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Properties

Label 16.4.24109722907...7601.3
Degree $16$
Signature $[4, 6]$
Discriminant $13^{10}\cdot 53^{10}$
Root discriminant $59.41$
Ramified primes $13, 53$
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![-29952, 9984, -13312, 12064, -75936, 29248, 39776, -56614, 5275, 19682, -5049, -2418, 776, 122, -46, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 46*x^14 + 122*x^13 + 776*x^12 - 2418*x^11 - 5049*x^10 + 19682*x^9 + 5275*x^8 - 56614*x^7 + 39776*x^6 + 29248*x^5 - 75936*x^4 + 12064*x^3 - 13312*x^2 + 9984*x - 29952)
 
gp: K = bnfinit(x^16 - 2*x^15 - 46*x^14 + 122*x^13 + 776*x^12 - 2418*x^11 - 5049*x^10 + 19682*x^9 + 5275*x^8 - 56614*x^7 + 39776*x^6 + 29248*x^5 - 75936*x^4 + 12064*x^3 - 13312*x^2 + 9984*x - 29952, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 46 x^{14} + 122 x^{13} + 776 x^{12} - 2418 x^{11} - 5049 x^{10} + 19682 x^{9} + 5275 x^{8} - 56614 x^{7} + 39776 x^{6} + 29248 x^{5} - 75936 x^{4} + 12064 x^{3} - 13312 x^{2} + 9984 x - 29952 \)

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:  \(24109722907876309716269637601=13^{10}\cdot 53^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{104} a^{11} - \frac{3}{26} a^{10} - \frac{3}{52} a^{9} + \frac{5}{52} a^{8} - \frac{1}{13} a^{7} + \frac{9}{52} a^{6} + \frac{23}{104} a^{5} + \frac{3}{26} a^{4} - \frac{1}{8} a^{3}$, $\frac{1}{208} a^{12} + \frac{3}{104} a^{10} - \frac{5}{104} a^{9} - \frac{11}{52} a^{8} - \frac{1}{8} a^{7} - \frac{21}{208} a^{6} - \frac{3}{26} a^{5} - \frac{25}{208} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{832} a^{13} - \frac{1}{416} a^{11} - \frac{9}{416} a^{10} + \frac{27}{208} a^{9} - \frac{1}{416} a^{8} - \frac{61}{832} a^{7} + \frac{9}{52} a^{6} + \frac{207}{832} a^{5} - \frac{3}{26} a^{4} - \frac{7}{16} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{43264} a^{14} + \frac{7}{21632} a^{13} - \frac{21}{21632} a^{12} + \frac{57}{21632} a^{11} + \frac{17}{169} a^{10} + \frac{2347}{21632} a^{9} + \frac{8679}{43264} a^{8} + \frac{565}{21632} a^{7} - \frac{3209}{43264} a^{6} + \frac{393}{21632} a^{5} + \frac{1759}{10816} a^{4} + \frac{177}{416} a^{3} + \frac{101}{208} a^{2} + \frac{23}{52} a - \frac{3}{52}$, $\frac{1}{16021946815780474543104} a^{15} + \frac{181997716506829}{308114361841932202752} a^{14} + \frac{2616501712238351845}{8010973407890237271552} a^{13} - \frac{318833812120896665}{616228723683864405504} a^{12} - \frac{670137090669401569}{308114361841932202752} a^{11} + \frac{97962779871236332057}{2670324469296745757184} a^{10} - \frac{38329778279605105805}{1780216312864497171456} a^{9} - \frac{398618466452529983503}{4005486703945118635776} a^{8} + \frac{2902628873562188681875}{16021946815780474543104} a^{7} + \frac{915748814600162597003}{4005486703945118635776} a^{6} - \frac{401134706638832467511}{2002743351972559317888} a^{5} - \frac{195685242056273298785}{1001371675986279658944} a^{4} + \frac{942616535433783233}{3209524602520127112} a^{3} - \frac{15832922727404054491}{38514295230241525344} a^{2} - \frac{655553912487999757}{1481319047316981744} a - \frac{1095462195952124279}{3209524602520127112}$

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:  \( 120347206.048 \) (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

\(\Q(\sqrt{13}) \), 4.4.8957.1, 8.4.225360027841.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ 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$13.2.1.1$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}$
$53$53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$