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Properties

Label 16.4.98340377973...249.28
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![21493, 266626, -522759, 313284, -9053, -221534, 200582, -73509, 1951, 12035, -3613, -1056, 597, -36, -3, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 3*x^14 - 36*x^13 + 597*x^12 - 1056*x^11 - 3613*x^10 + 12035*x^9 + 1951*x^8 - 73509*x^7 + 200582*x^6 - 221534*x^5 - 9053*x^4 + 313284*x^3 - 522759*x^2 + 266626*x + 21493)
 
gp: K = bnfinit(x^16 - 5*x^15 - 3*x^14 - 36*x^13 + 597*x^12 - 1056*x^11 - 3613*x^10 + 12035*x^9 + 1951*x^8 - 73509*x^7 + 200582*x^6 - 221534*x^5 - 9053*x^4 + 313284*x^3 - 522759*x^2 + 266626*x + 21493, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 3 x^{14} - 36 x^{13} + 597 x^{12} - 1056 x^{11} - 3613 x^{10} + 12035 x^{9} + 1951 x^{8} - 73509 x^{7} + 200582 x^{6} - 221534 x^{5} - 9053 x^{4} + 313284 x^{3} - 522759 x^{2} + 266626 x + 21493 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{12} a^{13} + \frac{1}{12} a^{12} - \frac{1}{4} a^{11} - \frac{1}{12} a^{10} - \frac{1}{12} a^{9} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{6} a^{5} + \frac{1}{6} a^{3} - \frac{1}{4} a^{2} + \frac{1}{12} a - \frac{1}{6}$, $\frac{1}{468} a^{14} + \frac{5}{468} a^{13} + \frac{11}{234} a^{12} - \frac{29}{234} a^{11} + \frac{1}{9} a^{10} + \frac{95}{468} a^{9} - \frac{17}{78} a^{8} + \frac{11}{52} a^{7} - \frac{125}{468} a^{6} + \frac{103}{468} a^{5} - \frac{5}{18} a^{4} - \frac{67}{468} a^{3} - \frac{155}{468} a^{2} + \frac{83}{468} a + \frac{139}{468}$, $\frac{1}{133935387124089695744847193774820508204} a^{15} + \frac{655959907704816158510231360179425}{826761648914133924350908603548274742} a^{14} - \frac{983704660385465554417355599751552821}{44645129041363231914949064591606836068} a^{13} + \frac{1081832294954567912475589914023718532}{11161282260340807978737266147901709017} a^{12} - \frac{487909526404980499538238580912895884}{3720427420113602659579088715967236339} a^{11} + \frac{6458377976584838741337902174210474603}{44645129041363231914949064591606836068} a^{10} + \frac{9741550773659645653401325677159816863}{133935387124089695744847193774820508204} a^{9} + \frac{4941769425108674795060291471678628701}{44645129041363231914949064591606836068} a^{8} + \frac{10095347686624705872674056243248216791}{66967693562044847872423596887410254102} a^{7} - \frac{14508036631768035440593715115869742551}{66967693562044847872423596887410254102} a^{6} + \frac{1874462989863605316369158031602209501}{44645129041363231914949064591606836068} a^{5} - \frac{35310278973858113414849974003972468253}{133935387124089695744847193774820508204} a^{4} + \frac{7029532347150546191393666501715253}{413380824457066962175454301774137371} a^{3} + \frac{7232885497047151473273807442136931209}{22322564520681615957474532295803418034} a^{2} - \frac{3269839946361022994594682385143203215}{22322564520681615957474532295803418034} a - \frac{20309968800585074706113188166686328387}{133935387124089695744847193774820508204}$

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:  \( 442568437.03 \) (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 ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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