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Properties

Label 16.0.52891851109...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{8}\cdot 13^{4}\cdot 41^{4}$
Root discriminant $30.39$
Ramified primes $2, 5, 13, 41$
Class number $32$ (GRH)
Class group $[2, 2, 8]$ (GRH)
Galois group $C_4^2:C_2^2$ (as 16T117)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8761, -27758, 55137, -75868, 77687, -64996, 46669, -28678, 15554, -7522, 3197, -1200, 405, -116, 29, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 29*x^14 - 116*x^13 + 405*x^12 - 1200*x^11 + 3197*x^10 - 7522*x^9 + 15554*x^8 - 28678*x^7 + 46669*x^6 - 64996*x^5 + 77687*x^4 - 75868*x^3 + 55137*x^2 - 27758*x + 8761)
 
gp: K = bnfinit(x^16 - 6*x^15 + 29*x^14 - 116*x^13 + 405*x^12 - 1200*x^11 + 3197*x^10 - 7522*x^9 + 15554*x^8 - 28678*x^7 + 46669*x^6 - 64996*x^5 + 77687*x^4 - 75868*x^3 + 55137*x^2 - 27758*x + 8761, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 29 x^{14} - 116 x^{13} + 405 x^{12} - 1200 x^{11} + 3197 x^{10} - 7522 x^{9} + 15554 x^{8} - 28678 x^{7} + 46669 x^{6} - 64996 x^{5} + 77687 x^{4} - 75868 x^{3} + 55137 x^{2} - 27758 x + 8761 \)

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:  \(528918511098265600000000=2^{24}\cdot 5^{8}\cdot 13^{4}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.39$
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:  $2, 5, 13, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{11} + \frac{1}{6} a^{10} - \frac{1}{2} a^{9} + \frac{1}{6} a^{8} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} + \frac{1}{3} a^{5} + \frac{1}{4} a^{3} + \frac{5}{12} a + \frac{1}{6}$, $\frac{1}{72} a^{14} + \frac{1}{36} a^{13} + \frac{1}{36} a^{12} - \frac{1}{12} a^{11} - \frac{17}{72} a^{10} - \frac{17}{36} a^{9} - \frac{2}{9} a^{8} + \frac{1}{18} a^{7} + \frac{1}{36} a^{6} - \frac{17}{36} a^{5} + \frac{5}{24} a^{4} + \frac{1}{4} a^{3} + \frac{1}{36} a^{2} + \frac{1}{4} a - \frac{29}{72}$, $\frac{1}{921282783576596259261048} a^{15} + \frac{548234579294809932859}{230320695894149064815262} a^{14} - \frac{104387579920707484389}{8530396144227743141306} a^{13} - \frac{29698311691800130552735}{460641391788298129630524} a^{12} - \frac{53844830986894714434935}{921282783576596259261048} a^{11} + \frac{25576351713047934850633}{230320695894149064815262} a^{10} - \frac{716684618149763076023}{12795594216341614711959} a^{9} + \frac{3221868244524273178801}{115160347947074532407631} a^{8} - \frac{186858579372720282637885}{460641391788298129630524} a^{7} - \frac{6907532982113004682299}{17060792288455486282612} a^{6} - \frac{253403966053819572392957}{921282783576596259261048} a^{5} - \frac{431286043927565205661}{38386782649024844135877} a^{4} + \frac{54162941669214402178132}{115160347947074532407631} a^{3} + \frac{30558437647222830094673}{460641391788298129630524} a^{2} + \frac{311554168616611232828233}{921282783576596259261048} a - \frac{108356651049004765044029}{460641391788298129630524}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{8}$, which has order $32$ (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:  \( 3710.59482488 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2:C_2^2$ (as 16T117):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $C_4^2:C_2^2$
Character table for $C_4^2:C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), 4.0.65600.5, \(\Q(\sqrt{2}, \sqrt{5})\), 4.0.65600.2, 8.0.4303360000.3, 8.8.432640000.1, 8.0.727267840000.21

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$41$41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$