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Properties

Label 16.0.14012498575...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{30}\cdot 5^{6}\cdot 17^{4}$
Root discriminant $13.62$
Ramified primes $2, 5, 17$
Class number $1$
Class group Trivial
Galois group 16T1276

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 14, -10, -30, 74, -18, -98, 120, -46, -22, 50, -38, 6, 10, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 10*x^14 + 6*x^13 - 38*x^12 + 50*x^11 - 22*x^10 - 46*x^9 + 120*x^8 - 98*x^7 - 18*x^6 + 74*x^5 - 30*x^4 - 10*x^3 + 14*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^16 - 6*x^15 + 10*x^14 + 6*x^13 - 38*x^12 + 50*x^11 - 22*x^10 - 46*x^9 + 120*x^8 - 98*x^7 - 18*x^6 + 74*x^5 - 30*x^4 - 10*x^3 + 14*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 10 x^{14} + 6 x^{13} - 38 x^{12} + 50 x^{11} - 22 x^{10} - 46 x^{9} + 120 x^{8} - 98 x^{7} - 18 x^{6} + 74 x^{5} - 30 x^{4} - 10 x^{3} + 14 x^{2} - 6 x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1401249857536000000=2^{30}\cdot 5^{6}\cdot 17^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.62$
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:  $2, 5, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{94195691} a^{15} - \frac{7973454}{94195691} a^{14} + \frac{46340008}{94195691} a^{13} + \frac{18516747}{94195691} a^{12} + \frac{6739706}{94195691} a^{11} + \frac{40584953}{94195691} a^{10} + \frac{31034327}{94195691} a^{9} - \frac{19423598}{94195691} a^{8} - \frac{72613}{321487} a^{7} + \frac{22041413}{94195691} a^{6} + \frac{21059663}{94195691} a^{5} + \frac{9124582}{94195691} a^{4} + \frac{16737359}{94195691} a^{3} + \frac{15255447}{94195691} a^{2} - \frac{38146684}{94195691} a - \frac{16667613}{94195691}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{100683}{18911} a^{15} - \frac{559370}{18911} a^{14} + \frac{759243}{18911} a^{13} + \frac{937971}{18911} a^{12} - \frac{3409905}{18911} a^{11} + \frac{3534732}{18911} a^{10} - \frac{664063}{18911} a^{9} - \frac{4924660}{18911} a^{8} + \frac{9931141}{18911} a^{7} - \frac{5518260}{18911} a^{6} - \frac{4220971}{18911} a^{5} + \frac{5615415}{18911} a^{4} - \frac{619795}{18911} a^{3} - \frac{1224928}{18911} a^{2} + \frac{889397}{18911} a - \frac{235223}{18911} \) (order $4$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 824.262669302 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1276:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1276
Character table for t16n1276 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.0.320.1, 8.0.6963200.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$5$5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$