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Properties

Label 16.0.59347523320...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 79^{6}$
Root discriminant $17.21$
Ramified primes $5, 79$
Class number $1$
Class group Trivial
Galois group $D_4:C_4$ (as 16T26)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, -56, 176, -284, 321, -342, 293, -227, 157, -89, 67, -46, 21, -8, 4, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 4*x^14 - 8*x^13 + 21*x^12 - 46*x^11 + 67*x^10 - 89*x^9 + 157*x^8 - 227*x^7 + 293*x^6 - 342*x^5 + 321*x^4 - 284*x^3 + 176*x^2 - 56*x + 16)
 
gp: K = bnfinit(x^16 - 3*x^15 + 4*x^14 - 8*x^13 + 21*x^12 - 46*x^11 + 67*x^10 - 89*x^9 + 157*x^8 - 227*x^7 + 293*x^6 - 342*x^5 + 321*x^4 - 284*x^3 + 176*x^2 - 56*x + 16, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 4 x^{14} - 8 x^{13} + 21 x^{12} - 46 x^{11} + 67 x^{10} - 89 x^{9} + 157 x^{8} - 227 x^{7} + 293 x^{6} - 342 x^{5} + 321 x^{4} - 284 x^{3} + 176 x^{2} - 56 x + 16 \)

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:  \(59347523320556640625=5^{12}\cdot 79^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.21$
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:  $5, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{10} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{25267944872} a^{15} + \frac{1575058559}{25267944872} a^{14} + \frac{485566519}{12633972436} a^{13} + \frac{525128229}{6316986218} a^{12} - \frac{4719940015}{25267944872} a^{11} - \frac{51116487}{3158493109} a^{10} + \frac{5579896867}{25267944872} a^{9} - \frac{8036058163}{25267944872} a^{8} + \frac{2274691851}{25267944872} a^{7} + \frac{6561857399}{25267944872} a^{6} - \frac{11230630957}{25267944872} a^{5} + \frac{3117316259}{6316986218} a^{4} - \frac{8905138343}{25267944872} a^{3} - \frac{5625835481}{12633972436} a^{2} - \frac{500392089}{6316986218} a + \frac{913334068}{3158493109}$

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:  \( \frac{27055331909}{25267944872} a^{15} - \frac{52325889703}{25267944872} a^{14} + \frac{13208098285}{6316986218} a^{13} - \frac{20091477810}{3158493109} a^{12} + \frac{397470085801}{25267944872} a^{11} - \frac{411294705483}{12633972436} a^{10} + \frac{939763492415}{25267944872} a^{9} - \frac{1416454493117}{25267944872} a^{8} + \frac{2748127814161}{25267944872} a^{7} - \frac{3225853136567}{25267944872} a^{6} + \frac{4513263408745}{25267944872} a^{5} - \frac{2236122731531}{12633972436} a^{4} + \frac{3976603345621}{25267944872} a^{3} - \frac{870033083155}{6316986218} a^{2} + \frac{134789557258}{3158493109} a - \frac{48302646998}{3158493109} \) (order $10$)
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:  \( 7222.77402778 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4:C_4$ (as 16T26):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $D_4:C_4$
Character table for $D_4:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.1975.1, \(\Q(\zeta_{5})\), 4.2.9875.1, 8.2.308149375.1, 8.2.7703734375.1, 8.0.97515625.1

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 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$79$79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$