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Properties

Label 16.16.5345972853...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{24}\cdot 5^{8}\cdot 13^{8}$
Root discriminant $22.80$
Ramified primes $2, 5, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $Q_8 : C_2$ (as 16T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -6, 41, 86, -459, -82, 1580, -936, -1453, 1370, 362, -624, 44, 100, -20, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 20*x^14 + 100*x^13 + 44*x^12 - 624*x^11 + 362*x^10 + 1370*x^9 - 1453*x^8 - 936*x^7 + 1580*x^6 - 82*x^5 - 459*x^4 + 86*x^3 + 41*x^2 - 6*x - 1)
 
gp: K = bnfinit(x^16 - 4*x^15 - 20*x^14 + 100*x^13 + 44*x^12 - 624*x^11 + 362*x^10 + 1370*x^9 - 1453*x^8 - 936*x^7 + 1580*x^6 - 82*x^5 - 459*x^4 + 86*x^3 + 41*x^2 - 6*x - 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 20 x^{14} + 100 x^{13} + 44 x^{12} - 624 x^{11} + 362 x^{10} + 1370 x^{9} - 1453 x^{8} - 936 x^{7} + 1580 x^{6} - 82 x^{5} - 459 x^{4} + 86 x^{3} + 41 x^{2} - 6 x - 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5345972853145600000000=2^{24}\cdot 5^{8}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.80$
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:  $2, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6} a^{12} - \frac{1}{2} a^{11} + \frac{1}{6} a^{10} - \frac{1}{6} a^{9} + \frac{1}{3} a^{8} - \frac{1}{6} a^{7} + \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} + \frac{1}{6} a^{2} + \frac{1}{6}$, $\frac{1}{102} a^{13} - \frac{1}{51} a^{12} - \frac{1}{51} a^{11} + \frac{4}{17} a^{10} + \frac{19}{102} a^{9} + \frac{49}{102} a^{8} + \frac{19}{102} a^{7} - \frac{1}{102} a^{6} - \frac{29}{102} a^{5} - \frac{35}{102} a^{4} + \frac{8}{51} a^{3} + \frac{43}{102} a^{2} + \frac{7}{102} a + \frac{37}{102}$, $\frac{1}{714} a^{14} + \frac{15}{238} a^{12} - \frac{19}{102} a^{11} - \frac{94}{357} a^{10} - \frac{45}{119} a^{9} + \frac{39}{238} a^{8} - \frac{58}{357} a^{7} - \frac{31}{714} a^{6} - \frac{24}{119} a^{5} + \frac{59}{119} a^{4} + \frac{4}{119} a^{3} - \frac{44}{119} a^{2} + \frac{1}{14} a - \frac{283}{714}$, $\frac{1}{110326566} a^{15} - \frac{36013}{110326566} a^{14} + \frac{88701}{18387761} a^{13} + \frac{3412639}{110326566} a^{12} - \frac{1391624}{55163283} a^{11} - \frac{44320859}{110326566} a^{10} + \frac{22571329}{110326566} a^{9} + \frac{17942725}{55163283} a^{8} - \frac{14563609}{110326566} a^{7} + \frac{13310173}{55163283} a^{6} - \frac{73573}{18387761} a^{5} - \frac{38103901}{110326566} a^{4} - \frac{916471}{2163266} a^{3} + \frac{326293}{6489798} a^{2} - \frac{2407957}{110326566} a - \frac{5324727}{36775522}$

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:  $15$
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:  \( 316695.993468 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4:C_2$ (as 16T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $Q_8 : C_2$
Character table for $Q_8 : C_2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{130}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{26}) \), \(\Q(\sqrt{2}, \sqrt{65})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{13})\), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{10}, \sqrt{26})\), \(\Q(\sqrt{5}, \sqrt{26})\), \(\Q(\sqrt{10}, \sqrt{13})\), 8.8.73116160000.2, 8.8.432640000.1 x2, 8.8.2924646400.1 x2, 8.8.1142440000.1 x2

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$5.8.4.1$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}$
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$