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Properties

Label 16.0.84414841594...0641.2
Degree $16$
Signature $[0, 8]$
Discriminant $13^{6}\cdot 53^{10}$
Root discriminant $31.29$
Ramified primes $13, 53$
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![7039, 5173, -5129, -102, 352, -1099, 1598, -1134, 556, -475, 436, -264, 135, -70, 30, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 30*x^14 - 70*x^13 + 135*x^12 - 264*x^11 + 436*x^10 - 475*x^9 + 556*x^8 - 1134*x^7 + 1598*x^6 - 1099*x^5 + 352*x^4 - 102*x^3 - 5129*x^2 + 5173*x + 7039)
 
gp: K = bnfinit(x^16 - 8*x^15 + 30*x^14 - 70*x^13 + 135*x^12 - 264*x^11 + 436*x^10 - 475*x^9 + 556*x^8 - 1134*x^7 + 1598*x^6 - 1099*x^5 + 352*x^4 - 102*x^3 - 5129*x^2 + 5173*x + 7039, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 30 x^{14} - 70 x^{13} + 135 x^{12} - 264 x^{11} + 436 x^{10} - 475 x^{9} + 556 x^{8} - 1134 x^{7} + 1598 x^{6} - 1099 x^{5} + 352 x^{4} - 102 x^{3} - 5129 x^{2} + 5173 x + 7039 \)

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:  \(844148415947491674530641=13^{6}\cdot 53^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.29$
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:  $13, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{47} a^{10} - \frac{5}{47} a^{9} - \frac{17}{47} a^{7} + \frac{15}{47} a^{5} - \frac{1}{47} a^{4} + \frac{22}{47} a^{3} - \frac{14}{47} a^{2} - \frac{1}{47} a + \frac{15}{47}$, $\frac{1}{47} a^{11} + \frac{22}{47} a^{9} - \frac{17}{47} a^{8} + \frac{9}{47} a^{7} + \frac{15}{47} a^{6} - \frac{20}{47} a^{5} + \frac{17}{47} a^{4} + \frac{2}{47} a^{3} + \frac{23}{47} a^{2} + \frac{10}{47} a - \frac{19}{47}$, $\frac{1}{2491} a^{12} - \frac{6}{2491} a^{11} - \frac{10}{2491} a^{10} + \frac{105}{2491} a^{9} - \frac{77}{2491} a^{8} - \frac{388}{2491} a^{7} - \frac{1238}{2491} a^{6} + \frac{597}{2491} a^{5} + \frac{261}{2491} a^{4} - \frac{552}{2491} a^{3} + \frac{273}{2491} a^{2} + \frac{22}{53} a - \frac{319}{2491}$, $\frac{1}{7473} a^{13} + \frac{1}{7473} a^{12} + \frac{18}{2491} a^{11} - \frac{71}{7473} a^{10} - \frac{1462}{7473} a^{9} + \frac{751}{2491} a^{8} + \frac{431}{2491} a^{7} - \frac{499}{2491} a^{6} + \frac{730}{7473} a^{5} + \frac{1061}{2491} a^{4} - \frac{3220}{7473} a^{3} + \frac{1885}{7473} a^{2} + \frac{3103}{7473} a + \frac{1636}{7473}$, $\frac{1}{72704817} a^{14} - \frac{7}{72704817} a^{13} + \frac{13858}{72704817} a^{12} - \frac{83057}{72704817} a^{11} - \frac{48571}{8078313} a^{10} + \frac{2946884}{72704817} a^{9} - \frac{3094883}{8078313} a^{8} + \frac{6705442}{24234939} a^{7} + \frac{27345205}{72704817} a^{6} - \frac{30095291}{72704817} a^{5} - \frac{15198790}{72704817} a^{4} - \frac{1264690}{8078313} a^{3} - \frac{15525457}{72704817} a^{2} - \frac{22551193}{72704817} a + \frac{24925765}{72704817}$, $\frac{1}{10687608099} a^{15} + \frac{22}{3562536033} a^{14} - \frac{7664}{169644573} a^{13} + \frac{286928}{1526801157} a^{12} - \frac{106173905}{10687608099} a^{11} + \frac{7519487}{10687608099} a^{10} - \frac{5528167}{1526801157} a^{9} - \frac{1330632032}{3562536033} a^{8} + \frac{410357176}{1526801157} a^{7} + \frac{76196174}{1526801157} a^{6} - \frac{1611846989}{3562536033} a^{5} - \frac{2293510792}{10687608099} a^{4} - \frac{4669748665}{10687608099} a^{3} + \frac{5293979641}{10687608099} a^{2} - \frac{242807540}{3562536033} a - \frac{2033424212}{10687608099}$

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:  \( -1 \) (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:  \( 449119.359225 \) (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

\(\Q(\sqrt{53}) \), 4.4.36517.1, 8.0.17335386757.1, 8.4.70675038317.1, 8.4.918775498121.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$13$13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53Data not computed