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Properties

Label 16.4.14815515721...5625.1
Degree $16$
Signature $[4, 6]$
Discriminant $5^{10}\cdot 79^{8}$
Root discriminant $24.30$
Ramified primes $5, 79$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4.D_4$ (as 16T339)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, -352, -1012, 2770, -2443, 1240, -1297, 1476, -892, 388, -166, 11, 33, -21, 16, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 16*x^14 - 21*x^13 + 33*x^12 + 11*x^11 - 166*x^10 + 388*x^9 - 892*x^8 + 1476*x^7 - 1297*x^6 + 1240*x^5 - 2443*x^4 + 2770*x^3 - 1012*x^2 - 352*x + 256)
 
gp: K = bnfinit(x^16 - 7*x^15 + 16*x^14 - 21*x^13 + 33*x^12 + 11*x^11 - 166*x^10 + 388*x^9 - 892*x^8 + 1476*x^7 - 1297*x^6 + 1240*x^5 - 2443*x^4 + 2770*x^3 - 1012*x^2 - 352*x + 256, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 16 x^{14} - 21 x^{13} + 33 x^{12} + 11 x^{11} - 166 x^{10} + 388 x^{9} - 892 x^{8} + 1476 x^{7} - 1297 x^{6} + 1240 x^{5} - 2443 x^{4} + 2770 x^{3} - 1012 x^{2} - 352 x + 256 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(14815515721743759765625=5^{10}\cdot 79^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.30$
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:  $5, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{62} a^{12} + \frac{7}{62} a^{11} - \frac{25}{62} a^{10} - \frac{1}{62} a^{9} + \frac{10}{31} a^{8} + \frac{17}{62} a^{7} + \frac{6}{31} a^{6} + \frac{15}{62} a^{5} + \frac{7}{62} a^{4} + \frac{14}{31} a^{3} + \frac{6}{31} a^{2} + \frac{9}{62} a - \frac{12}{31}$, $\frac{1}{62} a^{13} - \frac{6}{31} a^{11} - \frac{6}{31} a^{10} + \frac{27}{62} a^{9} + \frac{1}{62} a^{8} + \frac{17}{62} a^{7} - \frac{7}{62} a^{6} + \frac{13}{31} a^{5} - \frac{21}{62} a^{4} + \frac{1}{31} a^{3} - \frac{13}{62} a^{2} - \frac{25}{62} a - \frac{9}{31}$, $\frac{1}{3844} a^{14} - \frac{1}{124} a^{13} + \frac{1}{1922} a^{12} + \frac{365}{3844} a^{11} - \frac{633}{3844} a^{10} - \frac{75}{3844} a^{9} - \frac{487}{1922} a^{8} - \frac{55}{1922} a^{7} - \frac{463}{961} a^{6} + \frac{265}{1922} a^{5} + \frac{193}{3844} a^{4} - \frac{378}{961} a^{3} + \frac{949}{3844} a^{2} + \frac{430}{961} a + \frac{350}{961}$, $\frac{1}{114772731657376} a^{15} - \frac{13755358983}{114772731657376} a^{14} + \frac{167160469}{43212624871} a^{13} + \frac{497378829307}{114772731657376} a^{12} + \frac{1168215354101}{8828671665952} a^{11} - \frac{20163578072965}{114772731657376} a^{10} + \frac{25900448209405}{57386365828688} a^{9} + \frac{6058733154637}{28693182914344} a^{8} + \frac{2397517519521}{28693182914344} a^{7} + \frac{7712980809197}{28693182914344} a^{6} - \frac{29410014415681}{114772731657376} a^{5} + \frac{170873958439}{14346591457172} a^{4} + \frac{59000523405}{160971573152} a^{3} - \frac{24185111083775}{57386365828688} a^{2} - \frac{10537919325621}{28693182914344} a + \frac{532971142575}{3586647864293}$

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:  $9$
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:  \( 156067.71042 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.D_4$ (as 16T339):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $C_2^4.D_4$
Character table for $C_2^4.D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.1975.1, 8.4.121719003125.1, 8.2.308149375.1, 8.2.1540746875.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.4.2.1$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}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$79$79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
79.8.6.1$x^{8} - 553 x^{4} + 505521$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$