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Properties

Label 16.0.10505857648...7237.3
Degree $16$
Signature $[0, 8]$
Discriminant $3^{10}\cdot 37^{11}$
Root discriminant $23.79$
Ramified primes $3, 37$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2^3\times C_4).D_4$ (as 16T675)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15613, 14440, -1212, -8046, -7132, -5323, 2326, 5673, 1141, -1987, -827, 307, 198, -18, -21, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 21*x^14 - 18*x^13 + 198*x^12 + 307*x^11 - 827*x^10 - 1987*x^9 + 1141*x^8 + 5673*x^7 + 2326*x^6 - 5323*x^5 - 7132*x^4 - 8046*x^3 - 1212*x^2 + 14440*x + 15613)
 
gp: K = bnfinit(x^16 - 21*x^14 - 18*x^13 + 198*x^12 + 307*x^11 - 827*x^10 - 1987*x^9 + 1141*x^8 + 5673*x^7 + 2326*x^6 - 5323*x^5 - 7132*x^4 - 8046*x^3 - 1212*x^2 + 14440*x + 15613, 1)
 

Normalized defining polynomial

\( x^{16} - 21 x^{14} - 18 x^{13} + 198 x^{12} + 307 x^{11} - 827 x^{10} - 1987 x^{9} + 1141 x^{8} + 5673 x^{7} + 2326 x^{6} - 5323 x^{5} - 7132 x^{4} - 8046 x^{3} - 1212 x^{2} + 14440 x + 15613 \)

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:  \(10505857648455357927237=3^{10}\cdot 37^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.79$
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:  $3, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{16754257869586685866264545991} a^{15} - \frac{2531359818987833784963611837}{16754257869586685866264545991} a^{14} - \frac{2896699456652765467278406538}{16754257869586685866264545991} a^{13} - \frac{2555689201104958604776142992}{16754257869586685866264545991} a^{12} + \frac{1487899917626815443172127463}{16754257869586685866264545991} a^{11} + \frac{4942449168688500573528346574}{16754257869586685866264545991} a^{10} + \frac{8099831945051761768046133684}{16754257869586685866264545991} a^{9} + \frac{2483498417974905376737184184}{16754257869586685866264545991} a^{8} + \frac{5404886017297954198872798124}{16754257869586685866264545991} a^{7} - \frac{6365965199311082798121898926}{16754257869586685866264545991} a^{6} - \frac{6955681529397965287050623591}{16754257869586685866264545991} a^{5} - \frac{3443551290909715875741249243}{16754257869586685866264545991} a^{4} - \frac{3602697132432509507733753266}{16754257869586685866264545991} a^{3} + \frac{4942574352516812641824009045}{16754257869586685866264545991} a^{2} - \frac{2445514282803487804867689969}{16754257869586685866264545991} a - \frac{5556445597512573887445135371}{16754257869586685866264545991}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( \frac{27090287208059}{420877530703281937} a^{15} + \frac{225972348211641}{420877530703281937} a^{14} - \frac{159672921765867}{420877530703281937} a^{13} - \frac{4920426238551730}{420877530703281937} a^{12} - \frac{5744585827695910}{420877530703281937} a^{11} + \frac{40576099477058446}{420877530703281937} a^{10} + \frac{92798263130074909}{420877530703281937} a^{9} - \frac{103646249497499257}{420877530703281937} a^{8} - \frac{438756195889218427}{420877530703281937} a^{7} - \frac{90835799299341562}{420877530703281937} a^{6} + \frac{738400391950402617}{420877530703281937} a^{5} + \frac{695139109457382231}{420877530703281937} a^{4} + \frac{229375687559108869}{420877530703281937} a^{3} - \frac{181459688065996968}{420877530703281937} a^{2} - \frac{1532547884274884592}{420877530703281937} a - \frac{1558127497796580448}{420877530703281937} \) (order $6$)
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:  \( 30987.5420577 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^3\times C_4).D_4$ (as 16T675):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 31 conjugacy class representatives for $(C_2^3\times C_4).D_4$
Character table for $(C_2^3\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.333.1, 8.0.4102893.1

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 $16$ R $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ $16$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ $16$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$37$37.4.3.4$x^{4} + 296$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.2.2$x^{4} - 37 x^{2} + 6845$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
37.8.6.2$x^{8} + 333 x^{4} + 34225$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$