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Properties

Label 16.8.18953736582...5625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{8}\cdot 13^{8}\cdot 29^{6}$
Root discriminant $28.50$
Ramified primes $5, 13, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\wr C_2^2$ (as 16T128)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, -78, 148, 205, -300, -342, -167, -46, -72, -227, -120, -30, -17, -8, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 8*x^14 - 17*x^13 - 30*x^12 - 120*x^11 - 227*x^10 - 72*x^9 - 46*x^8 - 167*x^7 - 342*x^6 - 300*x^5 + 205*x^4 + 148*x^3 - 78*x^2 + 4*x + 1)
 
gp: K = bnfinit(x^16 - x^15 - 8*x^14 - 17*x^13 - 30*x^12 - 120*x^11 - 227*x^10 - 72*x^9 - 46*x^8 - 167*x^7 - 342*x^6 - 300*x^5 + 205*x^4 + 148*x^3 - 78*x^2 + 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 8 x^{14} - 17 x^{13} - 30 x^{12} - 120 x^{11} - 227 x^{10} - 72 x^{9} - 46 x^{8} - 167 x^{7} - 342 x^{6} - 300 x^{5} + 205 x^{4} + 148 x^{3} - 78 x^{2} + 4 x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(189537365823025172265625=5^{8}\cdot 13^{8}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.50$
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:  $5, 13, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{12} a^{12} + \frac{1}{6} a^{11} - \frac{1}{2} a^{10} + \frac{1}{12} a^{9} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} + \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{1}{12} a^{3} + \frac{1}{6} a^{2} + \frac{1}{12}$, $\frac{1}{12} a^{13} + \frac{1}{6} a^{11} + \frac{1}{12} a^{10} - \frac{1}{6} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{5}{12} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{12} a - \frac{1}{6}$, $\frac{1}{204} a^{14} - \frac{5}{204} a^{13} - \frac{1}{34} a^{12} - \frac{73}{204} a^{11} - \frac{55}{204} a^{10} + \frac{5}{51} a^{9} + \frac{1}{68} a^{8} + \frac{83}{204} a^{7} - \frac{2}{17} a^{6} + \frac{67}{204} a^{5} - \frac{59}{204} a^{4} + \frac{14}{51} a^{3} - \frac{79}{204} a^{2} + \frac{1}{12} a + \frac{43}{102}$, $\frac{1}{147757105659588} a^{15} - \frac{31794195785}{36939276414897} a^{14} + \frac{1191580576304}{36939276414897} a^{13} - \frac{771231222703}{73878552829794} a^{12} + \frac{453048599578}{36939276414897} a^{11} - \frac{16509584064055}{73878552829794} a^{10} - \frac{1293698011999}{73878552829794} a^{9} - \frac{1202766886180}{36939276414897} a^{8} + \frac{4039303290395}{12313092138299} a^{7} + \frac{13600480374791}{36939276414897} a^{6} + \frac{29135650186031}{73878552829794} a^{5} + \frac{22011112334899}{73878552829794} a^{4} - \frac{396203628646}{36939276414897} a^{3} - \frac{5655960533912}{36939276414897} a^{2} - \frac{5066920303015}{73878552829794} a - \frac{29643232236767}{147757105659588}$

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:  $11$
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:  \( 454688.665861 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_2^2$ (as 16T128):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 16 conjugacy class representatives for $C_2\wr C_2^2$
Character table for $C_2\wr C_2^2$

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{5}) \), 4.4.122525.1, 4.4.725.1, \(\Q(\sqrt{5}, \sqrt{13})\), 8.8.15012375625.1, 8.4.517668125.1, 8.4.435358893125.1

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
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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$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$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$