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Properties

Label 16.0.12985283690...2129.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 367^{6}$
Root discriminant $20.87$
Ramified primes $3, 367$
Class number $1$
Class group Trivial
Galois group $\GL(2,Z/4)$ (as 16T186)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![619, -2279, 3602, -3628, 3134, -2557, 1776, -1077, 763, -513, 252, -85, 32, -22, 14, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 14*x^14 - 22*x^13 + 32*x^12 - 85*x^11 + 252*x^10 - 513*x^9 + 763*x^8 - 1077*x^7 + 1776*x^6 - 2557*x^5 + 3134*x^4 - 3628*x^3 + 3602*x^2 - 2279*x + 619)
 
gp: K = bnfinit(x^16 - 5*x^15 + 14*x^14 - 22*x^13 + 32*x^12 - 85*x^11 + 252*x^10 - 513*x^9 + 763*x^8 - 1077*x^7 + 1776*x^6 - 2557*x^5 + 3134*x^4 - 3628*x^3 + 3602*x^2 - 2279*x + 619, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 14 x^{14} - 22 x^{13} + 32 x^{12} - 85 x^{11} + 252 x^{10} - 513 x^{9} + 763 x^{8} - 1077 x^{7} + 1776 x^{6} - 2557 x^{5} + 3134 x^{4} - 3628 x^{3} + 3602 x^{2} - 2279 x + 619 \)

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:  \(1298528369092716162129=3^{12}\cdot 367^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.87$
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:  $3, 367$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} - \frac{1}{8} a^{5} + \frac{3}{8} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a + \frac{3}{8}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{3}{8} a^{5} + \frac{3}{8} a^{4} - \frac{1}{2} a^{2} + \frac{3}{8} a$, $\frac{1}{48} a^{12} - \frac{1}{48} a^{11} + \frac{1}{48} a^{10} - \frac{5}{48} a^{9} - \frac{1}{24} a^{8} + \frac{5}{48} a^{7} + \frac{5}{48} a^{6} - \frac{13}{48} a^{5} + \frac{1}{3} a^{4} + \frac{7}{48} a^{3} - \frac{5}{48} a^{2} + \frac{5}{48} a + \frac{19}{48}$, $\frac{1}{48} a^{13} + \frac{1}{24} a^{10} + \frac{5}{48} a^{9} + \frac{1}{16} a^{8} - \frac{1}{6} a^{7} - \frac{1}{24} a^{6} + \frac{3}{16} a^{5} - \frac{19}{48} a^{4} - \frac{1}{12} a^{3} + \frac{1}{4} a^{2} - \frac{23}{48}$, $\frac{1}{576} a^{14} - \frac{1}{96} a^{13} + \frac{1}{576} a^{12} + \frac{19}{576} a^{11} - \frac{1}{96} a^{10} - \frac{1}{72} a^{9} - \frac{29}{288} a^{8} - \frac{29}{192} a^{7} - \frac{17}{72} a^{6} - \frac{1}{72} a^{5} - \frac{5}{12} a^{4} - \frac{77}{576} a^{3} + \frac{67}{576} a^{2} + \frac{1}{8} a + \frac{241}{576}$, $\frac{1}{5756463559872} a^{15} - \frac{12431779}{639607062208} a^{14} + \frac{19519504675}{5756463559872} a^{13} - \frac{8114764951}{2878231779936} a^{12} - \frac{30174119497}{639607062208} a^{11} + \frac{114462478361}{2878231779936} a^{10} + \frac{298805533639}{2878231779936} a^{9} - \frac{176174290579}{1918821186624} a^{8} - \frac{1260005585161}{5756463559872} a^{7} - \frac{63153903575}{1439115889968} a^{6} + \frac{12816591571}{39975441388} a^{5} - \frac{784648187201}{5756463559872} a^{4} + \frac{23448319657}{359778972492} a^{3} + \frac{195482000855}{1918821186624} a^{2} - \frac{2274442535387}{5756463559872} a - \frac{750629739095}{1918821186624}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{16322959}{1030885308} a^{15} - \frac{542536277}{8247082464} a^{14} + \frac{676412809}{4123541232} a^{13} - \frac{186620675}{916342496} a^{12} + \frac{2657143027}{8247082464} a^{11} - \frac{1099973333}{1030885308} a^{10} + \frac{12654466597}{4123541232} a^{9} - \frac{22401625001}{4123541232} a^{8} + \frac{59789050681}{8247082464} a^{7} - \frac{43795109245}{4123541232} a^{6} + \frac{77718315227}{4123541232} a^{5} - \frac{24691070087}{1030885308} a^{4} + \frac{77543880029}{2749027488} a^{3} - \frac{270759809849}{8247082464} a^{2} + \frac{117352865119}{4123541232} a - \frac{85089406715}{8247082464} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 48883.6384753 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$\GL(2,Z/4)$ (as 16T186):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 96
The 14 conjugacy class representatives for $\GL(2,Z/4)$
Character table for $\GL(2,Z/4)$

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.4.9909.1, 4.0.9909.1, 8.0.98188281.1

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

Sibling fields

Degree 12 siblings: 12.6.5343738144414469803.1, 12.0.14560594398949509.3, 12.0.14560594398949509.2, 12.0.14560594398949509.4
Degree 16 sibling: Deg 16
Degree 24 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
367Data not computed