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Properties

Label 16.0.23045377697175681.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 37^{8}$
Root discriminant $10.54$
Ramified primes $3, 37$
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![1, -1, 0, 11, -14, 6, 23, -44, 45, -44, 23, 6, -14, 11, 0, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 11*x^13 - 14*x^12 + 6*x^11 + 23*x^10 - 44*x^9 + 45*x^8 - 44*x^7 + 23*x^6 + 6*x^5 - 14*x^4 + 11*x^3 - x + 1)
 
gp: K = bnfinit(x^16 - x^15 + 11*x^13 - 14*x^12 + 6*x^11 + 23*x^10 - 44*x^9 + 45*x^8 - 44*x^7 + 23*x^6 + 6*x^5 - 14*x^4 + 11*x^3 - x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 11 x^{13} - 14 x^{12} + 6 x^{11} + 23 x^{10} - 44 x^{9} + 45 x^{8} - 44 x^{7} + 23 x^{6} + 6 x^{5} - 14 x^{4} + 11 x^{3} - x + 1 \)

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:  \(23045377697175681=3^{8}\cdot 37^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $10.54$
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:  $3, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{5} + \frac{1}{3} a$, $\frac{1}{9} a^{10} + \frac{1}{9} a^{9} - \frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{2}{9} a^{4} + \frac{1}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{9} a - \frac{2}{9}$, $\frac{1}{9} a^{11} + \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{1}{9} a^{6} + \frac{1}{9} a^{5} + \frac{1}{9} a^{3} - \frac{1}{9} a^{2} - \frac{1}{9}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{1}{9} a^{3} + \frac{1}{9} a^{2} + \frac{1}{9} a - \frac{1}{9}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} - \frac{1}{9} a^{2} - \frac{2}{9} a - \frac{1}{9}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{1}{9} a^{3} - \frac{2}{9} a^{2} - \frac{1}{9} a$, $\frac{1}{99} a^{15} - \frac{2}{99} a^{14} + \frac{2}{99} a^{13} - \frac{2}{99} a^{12} - \frac{1}{99} a^{11} - \frac{4}{99} a^{10} + \frac{16}{99} a^{9} - \frac{5}{99} a^{8} - \frac{5}{99} a^{7} + \frac{5}{99} a^{6} + \frac{29}{99} a^{5} - \frac{1}{99} a^{4} - \frac{8}{33} a^{3} - \frac{14}{33} a^{2} - \frac{13}{99} a + \frac{1}{99}$

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:  \( -\frac{67}{33} a^{15} + \frac{79}{33} a^{14} - \frac{13}{33} a^{13} - \frac{724}{33} a^{12} + \frac{1057}{33} a^{11} - \frac{193}{11} a^{10} - \frac{1325}{33} a^{9} + \frac{3041}{33} a^{8} - \frac{1157}{11} a^{7} + \frac{3779}{33} a^{6} - \frac{2636}{33} a^{5} + \frac{169}{11} a^{4} + \frac{365}{33} a^{3} - \frac{497}{33} a^{2} + \frac{112}{33} a - \frac{4}{11} \) (order $6$)
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:  \( 82.0564715035 \)
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

\(\Q(\sqrt{-111}) \), \(\Q(\sqrt{37}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{37})\), 4.2.4107.1 x2, 4.0.333.1 x2, 8.0.151807041.1, 8.2.50602347.1 x4, 8.0.4102893.1 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$37$37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$