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Properties

Label 16.0.26382542843...5141.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{7}\cdot 47^{15}$
Root discriminant $59.75$
Ramified primes $3, 47$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $D_{16}$ (as 16T56)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![360000, -666000, 572700, -353640, 178753, -66537, 24446, -8051, 4036, -2969, 1364, -717, 392, -83, 22, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 22*x^14 - 83*x^13 + 392*x^12 - 717*x^11 + 1364*x^10 - 2969*x^9 + 4036*x^8 - 8051*x^7 + 24446*x^6 - 66537*x^5 + 178753*x^4 - 353640*x^3 + 572700*x^2 - 666000*x + 360000)
 
gp: K = bnfinit(x^16 - 5*x^15 + 22*x^14 - 83*x^13 + 392*x^12 - 717*x^11 + 1364*x^10 - 2969*x^9 + 4036*x^8 - 8051*x^7 + 24446*x^6 - 66537*x^5 + 178753*x^4 - 353640*x^3 + 572700*x^2 - 666000*x + 360000, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 22 x^{14} - 83 x^{13} + 392 x^{12} - 717 x^{11} + 1364 x^{10} - 2969 x^{9} + 4036 x^{8} - 8051 x^{7} + 24446 x^{6} - 66537 x^{5} + 178753 x^{4} - 353640 x^{3} + 572700 x^{2} - 666000 x + 360000 \)

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:  \(26382542843244145338070455141=3^{7}\cdot 47^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.75$
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:  $3, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} + \frac{1}{8} a^{4} + \frac{1}{8} a^{3} - \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{8} - \frac{1}{16} a^{7} - \frac{3}{16} a^{6} + \frac{3}{16} a^{5} - \frac{5}{16} a^{3} + \frac{3}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{16} a^{7} + \frac{1}{16} a^{6} + \frac{1}{8} a^{5} - \frac{3}{16} a^{4} - \frac{1}{2} a^{3} + \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{48} a^{12} + \frac{1}{48} a^{11} - \frac{1}{48} a^{8} - \frac{1}{16} a^{7} + \frac{1}{8} a^{6} - \frac{1}{6} a^{5} + \frac{7}{48} a^{4} - \frac{5}{16} a^{3} + \frac{1}{3} a^{2} - \frac{1}{4} a$, $\frac{1}{960} a^{13} - \frac{1}{192} a^{12} - \frac{3}{160} a^{11} - \frac{3}{160} a^{10} - \frac{13}{960} a^{9} + \frac{11}{320} a^{8} - \frac{1}{160} a^{7} + \frac{53}{480} a^{6} + \frac{61}{960} a^{5} + \frac{53}{320} a^{4} - \frac{29}{60} a^{3} - \frac{11}{80} a^{2} - \frac{1}{5} a$, $\frac{1}{139200} a^{14} + \frac{11}{27840} a^{13} - \frac{113}{23200} a^{12} - \frac{323}{23200} a^{11} - \frac{3013}{139200} a^{10} + \frac{1951}{46400} a^{9} + \frac{449}{23200} a^{8} + \frac{1973}{69600} a^{7} + \frac{24421}{139200} a^{6} + \frac{2153}{46400} a^{5} + \frac{1759}{34800} a^{4} - \frac{1471}{11600} a^{3} - \frac{683}{5800} a^{2} + \frac{151}{580} a + \frac{6}{29}$, $\frac{1}{248401544940652622189544000} a^{15} + \frac{70757182049688745981}{24840154494065262218954400} a^{14} + \frac{110522885789796646754647}{248401544940652622189544000} a^{13} + \frac{80201461914720161154809}{31050193117581577773693000} a^{12} + \frac{2248769998385407096868897}{248401544940652622189544000} a^{11} + \frac{663034576638248907549173}{41400257490108770364924000} a^{10} - \frac{311588697492743369649091}{248401544940652622189544000} a^{9} - \frac{781862810519096260630349}{15525096558790788886846500} a^{8} - \frac{9018840056534669026732649}{248401544940652622189544000} a^{7} - \frac{24012217384606316412861943}{124200772470326311094772000} a^{6} + \frac{8391007119866520029661581}{248401544940652622189544000} a^{5} - \frac{64961372845505076206759}{350849639746684494618000} a^{4} + \frac{8152785647809791358145381}{31050193117581577773693000} a^{3} - \frac{62318225332922474489882}{258751609313179814780775} a^{2} - \frac{44795128993980358045039}{207001287450543851824620} a - \frac{479867373341992587226}{3450021457509064197077}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  $7$
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:  \( 397627138.337 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{16}$ (as 16T56):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $D_{16}$
Character table for $D_{16}$

Intermediate fields

\(\Q(\sqrt{-47}) \), 4.0.311469.1, 8.0.13678824252501.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ $16$ R ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47Data not computed