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Properties

Label 16.0.23789977600...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{12}\cdot 241^{2}$
Root discriminant $18.77$
Ramified primes $2, 5, 241$
Class number $1$
Class group Trivial
Galois group $C_2^4.C_2^3$ (as 16T208)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 6, 55, 90, 74, -4, -13, 0, 71, 16, -33, 12, 4, 2, 1, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + x^14 + 2*x^13 + 4*x^12 + 12*x^11 - 33*x^10 + 16*x^9 + 71*x^8 - 13*x^6 - 4*x^5 + 74*x^4 + 90*x^3 + 55*x^2 + 6*x + 1)
 
gp: K = bnfinit(x^16 - 2*x^15 + x^14 + 2*x^13 + 4*x^12 + 12*x^11 - 33*x^10 + 16*x^9 + 71*x^8 - 13*x^6 - 4*x^5 + 74*x^4 + 90*x^3 + 55*x^2 + 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + x^{14} + 2 x^{13} + 4 x^{12} + 12 x^{11} - 33 x^{10} + 16 x^{9} + 71 x^{8} - 13 x^{6} - 4 x^{5} + 74 x^{4} + 90 x^{3} + 55 x^{2} + 6 x + 1 \)

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:  \(237899776000000000000=2^{24}\cdot 5^{12}\cdot 241^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.77$
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:  $2, 5, 241$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{93} a^{14} - \frac{6}{31} a^{13} - \frac{34}{93} a^{12} + \frac{12}{31} a^{11} - \frac{2}{31} a^{10} + \frac{4}{31} a^{9} + \frac{13}{31} a^{8} - \frac{20}{93} a^{7} - \frac{23}{93} a^{6} + \frac{13}{31} a^{5} + \frac{1}{31} a^{4} - \frac{1}{93} a^{3} - \frac{14}{31} a^{2} - \frac{1}{3} a - \frac{19}{93}$, $\frac{1}{14483303013} a^{15} + \frac{10722314}{4827767671} a^{14} + \frac{915878174}{14483303013} a^{13} - \frac{2041257170}{4827767671} a^{12} - \frac{1406577682}{4827767671} a^{11} + \frac{2029624386}{4827767671} a^{10} - \frac{697526446}{4827767671} a^{9} + \frac{7020368107}{14483303013} a^{8} - \frac{3171168038}{14483303013} a^{7} + \frac{2245624864}{4827767671} a^{6} + \frac{1237102193}{4827767671} a^{5} - \frac{502003411}{14483303013} a^{4} - \frac{1043750927}{4827767671} a^{3} + \frac{4105901498}{14483303013} a^{2} + \frac{6099015218}{14483303013} a - \frac{1414883385}{4827767671}$

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:  \( -\frac{604534222}{4827767671} a^{15} + \frac{4266132967}{14483303013} a^{14} - \frac{37656654}{155734441} a^{13} - \frac{2148019603}{14483303013} a^{12} - \frac{2259908890}{4827767671} a^{11} - \frac{6591203668}{4827767671} a^{10} + \frac{21980731982}{4827767671} a^{9} - \frac{18235654914}{4827767671} a^{8} - \frac{106545711386}{14483303013} a^{7} + \frac{33063665359}{14483303013} a^{6} - \frac{485615410}{4827767671} a^{5} + \frac{2573293257}{4827767671} a^{4} - \frac{138983310820}{14483303013} a^{3} - \frac{41761804800}{4827767671} a^{2} - \frac{73899660052}{14483303013} a + \frac{6488871659}{14483303013} \) (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:  \( 17723.6658829 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T208):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\zeta_{5})\), 4.0.8000.2, \(\Q(\sqrt{2}, \sqrt{5})\), 8.4.15424000000.4, 8.4.616960000.2, 8.0.64000000.2

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$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}$
241Data not computed