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Properties

Label 16.4.98340377973...9249.6
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![-138069, -1079850, -500920, 222204, 137646, 1820, 28831, 27549, 3181, -4987, -2195, 402, 333, -4, -23, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 23*x^14 - 4*x^13 + 333*x^12 + 402*x^11 - 2195*x^10 - 4987*x^9 + 3181*x^8 + 27549*x^7 + 28831*x^6 + 1820*x^5 + 137646*x^4 + 222204*x^3 - 500920*x^2 - 1079850*x - 138069)
 
gp: K = bnfinit(x^16 - 23*x^14 - 4*x^13 + 333*x^12 + 402*x^11 - 2195*x^10 - 4987*x^9 + 3181*x^8 + 27549*x^7 + 28831*x^6 + 1820*x^5 + 137646*x^4 + 222204*x^3 - 500920*x^2 - 1079850*x - 138069, 1)
 

Normalized defining polynomial

\( x^{16} - 23 x^{14} - 4 x^{13} + 333 x^{12} + 402 x^{11} - 2195 x^{10} - 4987 x^{9} + 3181 x^{8} + 27549 x^{7} + 28831 x^{6} + 1820 x^{5} + 137646 x^{4} + 222204 x^{3} - 500920 x^{2} - 1079850 x - 138069 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{87} a^{14} + \frac{7}{87} a^{13} - \frac{35}{87} a^{12} - \frac{38}{87} a^{11} - \frac{2}{87} a^{10} + \frac{3}{29} a^{9} + \frac{20}{87} a^{8} - \frac{31}{87} a^{7} - \frac{25}{87} a^{6} - \frac{25}{87} a^{5} - \frac{8}{87} a^{4} + \frac{41}{87} a^{3} + \frac{11}{29} a^{2} - \frac{31}{87} a$, $\frac{1}{3087104925234142670611568755504608531248127} a^{15} - \frac{386089562150400139871893699663858804730}{134221953271049681330937771978461240489049} a^{14} - \frac{12171285224295767848784350585380377536111}{134221953271049681330937771978461240489049} a^{13} - \frac{353631180914746429683877820005671474028494}{1029034975078047556870522918501536177082709} a^{12} + \frac{894281767893544430712737588033139978813401}{3087104925234142670611568755504608531248127} a^{11} - \frac{415997915278907842905335165343710996985372}{1029034975078047556870522918501536177082709} a^{10} - \frac{144713737180407549275196793433751067308504}{1029034975078047556870522918501536177082709} a^{9} - \frac{415700828586313062577467196625502793002235}{3087104925234142670611568755504608531248127} a^{8} - \frac{9858336143862524776133958485109878677266}{35483964657863708857604238569018488864921} a^{7} + \frac{1417385720772538814323194023768965173080905}{3087104925234142670611568755504608531248127} a^{6} + \frac{34529485886655033295991150564450309214350}{106451893973591126572812715707055466594763} a^{5} - \frac{811980951327697626242671916149100734839673}{3087104925234142670611568755504608531248127} a^{4} - \frac{173706567907737802557848535674093102158466}{1029034975078047556870522918501536177082709} a^{3} + \frac{944077258462084963204646576286200885712271}{3087104925234142670611568755504608531248127} a^{2} + \frac{136700083713845861465211359129971561008613}{3087104925234142670611568755504608531248127} a + \frac{176690169661299984920833668977023820469}{1542781072081030819895836459522542994127}$

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:  \( 180663763.783 \) (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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$$[\ ]_{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61Data not computed