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Properties

Label 16.0.15251399075...5561.1
Degree $16$
Signature $[0, 8]$
Discriminant $23^{8}\cdot 41^{7}$
Root discriminant $24.35$
Ramified primes $23, 41$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $D_{16}$ (as 16T56)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![213, -347, 432, -78, -194, 248, 116, -548, 816, -812, 598, -352, 180, -84, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 180*x^12 - 352*x^11 + 598*x^10 - 812*x^9 + 816*x^8 - 548*x^7 + 116*x^6 + 248*x^5 - 194*x^4 - 78*x^3 + 432*x^2 - 347*x + 213)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 180*x^12 - 352*x^11 + 598*x^10 - 812*x^9 + 816*x^8 - 548*x^7 + 116*x^6 + 248*x^5 - 194*x^4 - 78*x^3 + 432*x^2 - 347*x + 213, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 84 x^{13} + 180 x^{12} - 352 x^{11} + 598 x^{10} - 812 x^{9} + 816 x^{8} - 548 x^{7} + 116 x^{6} + 248 x^{5} - 194 x^{4} - 78 x^{3} + 432 x^{2} - 347 x + 213 \)

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:  \(15251399075306833745561=23^{8}\cdot 41^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.35$
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:  $23, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{45} a^{12} - \frac{2}{15} a^{11} - \frac{2}{15} a^{10} - \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{16}{45} a^{7} + \frac{22}{45} a^{6} + \frac{8}{45} a^{5} - \frac{1}{3} a^{4} + \frac{4}{9} a^{3} - \frac{11}{45} a^{2} + \frac{13}{45} a - \frac{4}{15}$, $\frac{1}{45} a^{13} + \frac{1}{15} a^{11} + \frac{4}{45} a^{10} - \frac{1}{9} a^{9} - \frac{1}{45} a^{8} + \frac{16}{45} a^{7} + \frac{1}{9} a^{6} - \frac{4}{15} a^{5} + \frac{4}{9} a^{4} + \frac{19}{45} a^{3} - \frac{8}{45} a^{2} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{644355} a^{14} - \frac{7}{644355} a^{13} - \frac{988}{644355} a^{12} + \frac{6019}{644355} a^{11} - \frac{22294}{214785} a^{10} + \frac{21428}{214785} a^{9} - \frac{55712}{644355} a^{8} - \frac{96361}{644355} a^{7} + \frac{10438}{23865} a^{6} - \frac{5221}{71595} a^{5} - \frac{87346}{644355} a^{4} - \frac{169511}{644355} a^{3} - \frac{179627}{644355} a^{2} - \frac{293062}{644355} a + \frac{58732}{214785}$, $\frac{1}{297047655} a^{15} + \frac{223}{297047655} a^{14} + \frac{164018}{19803177} a^{13} - \frac{605549}{59409531} a^{12} - \frac{16337839}{297047655} a^{11} - \frac{205496}{6601059} a^{10} + \frac{40718593}{297047655} a^{9} + \frac{14124151}{297047655} a^{8} + \frac{59207293}{297047655} a^{7} + \frac{1666079}{11001765} a^{6} + \frac{134915561}{297047655} a^{5} + \frac{894032}{8028315} a^{4} + \frac{2221996}{19803177} a^{3} + \frac{38507533}{297047655} a^{2} - \frac{134040518}{297047655} a + \frac{19966229}{99015885}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 36538.9775693 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{16}$ (as 16T56):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $D_{16}$
Character table for $D_{16}$

Intermediate fields

\(\Q(\sqrt{-23}) \), 4.0.21689.1, 8.0.19286921561.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ $16$ $16$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $16$ $16$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$23$23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$