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Properties

Label 16.8.13895289211...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{16}\cdot 5^{12}\cdot 61^{2}\cdot 123601^{4}$
Root discriminant $209.62$
Ramified primes $2, 5, 61, 123601$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1392

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![952576, 0, 2334592, 0, -2111568, 0, -444920, 0, 219641, 0, 3380, 0, -1122, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 - 1122*x^12 + 3380*x^10 + 219641*x^8 - 444920*x^6 - 2111568*x^4 + 2334592*x^2 + 952576)
 
gp: K = bnfinit(x^16 - 4*x^14 - 1122*x^12 + 3380*x^10 + 219641*x^8 - 444920*x^6 - 2111568*x^4 + 2334592*x^2 + 952576, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{14} - 1122 x^{12} + 3380 x^{10} + 219641 x^{8} - 444920 x^{6} - 2111568 x^{4} + 2334592 x^{2} + 952576 \)

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:  \(13895289211961074442017936000000000000=2^{16}\cdot 5^{12}\cdot 61^{2}\cdot 123601^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $209.62$
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:  $2, 5, 61, 123601$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{3}$, $\frac{1}{32} a^{8} - \frac{1}{16} a^{6} - \frac{3}{32} a^{4} + \frac{1}{8} a^{2} - \frac{1}{2}$, $\frac{1}{64} a^{9} - \frac{1}{64} a^{8} - \frac{1}{32} a^{7} + \frac{1}{32} a^{6} - \frac{3}{64} a^{5} + \frac{3}{64} a^{4} - \frac{3}{16} a^{3} - \frac{1}{16} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{704} a^{10} + \frac{3}{704} a^{8} - \frac{13}{704} a^{6} - \frac{1}{64} a^{4} - \frac{1}{4} a^{3} + \frac{1}{176} a^{2} - \frac{1}{4} a + \frac{17}{44}$, $\frac{1}{1408} a^{11} + \frac{3}{1408} a^{9} - \frac{1}{64} a^{8} - \frac{13}{1408} a^{7} + \frac{1}{32} a^{6} + \frac{15}{128} a^{5} - \frac{5}{64} a^{4} - \frac{43}{352} a^{3} + \frac{1}{16} a^{2} + \frac{39}{88} a - \frac{1}{4}$, $\frac{1}{134930048} a^{12} - \frac{3}{134930048} a^{10} + \frac{1866207}{134930048} a^{8} - \frac{3732409}{134930048} a^{6} + \frac{763379}{33732512} a^{4} - \frac{74207}{8433128} a^{2} + \frac{35}{17281}$, $\frac{1}{269860096} a^{13} - \frac{1}{269860096} a^{12} - \frac{3}{269860096} a^{11} + \frac{3}{269860096} a^{10} + \frac{1866207}{269860096} a^{9} - \frac{1866207}{269860096} a^{8} + \frac{13133847}{269860096} a^{7} + \frac{3732409}{269860096} a^{6} + \frac{763379}{67465024} a^{5} + \frac{7669749}{67465024} a^{4} - \frac{282087}{4216564} a^{3} - \frac{2034075}{16866256} a^{2} - \frac{8623}{17281} a + \frac{8623}{17281}$, $\frac{1}{488716633856} a^{14} + \frac{41}{22214392448} a^{12} - \frac{11992157}{122179158464} a^{10} - \frac{2913216089}{244358316928} a^{8} - \frac{1}{16} a^{7} + \frac{22690488723}{488716633856} a^{6} - \frac{1}{8} a^{5} + \frac{9975435095}{122179158464} a^{4} + \frac{3}{16} a^{3} + \frac{3065958459}{30544789616} a^{2} - \frac{1}{2} a + \frac{4453127}{31295891}$, $\frac{1}{977433267712} a^{15} + \frac{41}{44428784896} a^{13} - \frac{1}{269860096} a^{12} - \frac{11992157}{244358316928} a^{11} + \frac{3}{269860096} a^{10} - \frac{2913216089}{488716633856} a^{9} + \frac{2350357}{269860096} a^{8} - \frac{38399090509}{977433267712} a^{7} - \frac{4700719}{269860096} a^{6} - \frac{20569354521}{244358316928} a^{5} - \frac{1962901}{33732512} a^{4} + \frac{14520254565}{61089579232} a^{3} + \frac{282087}{4216564} a^{2} - \frac{22389637}{125183564} a + \frac{17211}{69124}$

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:  \( 9525835804700 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1392:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 80 conjugacy class representatives for t16n1392 are not computed
Character table for t16n1392 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 4.4.247202000.1, 4.4.3090025.1, 8.8.61108828804000000.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
61Data not computed
123601Data not computed