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Properties

Label 16.0.51638717226...1049.9
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 53^{6}$
Root discriminant $30.34$
Ramified primes $13, 53$
Class number $4$ (GRH)
Class group $[4]$ (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![6561, -8748, 6561, -6075, 4941, -3213, 1971, -1230, 751, -410, 219, -119, 61, -25, 9, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 9*x^14 - 25*x^13 + 61*x^12 - 119*x^11 + 219*x^10 - 410*x^9 + 751*x^8 - 1230*x^7 + 1971*x^6 - 3213*x^5 + 4941*x^4 - 6075*x^3 + 6561*x^2 - 8748*x + 6561)
 
gp: K = bnfinit(x^16 - 4*x^15 + 9*x^14 - 25*x^13 + 61*x^12 - 119*x^11 + 219*x^10 - 410*x^9 + 751*x^8 - 1230*x^7 + 1971*x^6 - 3213*x^5 + 4941*x^4 - 6075*x^3 + 6561*x^2 - 8748*x + 6561, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 9 x^{14} - 25 x^{13} + 61 x^{12} - 119 x^{11} + 219 x^{10} - 410 x^{9} + 751 x^{8} - 1230 x^{7} + 1971 x^{6} - 3213 x^{5} + 4941 x^{4} - 6075 x^{3} + 6561 x^{2} - 8748 x + 6561 \)

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:  \(516387172268851080441049=13^{12}\cdot 53^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.34$
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:  $13, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{9} a^{7} + \frac{1}{9} a^{6} + \frac{1}{9} a^{5} + \frac{1}{3} a^{4} - \frac{2}{9} a^{3} + \frac{1}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{27} a^{11} - \frac{1}{27} a^{10} - \frac{1}{9} a^{9} + \frac{2}{27} a^{8} + \frac{4}{27} a^{7} - \frac{8}{27} a^{6} + \frac{2}{9} a^{5} + \frac{4}{27} a^{4} - \frac{11}{27} a^{3} - \frac{4}{9} a^{2}$, $\frac{1}{81} a^{12} - \frac{1}{81} a^{11} - \frac{1}{27} a^{10} + \frac{2}{81} a^{9} + \frac{13}{81} a^{8} + \frac{10}{81} a^{7} + \frac{8}{27} a^{6} + \frac{31}{81} a^{5} - \frac{20}{81} a^{4} + \frac{8}{27} a^{3} - \frac{2}{9} a^{2}$, $\frac{1}{243} a^{13} - \frac{1}{243} a^{12} - \frac{1}{81} a^{11} + \frac{2}{243} a^{10} - \frac{14}{243} a^{9} - \frac{17}{243} a^{8} + \frac{8}{81} a^{7} + \frac{4}{243} a^{6} - \frac{74}{243} a^{5} - \frac{37}{81} a^{4} + \frac{7}{27} a^{3} + \frac{1}{9} a^{2}$, $\frac{1}{461457} a^{14} - \frac{58}{461457} a^{13} - \frac{1}{17091} a^{12} - \frac{2824}{461457} a^{11} + \frac{15910}{461457} a^{10} - \frac{13961}{461457} a^{9} - \frac{11198}{153819} a^{8} + \frac{16051}{461457} a^{7} + \frac{210865}{461457} a^{6} + \frac{71002}{153819} a^{5} - \frac{9451}{51273} a^{4} - \frac{238}{1899} a^{3} + \frac{2240}{5697} a^{2} + \frac{881}{1899} a + \frac{214}{633}$, $\frac{1}{84446631} a^{15} - \frac{73}{84446631} a^{14} + \frac{28766}{28148877} a^{13} - \frac{241693}{84446631} a^{12} + \frac{929911}{84446631} a^{11} - \frac{2747897}{84446631} a^{10} + \frac{3761657}{28148877} a^{9} - \frac{10367006}{84446631} a^{8} + \frac{10378519}{84446631} a^{7} + \frac{9402308}{28148877} a^{6} - \frac{1386854}{3127653} a^{5} + \frac{812309}{3127653} a^{4} + \frac{402878}{1042551} a^{3} + \frac{123877}{347517} a^{2} + \frac{12107}{38613} a + \frac{13489}{38613}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 245119.613331 \) (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

\(\Q(\sqrt{13}) \), 4.0.2197.1, 4.4.8957.1, 4.0.116441.1, 8.8.718600843493.1, 8.0.4252075997.1, 8.0.13558506481.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$53$$\Q_{53}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{53}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{53}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{53}$$x + 2$$1$$1$$0$Trivial$[\ ]$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$