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Properties

Label 16.0.14505315669...641.16
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 53^{8}$
Root discriminant $49.84$
Ramified primes $13, 53$
Class number $64$ (GRH)
Class group $[2, 4, 8]$ (GRH)
Galois group $C_2^2.D_4$ (as 16T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![52092, -225810, 424309, -475942, 360296, -187868, 72233, -23710, 5385, 324, 765, -974, 161, 38, 8, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 8*x^14 + 38*x^13 + 161*x^12 - 974*x^11 + 765*x^10 + 324*x^9 + 5385*x^8 - 23710*x^7 + 72233*x^6 - 187868*x^5 + 360296*x^4 - 475942*x^3 + 424309*x^2 - 225810*x + 52092)
 
gp: K = bnfinit(x^16 - 8*x^15 + 8*x^14 + 38*x^13 + 161*x^12 - 974*x^11 + 765*x^10 + 324*x^9 + 5385*x^8 - 23710*x^7 + 72233*x^6 - 187868*x^5 + 360296*x^4 - 475942*x^3 + 424309*x^2 - 225810*x + 52092, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 8 x^{14} + 38 x^{13} + 161 x^{12} - 974 x^{11} + 765 x^{10} + 324 x^{9} + 5385 x^{8} - 23710 x^{7} + 72233 x^{6} - 187868 x^{5} + 360296 x^{4} - 475942 x^{3} + 424309 x^{2} - 225810 x + 52092 \)

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:  \(1450531566903202684958906641=13^{12}\cdot 53^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.84$
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 not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{7} - \frac{1}{4} a^{6} + \frac{1}{12} a^{5} - \frac{1}{4} a^{4} + \frac{1}{12} a^{3} - \frac{1}{4} a^{2} + \frac{1}{3} a$, $\frac{1}{12} a^{8} - \frac{1}{6} a^{6} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{5}{12} a^{2} - \frac{1}{2} a$, $\frac{1}{24} a^{9} - \frac{1}{24} a^{7} - \frac{1}{8} a^{6} + \frac{5}{24} a^{5} - \frac{1}{8} a^{4} - \frac{1}{6} a^{3} + \frac{3}{8} a^{2} + \frac{5}{12} a - \frac{1}{2}$, $\frac{1}{24} a^{10} - \frac{1}{24} a^{8} - \frac{1}{24} a^{7} - \frac{1}{24} a^{6} - \frac{1}{24} a^{5} + \frac{1}{12} a^{4} + \frac{11}{24} a^{3} + \frac{1}{6} a^{2} + \frac{1}{3} a$, $\frac{1}{48} a^{11} - \frac{1}{48} a^{10} - \frac{1}{24} a^{8} - \frac{1}{48} a^{7} + \frac{1}{48} a^{6} + \frac{1}{6} a^{5} - \frac{1}{24} a^{4} + \frac{1}{48} a^{3} + \frac{23}{48} a^{2} + \frac{1}{24} a + \frac{1}{4}$, $\frac{1}{144} a^{12} - \frac{1}{144} a^{11} - \frac{1}{72} a^{10} - \frac{1}{72} a^{9} - \frac{1}{48} a^{8} - \frac{5}{144} a^{7} - \frac{1}{24} a^{6} - \frac{1}{18} a^{5} + \frac{29}{144} a^{4} + \frac{17}{144} a^{3} - \frac{5}{72} a^{2} - \frac{5}{12} a$, $\frac{1}{2448} a^{13} + \frac{5}{816} a^{11} - \frac{11}{1224} a^{10} - \frac{29}{2448} a^{9} + \frac{11}{306} a^{8} + \frac{7}{2448} a^{7} - \frac{73}{306} a^{6} + \frac{91}{816} a^{5} - \frac{109}{1224} a^{4} - \frac{11}{2448} a^{3} - \frac{58}{153} a^{2} + \frac{4}{17} a - \frac{5}{34}$, $\frac{1}{5826240} a^{14} - \frac{271}{1942080} a^{13} + \frac{811}{416160} a^{12} + \frac{2377}{388416} a^{11} + \frac{35}{6936} a^{10} + \frac{4987}{1942080} a^{9} - \frac{1999}{72828} a^{8} - \frac{769}{18496} a^{7} - \frac{9}{68} a^{6} - \frac{24823}{388416} a^{5} + \frac{62317}{416160} a^{4} + \frac{541039}{1942080} a^{3} + \frac{1927309}{5826240} a^{2} + \frac{27725}{194208} a + \frac{78153}{161840}$, $\frac{1}{3901220620967678400} a^{15} - \frac{24397456131}{433468957885297600} a^{14} + \frac{97889465181593}{975305155241919600} a^{13} - \frac{161351401081939}{76494521979758400} a^{12} + \frac{80026973564675}{26008137473117856} a^{11} + \frac{5699269467406567}{1300406873655892800} a^{10} - \frac{7276181086185553}{1950610310483839200} a^{9} - \frac{2663650926282473}{260081374731178560} a^{8} - \frac{476069348640397}{18577241052227040} a^{7} + \frac{26280261084363229}{260081374731178560} a^{6} + \frac{4971669406131257}{60956572202619975} a^{5} - \frac{143098836684320417}{1300406873655892800} a^{4} + \frac{1213064399297813107}{3901220620967678400} a^{3} - \frac{3105665609827631}{46443102630567600} a^{2} - \frac{3015042242957553}{7740517105094600} a + \frac{22711722347448411}{54183619735662200}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{8}$, which has order $64$ (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:  \( 1434748.43032 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.D_4$ (as 16T33):

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 $C_2^2.D_4$
Character table for $C_2^2.D_4$

Intermediate fields

\(\Q(\sqrt{53}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{689}) \), 4.0.6171373.1 x2, \(\Q(\sqrt{13}, \sqrt{53})\), 4.0.116441.1 x2, 8.0.38085844705129.1 x2, 8.0.38085844705129.4, 8.4.2929680361933.4 x2

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 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ 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$13.4.3.2$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}$
$53$53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$