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Properties

Label 16.0.56019694551...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{10}\cdot 11^{2}\cdot 41^{4}$
Root discriminant $26.41$
Ramified primes $2, 5, 11, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T799

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8149, 7844, -762, -7528, -6040, 1108, 3530, -124, -507, 32, -14, 52, 48, -64, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 64*x^13 + 48*x^12 + 52*x^11 - 14*x^10 + 32*x^9 - 507*x^8 - 124*x^7 + 3530*x^6 + 1108*x^5 - 6040*x^4 - 7528*x^3 - 762*x^2 + 7844*x + 8149)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 64*x^13 + 48*x^12 + 52*x^11 - 14*x^10 + 32*x^9 - 507*x^8 - 124*x^7 + 3530*x^6 + 1108*x^5 - 6040*x^4 - 7528*x^3 - 762*x^2 + 7844*x + 8149, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 64 x^{13} + 48 x^{12} + 52 x^{11} - 14 x^{10} + 32 x^{9} - 507 x^{8} - 124 x^{7} + 3530 x^{6} + 1108 x^{5} - 6040 x^{4} - 7528 x^{3} - 762 x^{2} + 7844 x + 8149 \)

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:  \(56019694551040000000000=2^{24}\cdot 5^{10}\cdot 11^{2}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.41$
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:  $2, 5, 11, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{10} a^{10} + \frac{1}{5} a^{9} + \frac{2}{5} a^{7} - \frac{1}{2} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{10} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{10} a^{11} + \frac{1}{10} a^{9} - \frac{1}{10} a^{8} - \frac{3}{10} a^{7} - \frac{2}{5} a^{6} - \frac{3}{10} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{2}{5} a^{2} + \frac{1}{10} a + \frac{1}{10}$, $\frac{1}{10} a^{12} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a + \frac{3}{10}$, $\frac{1}{50} a^{13} - \frac{1}{50} a^{12} + \frac{1}{25} a^{11} + \frac{1}{25} a^{10} + \frac{1}{25} a^{9} - \frac{7}{50} a^{8} + \frac{7}{25} a^{7} + \frac{12}{25} a^{6} - \frac{12}{25} a^{5} + \frac{7}{50} a^{4} + \frac{11}{25} a^{3} + \frac{1}{5} a^{2} - \frac{23}{50} a - \frac{8}{25}$, $\frac{1}{250} a^{14} - \frac{9}{250} a^{12} - \frac{11}{250} a^{11} + \frac{2}{125} a^{10} - \frac{3}{50} a^{9} - \frac{24}{125} a^{8} + \frac{63}{250} a^{7} - \frac{1}{25} a^{6} - \frac{37}{250} a^{5} - \frac{28}{125} a^{4} + \frac{27}{250} a^{3} - \frac{103}{250} a^{2} - \frac{59}{250} a - \frac{61}{250}$, $\frac{1}{25500231853325370336250} a^{15} + \frac{3471470264711598298}{12750115926662685168125} a^{14} - \frac{35412732289590711117}{12750115926662685168125} a^{13} - \frac{20779065476478690508}{510004637066507406725} a^{12} - \frac{314794395705659787677}{25500231853325370336250} a^{11} - \frac{314535161516542641731}{25500231853325370336250} a^{10} - \frac{2918795533924629754913}{25500231853325370336250} a^{9} + \frac{344616813098448263383}{2550023185332537033625} a^{8} - \frac{4933391795358785786437}{25500231853325370336250} a^{7} - \frac{6754401847658840005947}{25500231853325370336250} a^{6} + \frac{6064591754088733006167}{25500231853325370336250} a^{5} - \frac{319269450230552264717}{1159101447878425924375} a^{4} + \frac{4853464644584515465357}{12750115926662685168125} a^{3} + \frac{11058546847508976962353}{25500231853325370336250} a^{2} + \frac{498541800847007411223}{1020009274133014813450} a + \frac{246040952489504467322}{12750115926662685168125}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{8860678453450310}{20400185482660296269} a^{15} + \frac{108655933233240167}{40800370965320592538} a^{14} - \frac{333049414569686137}{40800370965320592538} a^{13} + \frac{264067589238169691}{40800370965320592538} a^{12} + \frac{666090829095216465}{40800370965320592538} a^{11} - \frac{940864746066074860}{20400185482660296269} a^{10} - \frac{492751862500858562}{20400185482660296269} a^{9} - \frac{2042232154446996529}{40800370965320592538} a^{8} + \frac{5233858175473885525}{40800370965320592538} a^{7} + \frac{6905074292890104817}{20400185482660296269} a^{6} - \frac{23792976735814000133}{20400185482660296269} a^{5} - \frac{10390172393978812951}{3709124633210962958} a^{4} - \frac{6722096574199149797}{40800370965320592538} a^{3} + \frac{131333829026802259089}{40800370965320592538} a^{2} + \frac{187822591574933341339}{40800370965320592538} a + \frac{57357204535758793174}{20400185482660296269} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 175259.909113 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T799:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n799 are not computed
Character table for t16n799 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-5}) \), 4.4.16400.1, 4.0.1025.1, \(\Q(i, \sqrt{5})\), 8.0.268960000.3

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.12.20$x^{8} + 8 x^{6} + 12 x^{4} + 80$$4$$2$$12$$C_2^3: C_4$$[2, 2, 2]^{4}$
2.8.12.20$x^{8} + 8 x^{6} + 12 x^{4} + 80$$4$$2$$12$$C_2^3: C_4$$[2, 2, 2]^{4}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$