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Properties

Label 16.0.17422379862...1056.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 113^{8}$
Root discriminant $21.26$
Ramified primes $2, 113$
Class number $1$
Class group Trivial
Galois group $D_{8}$ (as 16T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 16, 32, 120, 265, 44, -144, -310, -16, -112, 162, 32, 36, -34, 8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 8*x^14 - 34*x^13 + 36*x^12 + 32*x^11 + 162*x^10 - 112*x^9 - 16*x^8 - 310*x^7 - 144*x^6 + 44*x^5 + 265*x^4 + 120*x^3 + 32*x^2 + 16*x + 4)
 
gp: K = bnfinit(x^16 - 4*x^15 + 8*x^14 - 34*x^13 + 36*x^12 + 32*x^11 + 162*x^10 - 112*x^9 - 16*x^8 - 310*x^7 - 144*x^6 + 44*x^5 + 265*x^4 + 120*x^3 + 32*x^2 + 16*x + 4, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 8 x^{14} - 34 x^{13} + 36 x^{12} + 32 x^{11} + 162 x^{10} - 112 x^{9} - 16 x^{8} - 310 x^{7} - 144 x^{6} + 44 x^{5} + 265 x^{4} + 120 x^{3} + 32 x^{2} + 16 x + 4 \)

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:  \(1742237986263159341056=2^{16}\cdot 113^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.26$
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:  $2, 113$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{20} a^{12} - \frac{1}{20} a^{11} - \frac{3}{20} a^{9} + \frac{3}{20} a^{8} + \frac{1}{5} a^{7} - \frac{9}{20} a^{6} + \frac{3}{20} a^{5} + \frac{2}{5} a^{4} - \frac{7}{20} a^{3} - \frac{1}{4} a^{2} + \frac{2}{5} a - \frac{1}{10}$, $\frac{1}{40} a^{13} - \frac{1}{40} a^{12} - \frac{1}{4} a^{11} + \frac{7}{40} a^{10} - \frac{7}{40} a^{9} + \frac{1}{10} a^{8} - \frac{9}{40} a^{7} + \frac{3}{40} a^{6} + \frac{9}{20} a^{5} + \frac{3}{40} a^{4} - \frac{3}{8} a^{3} - \frac{3}{10} a^{2} - \frac{1}{20} a$, $\frac{1}{10280} a^{14} + \frac{29}{10280} a^{13} - \frac{111}{5140} a^{12} - \frac{1691}{10280} a^{11} - \frac{937}{10280} a^{10} + \frac{77}{514} a^{9} - \frac{99}{2056} a^{8} - \frac{39}{2056} a^{7} - \frac{367}{5140} a^{6} + \frac{4417}{10280} a^{5} + \frac{2239}{10280} a^{4} - \frac{837}{2570} a^{3} - \frac{324}{1285} a^{2} + \frac{191}{2570} a - \frac{139}{2570}$, $\frac{1}{454732849640} a^{15} + \frac{8213171}{227366424820} a^{14} + \frac{4927044137}{454732849640} a^{13} - \frac{3708459901}{454732849640} a^{12} - \frac{21298034899}{227366424820} a^{11} - \frac{95826891747}{454732849640} a^{10} + \frac{80059535117}{454732849640} a^{9} + \frac{11988056564}{56841606205} a^{8} + \frac{41985797753}{90946569928} a^{7} - \frac{153563486061}{454732849640} a^{6} + \frac{5573620403}{45473284964} a^{5} + \frac{100937762189}{454732849640} a^{4} - \frac{3455502263}{113683212410} a^{3} - \frac{55753108733}{227366424820} a^{2} + \frac{55375498789}{113683212410} a + \frac{22297325467}{56841606205}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{43137951}{353877704} a^{15} + \frac{120177436}{221173565} a^{14} - \frac{432974203}{353877704} a^{13} + \frac{8296512151}{1769388520} a^{12} - \frac{5739801719}{884694260} a^{11} - \frac{361244035}{353877704} a^{10} - \frac{33320051031}{1769388520} a^{9} + \frac{9787834663}{442347130} a^{8} - \frac{15010016727}{1769388520} a^{7} + \frac{68436128347}{1769388520} a^{6} - \frac{278793777}{176938852} a^{5} - \frac{12318439479}{1769388520} a^{4} - \frac{11565976191}{442347130} a^{3} + \frac{249102091}{221173565} a^{2} + \frac{3888537}{221173565} a - \frac{345765887}{442347130} \) (order $4$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 45023.51136 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_8$ (as 16T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $D_{8}$
Character table for $D_{8}$

Intermediate fields

\(\Q(\sqrt{-113}) \), \(\Q(\sqrt{113}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{113})\), 4.2.51076.1 x2, 4.0.1808.1 x2, 8.0.41740124416.1, 8.2.10435031104.1 x4, 8.0.369381632.1 x4

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
$113$113.2.1.1$x^{2} - 113$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.1$x^{2} - 113$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.1$x^{2} - 113$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.1$x^{2} - 113$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.1$x^{2} - 113$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.1$x^{2} - 113$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.1$x^{2} - 113$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.1$x^{2} - 113$$2$$1$$1$$C_2$$[\ ]_{2}$