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Properties

Label 16.0.58929626493...3936.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 37^{8}$
Root discriminant $17.20$
Ramified primes $2, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times S_4$ (as 16T61)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 0, 88, 0, 166, 0, -244, 0, 207, 0, -108, 0, 38, 0, -8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 + 38*x^12 - 108*x^10 + 207*x^8 - 244*x^6 + 166*x^4 + 88*x^2 + 9)
 
gp: K = bnfinit(x^16 - 8*x^14 + 38*x^12 - 108*x^10 + 207*x^8 - 244*x^6 + 166*x^4 + 88*x^2 + 9, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{14} + 38 x^{12} - 108 x^{10} + 207 x^{8} - 244 x^{6} + 166 x^{4} + 88 x^{2} + 9 \)

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:  \(58929626493994663936=2^{24}\cdot 37^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.20$
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:  $2, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{20} a^{12} - \frac{1}{4} a^{10} + \frac{1}{5} a^{8} - \frac{1}{20} a^{6} - \frac{1}{2} a^{5} + \frac{2}{5} a^{4} - \frac{1}{2} a^{3} - \frac{1}{20} a^{2} - \frac{1}{2} a - \frac{9}{20}$, $\frac{1}{20} a^{13} - \frac{1}{4} a^{11} + \frac{1}{5} a^{9} - \frac{1}{20} a^{7} - \frac{1}{2} a^{6} + \frac{2}{5} a^{5} - \frac{1}{2} a^{4} - \frac{1}{20} a^{3} - \frac{1}{2} a^{2} - \frac{9}{20} a$, $\frac{1}{68980} a^{14} - \frac{657}{68980} a^{12} + \frac{551}{17245} a^{10} + \frac{831}{68980} a^{8} - \frac{1}{2} a^{7} - \frac{1433}{3449} a^{6} - \frac{1}{2} a^{5} - \frac{14157}{68980} a^{4} - \frac{1}{2} a^{3} - \frac{77}{68980} a^{2} - \frac{2143}{17245}$, $\frac{1}{206940} a^{15} - \frac{2053}{103470} a^{13} - \frac{49531}{206940} a^{11} + \frac{1435}{13796} a^{9} - \frac{31397}{68980} a^{7} + \frac{61721}{206940} a^{5} - \frac{1}{2} a^{4} + \frac{18088}{51735} a^{3} - \frac{1}{2} a^{2} + \frac{56959}{206940} a - \frac{1}{2}$

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:  \( \frac{611}{20694} a^{15} - \frac{2396}{10347} a^{13} + \frac{11114}{10347} a^{11} - \frac{10224}{3449} a^{9} + \frac{18855}{3449} a^{7} - \frac{63724}{10347} a^{5} + \frac{77117}{20694} a^{3} + \frac{33535}{10347} a \) (order $4$)
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:  \( 3490.02865985 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_4$ (as 16T61):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 48
The 10 conjugacy class representatives for $C_2\times S_4$
Character table for $C_2\times S_4$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-37}) \), \(\Q(\sqrt{37}) \), 4.0.592.1, \(\Q(i, \sqrt{37})\), 8.0.479785216.2, 8.0.5607424.1, 8.0.7676563456.4

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

Sibling fields

Degree 6 siblings: 6.4.350464.1, 6.4.12967168.3
Degree 8 siblings: 8.0.5607424.1, 8.0.7676563456.4
Degree 12 siblings: 12.0.491300061184.1, 12.8.168147445940224.1, 12.0.672589783760896.1, 12.4.18178102263808.1, 12.0.672589783760896.3, 12.0.672589783760896.2
Degree 24 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$37$37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$