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Properties

Label 16.0.44664132968...161.12
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 61^{8}$
Root discriminant $53.47$
Ramified primes $13, 61$
Class number $80$ (GRH)
Class group $[2, 40]$ (GRH)
Galois group $D_4:C_4$ (as 16T26)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![53821, -51585, 50866, 11928, -9718, 15969, 7968, -4821, -812, 1734, 6, -531, 9, 87, -7, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 7*x^14 + 87*x^13 + 9*x^12 - 531*x^11 + 6*x^10 + 1734*x^9 - 812*x^8 - 4821*x^7 + 7968*x^6 + 15969*x^5 - 9718*x^4 + 11928*x^3 + 50866*x^2 - 51585*x + 53821)
 
gp: K = bnfinit(x^16 - 6*x^15 - 7*x^14 + 87*x^13 + 9*x^12 - 531*x^11 + 6*x^10 + 1734*x^9 - 812*x^8 - 4821*x^7 + 7968*x^6 + 15969*x^5 - 9718*x^4 + 11928*x^3 + 50866*x^2 - 51585*x + 53821, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 7 x^{14} + 87 x^{13} + 9 x^{12} - 531 x^{11} + 6 x^{10} + 1734 x^{9} - 812 x^{8} - 4821 x^{7} + 7968 x^{6} + 15969 x^{5} - 9718 x^{4} + 11928 x^{3} + 50866 x^{2} - 51585 x + 53821 \)

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:  \(4466413296812760910104974161=13^{12}\cdot 61^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.47$
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:  $13, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{531170702900974310074395392882044349} a^{15} + \frac{215544439341843970981772935699457623}{531170702900974310074395392882044349} a^{14} + \frac{63335593065695928096403583760611680}{531170702900974310074395392882044349} a^{13} - \frac{227929688225380037136946984160035638}{531170702900974310074395392882044349} a^{12} - \frac{150768578403407710985021124716353440}{531170702900974310074395392882044349} a^{11} - \frac{13364742126946891254794939293581171}{531170702900974310074395392882044349} a^{10} + \frac{250598424987971821490904165524371447}{531170702900974310074395392882044349} a^{9} + \frac{180697059060455903845262912379048335}{531170702900974310074395392882044349} a^{8} - \frac{129626562236511539702006599662070918}{531170702900974310074395392882044349} a^{7} - \frac{142039096110709491937677926904299609}{531170702900974310074395392882044349} a^{6} + \frac{20378182484682016931544598918651291}{531170702900974310074395392882044349} a^{5} + \frac{198328096985067178339035428733843155}{531170702900974310074395392882044349} a^{4} - \frac{244054933156712659942438012859056657}{531170702900974310074395392882044349} a^{3} + \frac{115292724671481592653847290971554902}{531170702900974310074395392882044349} a^{2} + \frac{190989029877645073273741515160521631}{531170702900974310074395392882044349} a - \frac{265224533486285290261362685492205650}{531170702900974310074395392882044349}$

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

Class group and class number

$C_{2}\times C_{40}$, which has order $80$ (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:  \( 245689.718883 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4:C_4$ (as 16T26):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $D_4:C_4$
Character table for $D_4:C_4$

Intermediate fields

\(\Q(\sqrt{13}) \), 4.0.8175037.2, 4.4.10309.1, 4.0.134017.1, 8.0.6482804341.1, 8.8.1095593933629.1, 8.0.66831229951369.4

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

Sibling fields

Galois closure: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
$61$61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$