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Properties

Label 16.0.38982341165...0000.7
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{8}\cdot 29^{6}$
Root discriminant $22.36$
Ramified primes $2, 5, 29$
Class number $1$
Class group Trivial
Galois group $D_8:C_2$ (as 16T44)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31, -226, 877, -2200, 3793, -4596, 3921, -2244, 947, -380, 162, -74, 63, -32, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 10*x^14 - 32*x^13 + 63*x^12 - 74*x^11 + 162*x^10 - 380*x^9 + 947*x^8 - 2244*x^7 + 3921*x^6 - 4596*x^5 + 3793*x^4 - 2200*x^3 + 877*x^2 - 226*x + 31)
 
gp: K = bnfinit(x^16 - 4*x^15 + 10*x^14 - 32*x^13 + 63*x^12 - 74*x^11 + 162*x^10 - 380*x^9 + 947*x^8 - 2244*x^7 + 3921*x^6 - 4596*x^5 + 3793*x^4 - 2200*x^3 + 877*x^2 - 226*x + 31, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 10 x^{14} - 32 x^{13} + 63 x^{12} - 74 x^{11} + 162 x^{10} - 380 x^{9} + 947 x^{8} - 2244 x^{7} + 3921 x^{6} - 4596 x^{5} + 3793 x^{4} - 2200 x^{3} + 877 x^{2} - 226 x + 31 \)

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:  \(3898234116505600000000=2^{24}\cdot 5^{8}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.36$
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:  $2, 5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} + \frac{3}{7} a^{11} + \frac{3}{7} a^{9} + \frac{1}{7} a^{8} - \frac{3}{7} a^{6} - \frac{1}{7} a^{5} - \frac{1}{7} a^{4} + \frac{3}{7} a^{2} - \frac{1}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{13} - \frac{2}{7} a^{11} + \frac{3}{7} a^{10} - \frac{1}{7} a^{9} - \frac{3}{7} a^{8} - \frac{3}{7} a^{7} + \frac{1}{7} a^{6} + \frac{2}{7} a^{5} + \frac{3}{7} a^{4} + \frac{3}{7} a^{3} - \frac{3}{7} a^{2} - \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{14} + \frac{2}{7} a^{11} - \frac{1}{7} a^{10} + \frac{3}{7} a^{9} - \frac{1}{7} a^{8} + \frac{1}{7} a^{7} + \frac{3}{7} a^{6} + \frac{1}{7} a^{5} + \frac{1}{7} a^{4} - \frac{3}{7} a^{3} - \frac{3}{7} a^{2} - \frac{1}{7} a - \frac{3}{7}$, $\frac{1}{1813064000154709631} a^{15} + \frac{5414072147263411}{78828869571943897} a^{14} - \frac{14947864256439950}{259009142879244233} a^{13} + \frac{110824876088417795}{1813064000154709631} a^{12} + \frac{880841106605152731}{1813064000154709631} a^{11} - \frac{524370282888153290}{1813064000154709631} a^{10} - \frac{123004687909256010}{259009142879244233} a^{9} - \frac{796999848541357932}{1813064000154709631} a^{8} + \frac{9206051186406441}{259009142879244233} a^{7} + \frac{171203187636782829}{1813064000154709631} a^{6} + \frac{30103433971901987}{1813064000154709631} a^{5} + \frac{98803115330158266}{259009142879244233} a^{4} - \frac{679346387272239786}{1813064000154709631} a^{3} + \frac{695463034375020531}{1813064000154709631} a^{2} - \frac{101858700575932189}{1813064000154709631} a - \frac{25254538006737607}{58485935488861601}$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 9192.49654065 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_8:C_2$ (as 16T44):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $D_8:C_2$
Character table for $D_8:C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}) \), 4.4.46400.1, 4.4.725.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.2152960000.1

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 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$