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Properties

Label 16.0.14538313896...1969.1
Degree $16$
Signature $[0, 8]$
Discriminant $7^{14}\cdot 11^{8}$
Root discriminant $18.20$
Ramified primes $7, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![64, -144, 244, -308, 189, -91, 434, -809, 548, 25, -224, 77, 28, -21, 6, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 6*x^14 - 21*x^13 + 28*x^12 + 77*x^11 - 224*x^10 + 25*x^9 + 548*x^8 - 809*x^7 + 434*x^6 - 91*x^5 + 189*x^4 - 308*x^3 + 244*x^2 - 144*x + 64)
 
gp: K = bnfinit(x^16 - 3*x^15 + 6*x^14 - 21*x^13 + 28*x^12 + 77*x^11 - 224*x^10 + 25*x^9 + 548*x^8 - 809*x^7 + 434*x^6 - 91*x^5 + 189*x^4 - 308*x^3 + 244*x^2 - 144*x + 64, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 6 x^{14} - 21 x^{13} + 28 x^{12} + 77 x^{11} - 224 x^{10} + 25 x^{9} + 548 x^{8} - 809 x^{7} + 434 x^{6} - 91 x^{5} + 189 x^{4} - 308 x^{3} + 244 x^{2} - 144 x + 64 \)

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:  \(145383138964293121969=7^{14}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.20$
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:  $7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{5} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{10} - \frac{1}{4} a^{6} - \frac{1}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{11} + \frac{1}{32} a^{10} - \frac{1}{32} a^{9} - \frac{1}{16} a^{7} - \frac{3}{16} a^{6} - \frac{7}{32} a^{5} + \frac{1}{32} a^{4} - \frac{1}{32} a^{3} - \frac{5}{16} a^{2} + \frac{1}{4} a$, $\frac{1}{32} a^{12} - \frac{1}{16} a^{10} + \frac{1}{32} a^{9} - \frac{1}{16} a^{8} - \frac{1}{8} a^{7} - \frac{1}{32} a^{6} - \frac{1}{4} a^{5} - \frac{1}{16} a^{4} - \frac{9}{32} a^{3} + \frac{1}{16} a^{2} - \frac{1}{4} a$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{10} + \frac{3}{32} a^{7} - \frac{1}{8} a^{6} - \frac{3}{32} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{1408} a^{14} + \frac{5}{352} a^{13} - \frac{21}{1408} a^{12} + \frac{1}{176} a^{11} - \frac{3}{128} a^{10} - \frac{23}{704} a^{9} - \frac{83}{1408} a^{8} + \frac{29}{704} a^{7} - \frac{289}{1408} a^{6} - \frac{141}{704} a^{5} - \frac{125}{1408} a^{4} - \frac{123}{352} a^{3} + \frac{73}{352} a^{2} + \frac{39}{88} a - \frac{7}{22}$, $\frac{1}{656425088} a^{15} + \frac{32613}{328212544} a^{14} + \frac{6978339}{656425088} a^{13} - \frac{3578363}{328212544} a^{12} + \frac{3581771}{656425088} a^{11} + \frac{4923333}{82053136} a^{10} + \frac{12606141}{656425088} a^{9} - \frac{3405891}{82053136} a^{8} - \frac{18465885}{656425088} a^{7} + \frac{16540231}{82053136} a^{6} - \frac{117719229}{656425088} a^{5} - \frac{2153629}{29837504} a^{4} - \frac{41834}{466211} a^{3} - \frac{235363}{20513284} a^{2} - \frac{8498411}{20513284} a - \frac{938172}{5128321}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 13134.7691742 \) (assuming GRH)
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

\(\Q(\sqrt{-11}) \), \(\Q(\sqrt{77}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-7}, \sqrt{-11})\), 4.2.41503.1 x2, 4.0.3773.1 x2, 8.0.1722499009.1, 8.2.12057493063.1 x4, 8.0.1096135733.1 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
7Data not computed
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$