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Properties

Label 16.8.78946419571...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{6}\cdot 5^{4}\cdot 17^{8}\cdot 19^{2}\cdot 97^{8}$
Root discriminant $113.78$
Ramified primes $2, 5, 17, 19, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1123

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![92416, 0, -109744, 0, -113900, 0, 18367, 0, 22890, 0, 2243, 0, -323, 0, -14, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 14*x^14 - 323*x^12 + 2243*x^10 + 22890*x^8 + 18367*x^6 - 113900*x^4 - 109744*x^2 + 92416)
 
gp: K = bnfinit(x^16 - 14*x^14 - 323*x^12 + 2243*x^10 + 22890*x^8 + 18367*x^6 - 113900*x^4 - 109744*x^2 + 92416, 1)
 

Normalized defining polynomial

\( x^{16} - 14 x^{14} - 323 x^{12} + 2243 x^{10} + 22890 x^{8} + 18367 x^{6} - 113900 x^{4} - 109744 x^{2} + 92416 \)

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:  \(789464195714758583406110606440000=2^{6}\cdot 5^{4}\cdot 17^{8}\cdot 19^{2}\cdot 97^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $113.78$
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:  $2, 5, 17, 19, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{12} a^{8} + \frac{1}{6} a^{6} - \frac{1}{2} a^{3} - \frac{5}{12} a^{2} + \frac{1}{3}$, $\frac{1}{12} a^{9} + \frac{1}{6} a^{7} - \frac{5}{12} a^{3} - \frac{1}{6} a$, $\frac{1}{24} a^{10} - \frac{1}{24} a^{9} + \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{5}{24} a^{4} - \frac{7}{24} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{24} a^{11} - \frac{1}{24} a^{9} - \frac{5}{24} a^{5} + \frac{1}{24} a^{3} - \frac{1}{2} a$, $\frac{1}{48} a^{12} - \frac{1}{48} a^{10} - \frac{5}{48} a^{6} + \frac{1}{48} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{192} a^{13} + \frac{1}{64} a^{11} - \frac{15}{64} a^{7} - \frac{19}{192} a^{5} - \frac{13}{48} a^{3} - \frac{1}{2} a^{2} + \frac{1}{12} a$, $\frac{1}{1560045088417728} a^{14} - \frac{2397071947827}{520015029472576} a^{12} - \frac{16747262077}{1866082641648} a^{10} + \frac{27739940253475}{1560045088417728} a^{8} - \frac{48184235373965}{520015029472576} a^{6} - \frac{674157216709}{8125234835509} a^{4} - \frac{1}{2} a^{3} - \frac{36396405083249}{97502818026108} a^{2} + \frac{455670596483}{1282931816133}$, $\frac{1}{3120090176835456} a^{15} - \frac{1}{3120090176835456} a^{14} - \frac{2397071947827}{1040030058945152} a^{13} - \frac{25309723498555}{3120090176835456} a^{12} - \frac{16747262077}{3732165283296} a^{11} + \frac{9270663963}{622027547216} a^{10} + \frac{27739940253475}{3120090176835456} a^{9} + \frac{34087939038223}{1040030058945152} a^{8} + \frac{211823279362323}{1040030058945152} a^{7} - \frac{212957626640501}{3120090176835456} a^{6} - \frac{674157216709}{16250469671018} a^{5} + \frac{24234311566523}{780022544208864} a^{4} - \frac{36396405083249}{195005636052216} a^{3} - \frac{9535157866941}{65001878684072} a^{2} + \frac{455670596483}{2565863632266} a - \frac{14013328886}{1282931816133}$

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:  \( 1307647707310 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1123:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 55 conjugacy class representatives for t16n1123 are not computed
Character table for t16n1123 is not computed

Intermediate fields

\(\Q(\sqrt{1649}) \), \(\Q(\sqrt{97}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{17}, \sqrt{97})\), 8.8.184851351960025.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.3$x^{4} + 2 x^{2} + 4 x + 4$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
$5$5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
97Data not computed