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Properties

Label 16.0.117033789351264256.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 17^{8}$
Root discriminant $11.66$
Ramified primes $2, 17$
Class number $1$
Class group Trivial
Galois group $C_2^2.SD_{16}$ (as 16T163)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![29, -154, 372, -540, 607, -680, 742, -660, 466, -304, 204, -122, 60, -30, 16, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 16*x^14 - 30*x^13 + 60*x^12 - 122*x^11 + 204*x^10 - 304*x^9 + 466*x^8 - 660*x^7 + 742*x^6 - 680*x^5 + 607*x^4 - 540*x^3 + 372*x^2 - 154*x + 29)
 
gp: K = bnfinit(x^16 - 6*x^15 + 16*x^14 - 30*x^13 + 60*x^12 - 122*x^11 + 204*x^10 - 304*x^9 + 466*x^8 - 660*x^7 + 742*x^6 - 680*x^5 + 607*x^4 - 540*x^3 + 372*x^2 - 154*x + 29, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 16 x^{14} - 30 x^{13} + 60 x^{12} - 122 x^{11} + 204 x^{10} - 304 x^{9} + 466 x^{8} - 660 x^{7} + 742 x^{6} - 680 x^{5} + 607 x^{4} - 540 x^{3} + 372 x^{2} - 154 x + 29 \)

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:  \(117033789351264256=2^{24}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11.66$
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:  $2, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5}$, $\frac{1}{5702036005} a^{15} - \frac{463914881}{5702036005} a^{14} - \frac{3454853}{17021003} a^{13} + \frac{339005991}{1140407201} a^{12} - \frac{467262901}{1140407201} a^{11} + \frac{2144558698}{5702036005} a^{10} - \frac{1247359286}{5702036005} a^{9} - \frac{15449377}{5702036005} a^{8} + \frac{2256376174}{5702036005} a^{7} + \frac{387149486}{1140407201} a^{6} + \frac{838728038}{5702036005} a^{5} + \frac{670013796}{5702036005} a^{4} + \frac{179604538}{1140407201} a^{3} + \frac{1602417557}{5702036005} a^{2} - \frac{2107954041}{5702036005} a + \frac{772609146}{5702036005}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{1547985}{21517117} a^{15} + \frac{13600593}{21517117} a^{14} - \frac{590784}{321151} a^{13} + \frac{68225806}{21517117} a^{12} - \frac{131746677}{21517117} a^{11} + \frac{289598174}{21517117} a^{10} - \frac{479266799}{21517117} a^{9} + \frac{677542269}{21517117} a^{8} - \frac{1061982065}{21517117} a^{7} + \frac{1553991255}{21517117} a^{6} - \frac{1671467541}{21517117} a^{5} + \frac{1415330212}{21517117} a^{4} - \frac{1271713629}{21517117} a^{3} + \frac{1178187212}{21517117} a^{2} - \frac{724276338}{21517117} a + \frac{202158199}{21517117} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 114.462998418 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.SD_{16}$ (as 16T163):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 19 conjugacy class representatives for $C_2^2.SD_{16}$
Character table for $C_2^2.SD_{16}$

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.0.272.1, 8.0.1257728.1, 8.0.20123648.2, 8.0.342102016.6

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$