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Properties

Label 16.10.2384296512...1875.1
Degree $16$
Signature $[10, 3]$
Discriminant $-\,3^{12}\cdot 5^{12}\cdot 179^{5}$
Root discriminant $38.56$
Ramified primes $3, 5, 179$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1354

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2789, 23515, 6155, -86560, -48826, 27140, 22485, 3450, -3524, -2370, 120, 110, 29, 50, -10, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 10*x^14 + 50*x^13 + 29*x^12 + 110*x^11 + 120*x^10 - 2370*x^9 - 3524*x^8 + 3450*x^7 + 22485*x^6 + 27140*x^5 - 48826*x^4 - 86560*x^3 + 6155*x^2 + 23515*x - 2789)
 
gp: K = bnfinit(x^16 - 5*x^15 - 10*x^14 + 50*x^13 + 29*x^12 + 110*x^11 + 120*x^10 - 2370*x^9 - 3524*x^8 + 3450*x^7 + 22485*x^6 + 27140*x^5 - 48826*x^4 - 86560*x^3 + 6155*x^2 + 23515*x - 2789, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 10 x^{14} + 50 x^{13} + 29 x^{12} + 110 x^{11} + 120 x^{10} - 2370 x^{9} - 3524 x^{8} + 3450 x^{7} + 22485 x^{6} + 27140 x^{5} - 48826 x^{4} - 86560 x^{3} + 6155 x^{2} + 23515 x - 2789 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-23842965126465199951171875=-\,3^{12}\cdot 5^{12}\cdot 179^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.56$
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:  $3, 5, 179$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{292988876281948316076509930552547} a^{15} + \frac{9071628663636852527573822983754}{292988876281948316076509930552547} a^{14} + \frac{14148207013692384966091309654489}{97662958760649438692169976850849} a^{13} - \frac{5533581988103383449401249714590}{292988876281948316076509930552547} a^{12} + \frac{34148622516261168721657816866955}{97662958760649438692169976850849} a^{11} - \frac{116657764226704012929608940028931}{292988876281948316076509930552547} a^{10} - \frac{2034762498482836152077284522546}{26635352389268028734228175504777} a^{9} - \frac{42604589214239518383529765411278}{97662958760649438692169976850849} a^{8} + \frac{11475138700865106922134350061367}{26635352389268028734228175504777} a^{7} + \frac{86925600968691869168754176007269}{292988876281948316076509930552547} a^{6} - \frac{33492831866113902250394262396497}{97662958760649438692169976850849} a^{5} - \frac{29016406520473118033729828060249}{292988876281948316076509930552547} a^{4} - \frac{5575409806275631342802536202235}{97662958760649438692169976850849} a^{3} - \frac{20860442538646407142800546536810}{292988876281948316076509930552547} a^{2} + \frac{129466016247446737147379970322241}{292988876281948316076509930552547} a + \frac{47838258015566786190567252554515}{97662958760649438692169976850849}$

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:  $12$
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:  \( 7759805.06061 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1354:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 59 conjugacy class representatives for t16n1354 are not computed
Character table for t16n1354 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 8.6.226546875.1

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 $16$ R R $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$3$3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$179$$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.1.1$x^{2} - 179$$2$$1$$1$$C_2$$[\ ]_{2}$
179.2.1.1$x^{2} - 179$$2$$1$$1$$C_2$$[\ ]_{2}$
179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
179.4.3.2$x^{4} - 179$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$