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Properties

Label 16.0.47526285764...7681.5
Degree $16$
Signature $[0, 8]$
Discriminant $13^{8}\cdot 17^{12}$
Root discriminant $30.19$
Ramified primes $13, 17$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^2 : C_4$ (as 16T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4096, 0, -2112, 0, 2484, 0, -949, 0, 866, 0, 11, 0, -43, 0, -2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^14 - 43*x^12 + 11*x^10 + 866*x^8 - 949*x^6 + 2484*x^4 - 2112*x^2 + 4096)
 
gp: K = bnfinit(x^16 - 2*x^14 - 43*x^12 + 11*x^10 + 866*x^8 - 949*x^6 + 2484*x^4 - 2112*x^2 + 4096, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{14} - 43 x^{12} + 11 x^{10} + 866 x^{8} - 949 x^{6} + 2484 x^{4} - 2112 x^{2} + 4096 \)

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:  \(475262857646065983187681=13^{8}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.19$
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:  $13, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{4} + \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} - \frac{1}{4} a^{7} - \frac{1}{16} a^{5} + \frac{1}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{64} a^{12} - \frac{1}{64} a^{10} - \frac{1}{16} a^{8} - \frac{1}{4} a^{7} - \frac{1}{64} a^{6} + \frac{1}{64} a^{4} + \frac{1}{16} a^{2} - \frac{1}{4} a$, $\frac{1}{128} a^{13} - \frac{1}{128} a^{12} + \frac{3}{128} a^{11} - \frac{7}{128} a^{10} - \frac{1}{16} a^{9} + \frac{3}{32} a^{8} + \frac{15}{128} a^{7} + \frac{1}{128} a^{6} + \frac{29}{128} a^{5} - \frac{25}{128} a^{4} - \frac{7}{16} a^{3} + \frac{13}{32} a^{2} - \frac{1}{8} a$, $\frac{1}{28147325824} a^{14} - \frac{108096245}{14073662912} a^{12} - \frac{1}{32} a^{11} + \frac{100370189}{28147325824} a^{10} + \frac{1}{32} a^{9} + \frac{1007854075}{28147325824} a^{8} - \frac{1}{8} a^{7} + \frac{1416343525}{14073662912} a^{6} + \frac{1}{32} a^{5} - \frac{5922548781}{28147325824} a^{4} + \frac{15}{32} a^{3} - \frac{2537388207}{7036831456} a^{2} - \frac{3}{8} a - \frac{99456098}{219900983}$, $\frac{1}{225178606592} a^{15} - \frac{108096245}{112589303296} a^{13} - \frac{1}{128} a^{12} + \frac{3618785917}{225178606592} a^{11} - \frac{7}{128} a^{10} + \frac{4526269803}{225178606592} a^{9} - \frac{1}{32} a^{8} - \frac{12657319387}{112589303296} a^{7} + \frac{1}{128} a^{6} - \frac{23514627421}{225178606592} a^{5} - \frac{25}{128} a^{4} + \frac{17693502229}{56294651648} a^{3} + \frac{1}{32} a^{2} + \frac{20988787}{3518415728} a - \frac{1}{2}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 267164.850896 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_4$ (as 16T10):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_2^2 : C_4$
Character table for $C_2^2 : C_4$

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{221}) \), \(\Q(\sqrt{13}, \sqrt{17})\), 4.0.63869.1 x2, 4.0.830297.1 x2, 4.0.2873.1 x2, 4.0.3757.1 x2, 4.4.4913.1, 4.4.830297.1, 8.0.689393108209.3, 8.0.2385443281.2, 8.8.689393108209.1, 8.0.4079249161.1 x2, 8.0.689393108209.2 x2

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$17$17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$