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Properties

Label 16.0.18244960011...2161.3
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 23^{8}$
Root discriminant $32.83$
Ramified primes $13, 23$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $C_2^2 : C_4$ (as 16T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1296, -3384, -248, 7952, 6151, -4190, -708, 1926, -498, 824, 321, -90, 147, -44, 25, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 25*x^14 - 44*x^13 + 147*x^12 - 90*x^11 + 321*x^10 + 824*x^9 - 498*x^8 + 1926*x^7 - 708*x^6 - 4190*x^5 + 6151*x^4 + 7952*x^3 - 248*x^2 - 3384*x + 1296)
 
gp: K = bnfinit(x^16 - 4*x^15 + 25*x^14 - 44*x^13 + 147*x^12 - 90*x^11 + 321*x^10 + 824*x^9 - 498*x^8 + 1926*x^7 - 708*x^6 - 4190*x^5 + 6151*x^4 + 7952*x^3 - 248*x^2 - 3384*x + 1296, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 25 x^{14} - 44 x^{13} + 147 x^{12} - 90 x^{11} + 321 x^{10} + 824 x^{9} - 498 x^{8} + 1926 x^{7} - 708 x^{6} - 4190 x^{5} + 6151 x^{4} + 7952 x^{3} - 248 x^{2} - 3384 x + 1296 \)

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:  \(1824496001102094673202161=13^{12}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.83$
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:  $13, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{6} a^{8} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{6} a^{2} + \frac{1}{3} a$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{6} + \frac{1}{3} a^{3} - \frac{1}{2} a$, $\frac{1}{6} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{18} a^{13} + \frac{1}{18} a^{12} - \frac{1}{18} a^{11} + \frac{1}{18} a^{10} - \frac{1}{18} a^{8} - \frac{1}{9} a^{7} + \frac{5}{18} a^{6} - \frac{1}{6} a^{5} - \frac{1}{9} a^{4} + \frac{4}{9} a^{3} - \frac{5}{18} a^{2} - \frac{1}{9} a$, $\frac{1}{108} a^{14} + \frac{7}{108} a^{12} + \frac{1}{54} a^{11} - \frac{7}{108} a^{10} + \frac{2}{27} a^{9} - \frac{7}{108} a^{8} - \frac{7}{54} a^{7} + \frac{11}{54} a^{6} - \frac{5}{27} a^{5} + \frac{11}{54} a^{4} + \frac{1}{54} a^{3} + \frac{13}{36} a^{2} - \frac{23}{54} a - \frac{1}{3}$, $\frac{1}{302103987745244193888596856} a^{15} - \frac{136353072904856181071237}{50350664624207365648099476} a^{14} - \frac{3856972786060360994882015}{302103987745244193888596856} a^{13} - \frac{10364950448055421158155105}{151051993872622096944298428} a^{12} - \frac{21773007413617488014850265}{302103987745244193888596856} a^{11} + \frac{4249176000231946755019943}{75525996936311048472149214} a^{10} + \frac{19663883466634299155618705}{302103987745244193888596856} a^{9} - \frac{7549383397623209583898693}{151051993872622096944298428} a^{8} - \frac{948121499594963666875699}{151051993872622096944298428} a^{7} - \frac{20941378582946490531964783}{151051993872622096944298428} a^{6} - \frac{17758573633265704855119497}{75525996936311048472149214} a^{5} - \frac{27358522978699754591013317}{151051993872622096944298428} a^{4} - \frac{13790644222338543447244165}{33567109749471577098732984} a^{3} + \frac{55555131939561958074648577}{151051993872622096944298428} a^{2} - \frac{1389146311249453766811185}{8391777437367894274683246} a - \frac{110900463835378463124101}{466209857631549681926847}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  \( 111793.016463 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_4$ (as 16T10):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_2^2 : C_4$
Character table for $C_2^2 : C_4$

Intermediate fields

\(\Q(\sqrt{-23}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-299}) \), \(\Q(\sqrt{13}, \sqrt{-23})\), 4.2.50531.1 x2, 4.0.1162213.1 x2, 4.0.6877.1 x2, 4.2.3887.1 x2, 4.0.2197.1, 4.4.1162213.1, 8.0.1350739057369.2, 8.0.7992538801.1, 8.0.1350739057369.1, 8.0.2553381961.1 x2, 8.4.1350739057369.1 x2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
$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}$