a^{12} + \frac{1}{3} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{13} + \frac{1}{3} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{19026} a^{14} - \frac{13}{1359} a^{13} + \frac{41}{19026} a^{12} - \frac{149}{19026} a^{11} - \frac{19}{3171} a^{10} - \frac{146}{9513} a^{9} - \frac{577}{9513} a^{8} + \frac{4288}{9513} a^{7} - \frac{1}{21} a^{6} - \frac{4198}{9513} a^{5} + \frac{3179}{19026} a^{4} - \frac{1672}{9513} a^{3} - \frac{7361}{19026} a^{2} - \frac{5}{906} a + \frac{344}{1057}$, $\frac{1}{1206129263694834415794104394} a^{15} - \frac{2410738129616624034301}{201021543949139069299017399} a^{14} - \frac{96746629952054064392055353}{1206129263694834415794104394} a^{13} + \frac{19044896347362152105772112}{603064631847417207897052197} a^{12} - \frac{66157067942573389774911607}{1206129263694834415794104394} a^{11} + \frac{619242160484911523648195}{7987611017846585535060294} a^{10} + \frac{95436097433293277169796583}{1206129263694834415794104394} a^{9} - \frac{35812846507799382734465027}{1206129263694834415794104394} a^{8} + \frac{368639378397336012232701667}{1206129263694834415794104394} a^{7} - \frac{210186324662495779689923105}{1206129263694834415794104394} a^{6} - \frac{51181186978084514089553887}{603064631847417207897052197} a^{5} - \frac{522183323660431973674261681}{1206129263694834415794104394} a^{4} + \frac{31569194749878963370450517}{67007181316379689766339133} a^{3} + \frac{278385007826524017255044759}{1206129263694834415794104394} a^{2} - \frac{60886124645820449807136008}{201021543949139069299017399} a - \frac{4988780823768619237030135}{67007181316379689766339133}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{26}$, which has order $104$ (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:  \( 108588.634681 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^6.C_2^2$ (as 16T528):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $C_2^6.C_2^2$
Character table for $C_2^6.C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{85}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{5}, \sqrt{17})\), 8.0.15868990000.1, 8.0.25390384000000.1, 8.8.83521000000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.8.2$x^{8} + 2 x^{7} + 8 x^{2} + 48$$2$$4$$8$$C_2^2:C_4$$[2, 2]^{4}$
5Data not computed
$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}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$