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Properties

Label 16.0.11888044252...7313.1
Degree $16$
Signature $[0, 8]$
Discriminant $31^{15}\cdot 47^{7}$
Root discriminant $134.80$
Ramified primes $31, 47$
Class number $9$ (GRH)
Class group $[3, 3]$ (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![85351591, 54292082, 9240200, 7815964, -3351926, -7059308, -1426424, 872766, 409334, 2216, -36633, -5763, 1385, 275, -33, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 33*x^14 + 275*x^13 + 1385*x^12 - 5763*x^11 - 36633*x^10 + 2216*x^9 + 409334*x^8 + 872766*x^7 - 1426424*x^6 - 7059308*x^5 - 3351926*x^4 + 7815964*x^3 + 9240200*x^2 + 54292082*x + 85351591)
 
gp: K = bnfinit(x^16 - 2*x^15 - 33*x^14 + 275*x^13 + 1385*x^12 - 5763*x^11 - 36633*x^10 + 2216*x^9 + 409334*x^8 + 872766*x^7 - 1426424*x^6 - 7059308*x^5 - 3351926*x^4 + 7815964*x^3 + 9240200*x^2 + 54292082*x + 85351591, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 33 x^{14} + 275 x^{13} + 1385 x^{12} - 5763 x^{11} - 36633 x^{10} + 2216 x^{9} + 409334 x^{8} + 872766 x^{7} - 1426424 x^{6} - 7059308 x^{5} - 3351926 x^{4} + 7815964 x^{3} + 9240200 x^{2} + 54292082 x + 85351591 \)

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:  \(11888044252790185643475966652687313=31^{15}\cdot 47^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $134.80$
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:  $31, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{11} + \frac{2}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{4465} a^{13} - \frac{52}{4465} a^{12} - \frac{1696}{4465} a^{11} + \frac{231}{893} a^{10} - \frac{303}{893} a^{9} + \frac{117}{235} a^{8} + \frac{952}{4465} a^{7} - \frac{563}{4465} a^{6} + \frac{227}{893} a^{5} - \frac{102}{4465} a^{4} + \frac{804}{4465} a^{3} + \frac{233}{4465} a^{2} + \frac{864}{4465} a - \frac{106}{235}$, $\frac{1}{4465} a^{14} + \frac{13}{893} a^{12} - \frac{2202}{4465} a^{11} + \frac{100}{893} a^{10} - \frac{652}{4465} a^{9} + \frac{458}{4465} a^{8} - \frac{174}{4465} a^{7} - \frac{1351}{4465} a^{6} + \frac{873}{4465} a^{5} - \frac{7}{893} a^{4} + \frac{1856}{4465} a^{3} - \frac{83}{893} a^{2} - \frac{1736}{4465} a - \frac{107}{235}$, $\frac{1}{590869187482586784419471956480141908874175} a^{15} + \frac{9454830420893407707143349894189943398}{118173837496517356883894391296028381774835} a^{14} - \frac{12745795867396369199919338501682198313}{590869187482586784419471956480141908874175} a^{13} + \frac{7144830713489940361448004459562540038024}{590869187482586784419471956480141908874175} a^{12} + \frac{143045500877948771848646695580441832324883}{590869187482586784419471956480141908874175} a^{11} + \frac{172363153432771047013003855314028134056478}{590869187482586784419471956480141908874175} a^{10} + \frac{85336998977815176976642194726126503369178}{590869187482586784419471956480141908874175} a^{9} + \frac{166101825842369266438173085595446740644717}{590869187482586784419471956480141908874175} a^{8} + \frac{151656211490981340117274040632247811823943}{590869187482586784419471956480141908874175} a^{7} - \frac{172766618115926430821463734745499701821138}{590869187482586784419471956480141908874175} a^{6} + \frac{9412672455917364134107402433268964745069}{118173837496517356883894391296028381774835} a^{5} + \frac{129444316778757196317729983554999671173657}{590869187482586784419471956480141908874175} a^{4} - \frac{33763031168551821849610490142601849954032}{590869187482586784419471956480141908874175} a^{3} - \frac{236722897048866225228396789639593803537}{1818059038407959336675298327631205873459} a^{2} - \frac{11563078368038778211107551262373955872883}{118173837496517356883894391296028381774835} a - \frac{732395073977029252996202024088292681}{1709923477679507295319305450915320975}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$ (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:  \( 5904415332.32 \) (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

\(\Q(\sqrt{-31}) \), 4.0.1400177.1, 8.0.2856442134846353.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ R $16$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ R $16$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$

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
31Data not computed
$47$$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$