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Properties

Label 16.0.14727565440...4569.2
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 43^{6}$
Root discriminant $28.05$
Ramified primes $13, 43$
Class number $1$ (GRH)
Class group Trivial (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![729, -729, 810, -1836, 2538, -2361, 1168, -66, -240, 62, 87, -73, 41, -23, 14, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 14*x^14 - 23*x^13 + 41*x^12 - 73*x^11 + 87*x^10 + 62*x^9 - 240*x^8 - 66*x^7 + 1168*x^6 - 2361*x^5 + 2538*x^4 - 1836*x^3 + 810*x^2 - 729*x + 729)
 
gp: K = bnfinit(x^16 - 5*x^15 + 14*x^14 - 23*x^13 + 41*x^12 - 73*x^11 + 87*x^10 + 62*x^9 - 240*x^8 - 66*x^7 + 1168*x^6 - 2361*x^5 + 2538*x^4 - 1836*x^3 + 810*x^2 - 729*x + 729, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 14 x^{14} - 23 x^{13} + 41 x^{12} - 73 x^{11} + 87 x^{10} + 62 x^{9} - 240 x^{8} - 66 x^{7} + 1168 x^{6} - 2361 x^{5} + 2538 x^{4} - 1836 x^{3} + 810 x^{2} - 729 x + 729 \)

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:  \(147275654405708032604569=13^{12}\cdot 43^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.05$
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:  $13, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{4}{9} a^{6} - \frac{2}{9} a^{5} + \frac{1}{9} a^{4} - \frac{1}{9} a^{3} + \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{9} a^{7} + \frac{4}{9} a^{6} - \frac{2}{9} a^{5} - \frac{1}{9} a^{4} + \frac{1}{9} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{27} a^{13} + \frac{1}{27} a^{12} - \frac{1}{27} a^{11} - \frac{2}{27} a^{10} - \frac{4}{27} a^{9} - \frac{4}{27} a^{8} + \frac{11}{27} a^{6} - \frac{2}{9} a^{5} - \frac{2}{9} a^{4} + \frac{1}{27} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{243} a^{14} + \frac{1}{243} a^{13} + \frac{11}{243} a^{12} + \frac{7}{243} a^{11} - \frac{16}{243} a^{10} + \frac{38}{243} a^{9} - \frac{37}{243} a^{7} - \frac{37}{81} a^{6} - \frac{34}{81} a^{5} + \frac{70}{243} a^{4} - \frac{8}{81} a^{3} - \frac{5}{27} a^{2} - \frac{1}{9} a + \frac{1}{3}$, $\frac{1}{2150980454301597} a^{15} + \frac{2002972362910}{2150980454301597} a^{14} - \frac{22095537067747}{2150980454301597} a^{13} + \frac{109225617900706}{2150980454301597} a^{12} + \frac{110155069134746}{2150980454301597} a^{11} + \frac{343042901182337}{2150980454301597} a^{10} - \frac{103834215615380}{716993484767199} a^{9} + \frac{55836790318394}{2150980454301597} a^{8} + \frac{18712866408782}{238997828255733} a^{7} - \frac{197494275279244}{716993484767199} a^{6} - \frac{832201335388541}{2150980454301597} a^{5} + \frac{14839765822985}{238997828255733} a^{4} - \frac{214883863602}{983530157431} a^{3} + \frac{30474312760156}{79665942751911} a^{2} + \frac{429335431109}{26555314250637} a - \frac{1355714175805}{8851771416879}$

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:  \( 634632.996614 \) (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, 4.2.7267.1, 4.2.94471.1, 8.2.2270799427.1, 8.2.383765103163.1, 8.0.8924769841.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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}$
$43$43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$