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Properties

Label 16.0.18244960011...2161.1
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 $QD_{16}$ (as 16T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![347, -1258, 1297, -1080, 3832, -3358, 5757, -2042, 1021, -506, -273, 294, -48, -14, 15, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 15*x^14 - 14*x^13 - 48*x^12 + 294*x^11 - 273*x^10 - 506*x^9 + 1021*x^8 - 2042*x^7 + 5757*x^6 - 3358*x^5 + 3832*x^4 - 1080*x^3 + 1297*x^2 - 1258*x + 347)
 
gp: K = bnfinit(x^16 - 6*x^15 + 15*x^14 - 14*x^13 - 48*x^12 + 294*x^11 - 273*x^10 - 506*x^9 + 1021*x^8 - 2042*x^7 + 5757*x^6 - 3358*x^5 + 3832*x^4 - 1080*x^3 + 1297*x^2 - 1258*x + 347, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 15 x^{14} - 14 x^{13} - 48 x^{12} + 294 x^{11} - 273 x^{10} - 506 x^{9} + 1021 x^{8} - 2042 x^{7} + 5757 x^{6} - 3358 x^{5} + 3832 x^{4} - 1080 x^{3} + 1297 x^{2} - 1258 x + 347 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{46} a^{12} - \frac{5}{23} a^{11} - \frac{1}{23} a^{10} - \frac{1}{23} a^{9} - \frac{11}{46} a^{8} + \frac{8}{23} a^{7} - \frac{7}{23} a^{6} + \frac{1}{46} a^{5} - \frac{5}{46} a^{4} - \frac{9}{46} a^{3} - \frac{9}{46} a^{2} + \frac{3}{46} a - \frac{5}{46}$, $\frac{1}{46} a^{13} - \frac{5}{23} a^{11} + \frac{1}{46} a^{10} - \frac{4}{23} a^{9} - \frac{1}{23} a^{8} - \frac{15}{46} a^{7} + \frac{11}{23} a^{6} - \frac{9}{23} a^{5} - \frac{13}{46} a^{4} + \frac{8}{23} a^{3} - \frac{9}{23} a^{2} - \frac{21}{46} a - \frac{2}{23}$, $\frac{1}{47702} a^{14} - \frac{83}{23851} a^{13} - \frac{365}{47702} a^{12} + \frac{2479}{23851} a^{11} - \frac{1434}{23851} a^{10} - \frac{2087}{23851} a^{9} - \frac{1137}{47702} a^{8} - \frac{15151}{47702} a^{7} - \frac{59}{782} a^{6} + \frac{7151}{47702} a^{5} - \frac{590}{23851} a^{4} + \frac{9610}{23851} a^{3} + \frac{3725}{23851} a^{2} + \frac{2325}{47702} a + \frac{817}{23851}$, $\frac{1}{36988555738158498711122} a^{15} + \frac{190443803596830956}{18494277869079249355561} a^{14} - \frac{173501849003199993101}{36988555738158498711122} a^{13} - \frac{179708510854491660136}{18494277869079249355561} a^{12} - \frac{1822522680484952063124}{18494277869079249355561} a^{11} - \frac{721371948592066579713}{36988555738158498711122} a^{10} - \frac{1230686104745074327935}{18494277869079249355561} a^{9} + \frac{5526581396530102493893}{36988555738158498711122} a^{8} - \frac{8455616457967887096251}{18494277869079249355561} a^{7} - \frac{6446812616998567260859}{18494277869079249355561} a^{6} - \frac{6669318023107941862549}{36988555738158498711122} a^{5} - \frac{1421177712548160688943}{18494277869079249355561} a^{4} - \frac{2689324961980478147217}{36988555738158498711122} a^{3} + \frac{8532625476830609028001}{18494277869079249355561} a^{2} - \frac{361367100148242368920}{804099037786054319807} a - \frac{3290105974688831506030}{18494277869079249355561}$

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:  \( 64747.4698794 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SD_{16}$ (as 16T12):

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 $QD_{16}$
Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{-299}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{13}, \sqrt{-23})\), 4.2.3887.1 x2, 4.0.6877.1 x2, 8.0.7992538801.1, 8.2.58727785103.1 x4

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$