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Properties

Label 16.0.10366174140...5216.6
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{12}\cdot 13^{6}\cdot 97^{2}$
Root discriminant $42.26$
Ramified primes $2, 3, 13, 97$
Class number $144$ (GRH)
Class group $[2, 2, 36]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T707)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6133, -27024, 69862, -103040, 100450, -74928, 54430, -30648, 15655, -7360, 3694, -1472, 488, -140, 40, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 40*x^14 - 140*x^13 + 488*x^12 - 1472*x^11 + 3694*x^10 - 7360*x^9 + 15655*x^8 - 30648*x^7 + 54430*x^6 - 74928*x^5 + 100450*x^4 - 103040*x^3 + 69862*x^2 - 27024*x + 6133)
 
gp: K = bnfinit(x^16 - 8*x^15 + 40*x^14 - 140*x^13 + 488*x^12 - 1472*x^11 + 3694*x^10 - 7360*x^9 + 15655*x^8 - 30648*x^7 + 54430*x^6 - 74928*x^5 + 100450*x^4 - 103040*x^3 + 69862*x^2 - 27024*x + 6133, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 40 x^{14} - 140 x^{13} + 488 x^{12} - 1472 x^{11} + 3694 x^{10} - 7360 x^{9} + 15655 x^{8} - 30648 x^{7} + 54430 x^{6} - 74928 x^{5} + 100450 x^{4} - 103040 x^{3} + 69862 x^{2} - 27024 x + 6133 \)

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:  \(103661741408596533461385216=2^{32}\cdot 3^{12}\cdot 13^{6}\cdot 97^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.26$
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:  $2, 3, 13, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{481} a^{12} - \frac{6}{481} a^{11} - \frac{161}{481} a^{10} - \frac{102}{481} a^{9} + \frac{53}{481} a^{8} - \frac{147}{481} a^{7} + \frac{4}{13} a^{6} + \frac{141}{481} a^{5} + \frac{230}{481} a^{4} + \frac{2}{37} a^{3} + \frac{113}{481} a^{2} + \frac{5}{13} a + \frac{17}{481}$, $\frac{1}{481} a^{13} - \frac{197}{481} a^{11} - \frac{106}{481} a^{10} - \frac{6}{37} a^{9} + \frac{171}{481} a^{8} + \frac{228}{481} a^{7} + \frac{67}{481} a^{6} + \frac{114}{481} a^{5} - \frac{1}{13} a^{4} - \frac{212}{481} a^{3} - \frac{99}{481} a^{2} + \frac{165}{481} a + \frac{102}{481}$, $\frac{1}{19156639481629} a^{14} - \frac{7}{19156639481629} a^{13} - \frac{16709203468}{19156639481629} a^{12} + \frac{100255220899}{19156639481629} a^{11} + \frac{8284088670179}{19156639481629} a^{10} - \frac{4026170579378}{19156639481629} a^{9} + \frac{8896815380725}{19156639481629} a^{8} + \frac{8829208865172}{19156639481629} a^{7} - \frac{486988102344}{1473587652433} a^{6} - \frac{8850115470471}{19156639481629} a^{5} - \frac{7320891006588}{19156639481629} a^{4} + \frac{6790410803651}{19156639481629} a^{3} + \frac{1518381677139}{19156639481629} a^{2} - \frac{7874429027382}{19156639481629} a + \frac{7112855351185}{19156639481629}$, $\frac{1}{208290141083752117} a^{15} + \frac{5429}{208290141083752117} a^{14} + \frac{19777157333053}{208290141083752117} a^{13} - \frac{2873289391495}{208290141083752117} a^{12} + \frac{60282319265623859}{208290141083752117} a^{11} - \frac{11673779957375814}{208290141083752117} a^{10} + \frac{82959354357108724}{208290141083752117} a^{9} + \frac{66987591124899125}{208290141083752117} a^{8} + \frac{31809574735576652}{208290141083752117} a^{7} + \frac{37145839944756372}{208290141083752117} a^{6} - \frac{4775468208514166}{208290141083752117} a^{5} - \frac{31732035644198382}{208290141083752117} a^{4} + \frac{88640782166506703}{208290141083752117} a^{3} - \frac{46542776910118207}{208290141083752117} a^{2} + \frac{5609588791939700}{208290141083752117} a - \frac{97014105741394090}{208290141083752117}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{36}$, which has order $144$ (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:  \( -1 \) (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:  \( 20288.2882099 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T707):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 31 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 4.4.7488.1, 8.8.5438803968.1, 8.0.104963309568.1, 8.0.10181441028096.1

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
3Data not computed
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{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}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$