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Properties

Label 16.4.98340377973...249.24
Degree $16$
Signature $[4, 6]$
Discriminant $13^{10}\cdot 61^{10}$
Root discriminant $64.87$
Ramified primes $13, 61$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T875

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7472, 10360, -5472, 3548, 1261, -3270, 10465, -6069, 4881, -2057, 49, 98, -135, 54, -4, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 4*x^14 + 54*x^13 - 135*x^12 + 98*x^11 + 49*x^10 - 2057*x^9 + 4881*x^8 - 6069*x^7 + 10465*x^6 - 3270*x^5 + 1261*x^4 + 3548*x^3 - 5472*x^2 + 10360*x + 7472)
 
gp: K = bnfinit(x^16 - 5*x^15 - 4*x^14 + 54*x^13 - 135*x^12 + 98*x^11 + 49*x^10 - 2057*x^9 + 4881*x^8 - 6069*x^7 + 10465*x^6 - 3270*x^5 + 1261*x^4 + 3548*x^3 - 5472*x^2 + 10360*x + 7472, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 4 x^{14} + 54 x^{13} - 135 x^{12} + 98 x^{11} + 49 x^{10} - 2057 x^{9} + 4881 x^{8} - 6069 x^{7} + 10465 x^{6} - 3270 x^{5} + 1261 x^{4} + 3548 x^{3} - 5472 x^{2} + 10360 x + 7472 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(98340377973019428085802419249=13^{10}\cdot 61^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.87$
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:  $13, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} + \frac{3}{8} a^{8} + \frac{3}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{2} a^{5} + \frac{3}{8} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{8} a^{6} + \frac{3}{8} a^{5} - \frac{1}{2} a^{4} - \frac{3}{8} a^{3}$, $\frac{1}{24} a^{13} - \frac{1}{24} a^{12} - \frac{1}{24} a^{10} - \frac{5}{24} a^{9} - \frac{5}{24} a^{8} - \frac{5}{24} a^{6} + \frac{5}{24} a^{5} + \frac{1}{6} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{1560} a^{14} - \frac{1}{130} a^{13} + \frac{4}{195} a^{12} - \frac{49}{1560} a^{11} - \frac{31}{520} a^{10} + \frac{371}{1560} a^{9} - \frac{359}{1560} a^{8} + \frac{523}{1560} a^{7} - \frac{241}{520} a^{6} + \frac{3}{260} a^{5} + \frac{16}{195} a^{4} + \frac{33}{104} a^{3} - \frac{209}{780} a^{2} + \frac{1}{390} a - \frac{97}{195}$, $\frac{1}{2083626094025188359166573560} a^{15} + \frac{29585680763166325398409}{160278930309629873782044120} a^{14} + \frac{630880905529409482059689}{2083626094025188359166573560} a^{13} + \frac{7118468378513191344802219}{2083626094025188359166573560} a^{12} + \frac{95925851556905496821451401}{2083626094025188359166573560} a^{11} + \frac{65631745944105775292025449}{2083626094025188359166573560} a^{10} + \frac{16091376321506738812546239}{69454203134172945305552452} a^{9} - \frac{336944077566342031308628379}{1041813047012594179583286780} a^{8} + \frac{699626392471335756483084199}{2083626094025188359166573560} a^{7} - \frac{5477011107948585562853623}{26713155051604978963674020} a^{6} - \frac{1660379967630986633006435}{7184917565604097790229564} a^{5} - \frac{209265676198893205589403247}{520906523506297089791643390} a^{4} + \frac{519522011796407269918999637}{2083626094025188359166573560} a^{3} - \frac{148178947459305414376626693}{347271015670864726527762260} a^{2} - \frac{2596931088743817811619555}{8013946515481493689102206} a - \frac{16417176518048071852212328}{260453261753148544895821695}$

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:  $9$
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:  \( 356375611.783 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T875:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n875
Character table for t16n875 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 4.4.10309.1, 8.4.395451064801.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
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}$
61Data not computed