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Properties

Label 16.0.64793809550...6336.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{64}\cdot 37^{8}$
Root discriminant $97.32$
Ramified primes $2, 37$
Class number $161330$ (GRH)
Class group $[161330]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![701137441, 0, 306110016, 0, 178564176, 0, 39680928, 0, 4330260, 0, 256608, 0, 8424, 0, 144, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 144*x^14 + 8424*x^12 + 256608*x^10 + 4330260*x^8 + 39680928*x^6 + 178564176*x^4 + 306110016*x^2 + 701137441)
 
gp: K = bnfinit(x^16 + 144*x^14 + 8424*x^12 + 256608*x^10 + 4330260*x^8 + 39680928*x^6 + 178564176*x^4 + 306110016*x^2 + 701137441, 1)
 

Normalized defining polynomial

\( x^{16} + 144 x^{14} + 8424 x^{12} + 256608 x^{10} + 4330260 x^{8} + 39680928 x^{6} + 178564176 x^{4} + 306110016 x^{2} + 701137441 \)

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:  \(64793809550643768832929443086336=2^{64}\cdot 37^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.32$
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:  $2, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1184=2^{5}\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{1184}(1,·)$, $\chi_{1184}(1035,·)$, $\chi_{1184}(1037,·)$, $\chi_{1184}(591,·)$, $\chi_{1184}(593,·)$, $\chi_{1184}(147,·)$, $\chi_{1184}(149,·)$, $\chi_{1184}(1183,·)$, $\chi_{1184}(739,·)$, $\chi_{1184}(741,·)$, $\chi_{1184}(295,·)$, $\chi_{1184}(297,·)$, $\chi_{1184}(887,·)$, $\chi_{1184}(889,·)$, $\chi_{1184}(443,·)$, $\chi_{1184}(445,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3781} a^{8} + \frac{72}{3781} a^{6} + \frac{1620}{3781} a^{4} + \frac{321}{3781} a^{2} + \frac{1779}{3781}$, $\frac{1}{100117099} a^{9} + \frac{50018921}{100117099} a^{7} + \frac{47559038}{100117099} a^{5} - \frac{45008703}{100117099} a^{3} + \frac{47714218}{100117099} a$, $\frac{1}{100117099} a^{10} + \frac{90}{100117099} a^{8} - \frac{49698329}{100117099} a^{6} + \frac{19335267}{100117099} a^{4} + \frac{10405307}{100117099} a^{2} + \frac{778}{3781}$, $\frac{1}{100117099} a^{11} - \frac{46131764}{100117099} a^{7} + \frac{44057104}{100117099} a^{5} - \frac{43612482}{100117099} a^{3} + \frac{31356299}{100117099} a$, $\frac{1}{100117099} a^{12} - \frac{5346}{100117099} a^{8} - \frac{38822166}{100117099} a^{6} - \frac{6171176}{100117099} a^{4} + \frac{20605825}{100117099} a^{2} - \frac{1402}{3781}$, $\frac{1}{100117099} a^{13} + \frac{49675170}{100117099} a^{7} + \frac{47131611}{100117099} a^{5} - \frac{14531516}{100117099} a^{3} + \frac{44834717}{100117099} a$, $\frac{1}{100117099} a^{14} + \frac{566}{100117099} a^{8} - \frac{25341412}{100117099} a^{6} + \frac{6757600}{100117099} a^{4} + \frac{17905574}{100117099} a^{2} + \frac{1219}{3781}$, $\frac{1}{100117099} a^{15} - \frac{2911681}{100117099} a^{7} + \frac{19841723}{100117099} a^{5} - \frac{37028773}{100117099} a^{3} - \frac{42469856}{100117099} a$

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

Class group and class number

$C_{161330}$, which has order $161330$ (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:  \( 15753.94986242651 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{-37}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-74}) \), \(\Q(\sqrt{2}, \sqrt{-37})\), \(\Q(\zeta_{16})^+\), 4.0.2803712.2, 8.0.31443203915776.12, 8.0.4024730101219328.4, \(\Q(\zeta_{32})^+\)

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ R ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
37Data not computed