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Properties

Label 16.0.58966431579...9761.6
Degree $16$
Signature $[0, 8]$
Discriminant $13^{14}\cdot 157^{6}$
Root discriminant $62.83$
Ramified primes $13, 157$
Class number $26$ (GRH)
Class group $[26]$ (GRH)
Galois group $(C_2\times C_4).D_4$ (as 16T121)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5333168, -341152, 352784, -706086, 368815, -5820, 132380, -5457, 16260, -1780, 2629, -256, 188, 1, 16, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 16*x^14 + x^13 + 188*x^12 - 256*x^11 + 2629*x^10 - 1780*x^9 + 16260*x^8 - 5457*x^7 + 132380*x^6 - 5820*x^5 + 368815*x^4 - 706086*x^3 + 352784*x^2 - 341152*x + 5333168)
 
gp: K = bnfinit(x^16 - 2*x^15 + 16*x^14 + x^13 + 188*x^12 - 256*x^11 + 2629*x^10 - 1780*x^9 + 16260*x^8 - 5457*x^7 + 132380*x^6 - 5820*x^5 + 368815*x^4 - 706086*x^3 + 352784*x^2 - 341152*x + 5333168, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 16 x^{14} + x^{13} + 188 x^{12} - 256 x^{11} + 2629 x^{10} - 1780 x^{9} + 16260 x^{8} - 5457 x^{7} + 132380 x^{6} - 5820 x^{5} + 368815 x^{4} - 706086 x^{3} + 352784 x^{2} - 341152 x + 5333168 \)

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:  \(58966431579683595226807139761=13^{14}\cdot 157^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.83$
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:  $13, 157$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{4} a^{14} + \frac{1}{4} a^{11} - \frac{1}{2} a^{10} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{55501360485555181826739813680737778214354577352} a^{15} + \frac{1878907415910194383861556524984552904786723431}{27750680242777590913369906840368889107177288676} a^{14} - \frac{176196678258345011219104477388413947375670547}{816196477728752673922644318834379091387567314} a^{13} + \frac{4034755868569049466205581522915139770589158993}{55501360485555181826739813680737778214354577352} a^{12} - \frac{3680730863189168647328638455948683627365013323}{13875340121388795456684953420184444553588644338} a^{11} + \frac{3781972040118964286636326263437401802687438959}{13875340121388795456684953420184444553588644338} a^{10} + \frac{10548409240254380322760782211379197724123173645}{55501360485555181826739813680737778214354577352} a^{9} + \frac{76015870483945897399377257567771917317723943}{816196477728752673922644318834379091387567314} a^{8} - \frac{3062551476613740279224073885137436736045999857}{6937670060694397728342476710092222276794322169} a^{7} + \frac{289704030935609966439801627816822046460381445}{1290729313617562368063716597226459958473362264} a^{6} - \frac{2056071846997051867386947525033241024936695593}{13875340121388795456684953420184444553588644338} a^{5} - \frac{411467641655629657842423179092713056016082363}{6937670060694397728342476710092222276794322169} a^{4} + \frac{1155318440373172567375602106835630360823031823}{55501360485555181826739813680737778214354577352} a^{3} + \frac{10911699878569613185957582454266250063806206413}{27750680242777590913369906840368889107177288676} a^{2} - \frac{1219697487646971144145676010488268958067205209}{13875340121388795456684953420184444553588644338} a + \frac{950103823530377406269781037078611432893401036}{6937670060694397728342476710092222276794322169}$

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

Class group and class number

$C_{26}$, which has order $26$ (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:  \( 1814278.67304 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).D_4$ (as 16T121):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $(C_2\times C_4).D_4$
Character table for $(C_2\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 4.0.2197.1, 4.4.344929.1, 4.0.26533.1, 8.8.242830046698681.1, 8.0.242830046698681.1, 8.0.118976015041.1

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$13.8.7.2$x^{8} - 52$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
13.8.7.2$x^{8} - 52$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$157$157.4.0.1$x^{4} - x + 15$$1$$4$$0$$C_4$$[\ ]^{4}$
157.4.3.4$x^{4} + 19625$$4$$1$$3$$C_4$$[\ ]_{4}$
157.4.3.4$x^{4} + 19625$$4$$1$$3$$C_4$$[\ ]_{4}$
157.4.0.1$x^{4} - x + 15$$1$$4$$0$$C_4$$[\ ]^{4}$