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Properties

Label 16.8.53586372784...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 5^{8}\cdot 19^{4}\cdot 89^{4}$
Root discriminant $40.56$
Ramified primes $2, 5, 19, 89$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4.C_2^4$ (as 16T573)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15101, -6438, -111076, -107440, 89052, 106560, -60417, -40078, 24054, 8438, -5162, -1120, 632, 80, -41, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 41*x^14 + 80*x^13 + 632*x^12 - 1120*x^11 - 5162*x^10 + 8438*x^9 + 24054*x^8 - 40078*x^7 - 60417*x^6 + 106560*x^5 + 89052*x^4 - 107440*x^3 - 111076*x^2 - 6438*x + 15101)
 
gp: K = bnfinit(x^16 - 2*x^15 - 41*x^14 + 80*x^13 + 632*x^12 - 1120*x^11 - 5162*x^10 + 8438*x^9 + 24054*x^8 - 40078*x^7 - 60417*x^6 + 106560*x^5 + 89052*x^4 - 107440*x^3 - 111076*x^2 - 6438*x + 15101, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 41 x^{14} + 80 x^{13} + 632 x^{12} - 1120 x^{11} - 5162 x^{10} + 8438 x^{9} + 24054 x^{8} - 40078 x^{7} - 60417 x^{6} + 106560 x^{5} + 89052 x^{4} - 107440 x^{3} - 111076 x^{2} - 6438 x + 15101 \)

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:  \(53586372784036249600000000=2^{24}\cdot 5^{8}\cdot 19^{4}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.56$
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:  $2, 5, 19, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{10} + \frac{1}{8} a^{8} - \frac{3}{8} a^{6} + \frac{1}{4} a^{5} + \frac{3}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{3}{8}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{16} a^{11} - \frac{3}{16} a^{10} - \frac{7}{16} a^{9} - \frac{1}{16} a^{8} + \frac{5}{16} a^{7} + \frac{1}{16} a^{6} + \frac{1}{16} a^{5} + \frac{7}{16} a^{4} + \frac{1}{8} a^{3} - \frac{5}{16} a - \frac{7}{16}$, $\frac{1}{32} a^{14} - \frac{1}{16} a^{8} + \frac{1}{16} a^{7} - \frac{1}{2} a^{6} + \frac{1}{8} a^{5} + \frac{15}{32} a^{4} + \frac{1}{16} a^{3} - \frac{1}{32} a^{2} + \frac{3}{32}$, $\frac{1}{187467103864093621280606481770848} a^{15} - \frac{1289969389004514207143440588341}{93733551932046810640303240885424} a^{14} + \frac{299720027713368549035114799349}{23433387983011702660075810221356} a^{13} + \frac{1753777398858108303830595900629}{46866775966023405320151620442712} a^{12} + \frac{50030641192887740870251187773}{5858346995752925665018952555339} a^{11} - \frac{3571584869367904369557639676699}{46866775966023405320151620442712} a^{10} - \frac{42098183899100716395295204769493}{93733551932046810640303240885424} a^{9} - \frac{17613821502087642894190473794675}{93733551932046810640303240885424} a^{8} + \frac{5014545972390751456152029804943}{46866775966023405320151620442712} a^{7} - \frac{4446070990999093471152446395977}{11716693991505851330037905110678} a^{6} - \frac{40224684733508106995151830032265}{187467103864093621280606481770848} a^{5} - \frac{1431594367164169086185965385389}{23433387983011702660075810221356} a^{4} - \frac{8800345459963505238374185618901}{187467103864093621280606481770848} a^{3} + \frac{687138259721406444370957002513}{93733551932046810640303240885424} a^{2} + \frac{171678304928832074376357252347}{4572368386929112714161133701728} a + \frac{24715974087271413548025589870683}{93733551932046810640303240885424}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 5313442.23444 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^4$ (as 16T573):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $C_2^4.C_2^4$
Character table for $C_2^4.C_2^4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.7600.1, 4.4.2225.1, 4.4.676400.1, 8.8.457516960000.1, 8.4.316840000.1, 8.4.1830067840000.2

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.8.16.12$x^{8} + 6 x^{6} + 4 x^{5} + 2 x^{4} + 12$$4$$2$$16$$C_2^2:C_4$$[2, 2, 3]^{2}$
2.8.8.2$x^{8} + 2 x^{7} + 8 x^{2} + 48$$2$$4$$8$$C_2^2:C_4$$[2, 2]^{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}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$89$89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$