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Properties

Label 16.0.281792804290560000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 3^{8}\cdot 5^{4}$
Root discriminant $12.32$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois group $C_2^2 \times D_4$ (as 16T25)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 6, -16, 44, -56, 60, -88, 83, -52, 36, -12, -4, 0, 2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 2*x^14 - 4*x^12 - 12*x^11 + 36*x^10 - 52*x^9 + 83*x^8 - 88*x^7 + 60*x^6 - 56*x^5 + 44*x^4 - 16*x^3 + 6*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^16 + 2*x^14 - 4*x^12 - 12*x^11 + 36*x^10 - 52*x^9 + 83*x^8 - 88*x^7 + 60*x^6 - 56*x^5 + 44*x^4 - 16*x^3 + 6*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} + 2 x^{14} - 4 x^{12} - 12 x^{11} + 36 x^{10} - 52 x^{9} + 83 x^{8} - 88 x^{7} + 60 x^{6} - 56 x^{5} + 44 x^{4} - 16 x^{3} + 6 x^{2} - 4 x + 1 \)

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:  \(281792804290560000=2^{36}\cdot 3^{8}\cdot 5^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.32$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{55} a^{14} - \frac{16}{55} a^{13} - \frac{9}{55} a^{12} + \frac{16}{55} a^{11} - \frac{2}{55} a^{10} - \frac{17}{55} a^{9} + \frac{17}{55} a^{8} - \frac{4}{11} a^{7} - \frac{1}{5} a^{6} - \frac{17}{55} a^{5} + \frac{24}{55} a^{4} - \frac{26}{55} a^{3} - \frac{8}{55} a^{2} + \frac{14}{55} a - \frac{7}{55}$, $\frac{1}{5383015} a^{15} + \frac{45429}{5383015} a^{14} + \frac{1218441}{5383015} a^{13} + \frac{2340626}{5383015} a^{12} - \frac{1235062}{5383015} a^{11} + \frac{1589068}{5383015} a^{10} - \frac{82243}{5383015} a^{9} - \frac{315873}{1076603} a^{8} + \frac{2408909}{5383015} a^{7} - \frac{899982}{5383015} a^{6} - \frac{214536}{489365} a^{5} - \frac{763176}{5383015} a^{4} + \frac{2068647}{5383015} a^{3} + \frac{1258899}{5383015} a^{2} - \frac{2347857}{5383015} a + \frac{416120}{1076603}$

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:  \( \frac{7335514}{1076603} a^{15} + \frac{21771183}{5383015} a^{14} + \frac{88719952}{5383015} a^{13} + \frac{54921153}{5383015} a^{12} - \frac{107077247}{5383015} a^{11} - \frac{496850306}{5383015} a^{10} + \frac{1021962669}{5383015} a^{9} - \frac{1332800764}{5383015} a^{8} + \frac{462145676}{1076603} a^{7} - \frac{1932399318}{5383015} a^{6} + \frac{107675024}{489365} a^{5} - \frac{1432973573}{5383015} a^{4} + \frac{817209282}{5383015} a^{3} - \frac{168754859}{5383015} a^{2} + \frac{144507767}{5383015} a - \frac{68578216}{5383015} \) (order $24$)
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:  \( 1291.96804794 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times D_4$ (as 16T25):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 \times D_4$
Character table for $C_2^2 \times D_4$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-3}) \), 4.0.1280.1, 4.0.2880.1, 4.0.320.1, 4.0.11520.1, \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(i, \sqrt{6})\), \(\Q(\zeta_{8})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\zeta_{12})\), \(\Q(\zeta_{24})\), 8.0.530841600.2, 8.0.530841600.4, 8.0.6553600.1, 8.0.530841600.3, 8.0.132710400.4, 8.0.8294400.2

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

Sibling fields

Galois closure: data not computed
Degree 16 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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$