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Properties

Label 16.0.24790578076...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 5^{12}\cdot 31^{4}$
Root discriminant $44.63$
Ramified primes $2, 5, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4.C_2^3$ (as 16T268)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![923521, 0, 476656, 0, 232562, 0, 54498, 0, 10940, 0, 1092, 0, 77, 0, 14, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 14*x^14 + 77*x^12 + 1092*x^10 + 10940*x^8 + 54498*x^6 + 232562*x^4 + 476656*x^2 + 923521)
 
gp: K = bnfinit(x^16 + 14*x^14 + 77*x^12 + 1092*x^10 + 10940*x^8 + 54498*x^6 + 232562*x^4 + 476656*x^2 + 923521, 1)
 

Normalized defining polynomial

\( x^{16} + 14 x^{14} + 77 x^{12} + 1092 x^{10} + 10940 x^{8} + 54498 x^{6} + 232562 x^{4} + 476656 x^{2} + 923521 \)

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:  \(247905780760576000000000000=2^{40}\cdot 5^{12}\cdot 31^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.63$
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:  $2, 5, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{155} a^{8} - \frac{58}{155} a^{6} - \frac{56}{155} a^{4} - \frac{22}{155} a^{2} + \frac{1}{5}$, $\frac{1}{155} a^{9} - \frac{58}{155} a^{7} - \frac{56}{155} a^{5} - \frac{22}{155} a^{3} + \frac{1}{5} a$, $\frac{1}{155} a^{10} - \frac{2}{31} a^{6} - \frac{3}{31} a^{4} - \frac{1}{31} a^{2} - \frac{2}{5}$, $\frac{1}{155} a^{11} - \frac{2}{31} a^{7} - \frac{3}{31} a^{5} - \frac{1}{31} a^{3} - \frac{2}{5} a$, $\frac{1}{33635} a^{12} + \frac{76}{33635} a^{10} - \frac{16}{33635} a^{8} - \frac{1822}{33635} a^{6} + \frac{1213}{4805} a^{4} - \frac{96}{217} a^{2} + \frac{17}{35}$, $\frac{1}{33635} a^{13} + \frac{76}{33635} a^{11} - \frac{16}{33635} a^{9} - \frac{1822}{33635} a^{7} + \frac{1213}{4805} a^{5} - \frac{96}{217} a^{3} + \frac{17}{35} a$, $\frac{1}{98795446435} a^{14} - \frac{1056869}{98795446435} a^{12} + \frac{188583989}{98795446435} a^{10} + \frac{12489777}{14113635205} a^{8} + \frac{9899652093}{98795446435} a^{6} - \frac{1505429811}{3186949885} a^{4} - \frac{2434475}{20560967} a^{2} - \frac{1438949}{3316285}$, $\frac{1}{98795446435} a^{15} - \frac{1056869}{98795446435} a^{13} + \frac{188583989}{98795446435} a^{11} + \frac{12489777}{14113635205} a^{9} + \frac{9899652093}{98795446435} a^{7} - \frac{1505429811}{3186949885} a^{5} - \frac{2434475}{20560967} a^{3} - \frac{1438949}{3316285} a$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( \frac{1081204}{98795446435} a^{14} + \frac{8248997}{98795446435} a^{12} + \frac{16978862}{98795446435} a^{10} + \frac{167694685}{19759089287} a^{8} + \frac{1181737976}{19759089287} a^{6} + \frac{325143197}{3186949885} a^{4} + \frac{4078688}{14686405} a^{2} - \frac{491499}{3316285} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 15745481.3625 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T268):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 29 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.0.8000.2, \(\Q(\zeta_{5})\), \(\Q(\sqrt{2}, \sqrt{5})\), 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$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}$
31Data not computed