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Properties

Label 16.0.19657182513...5696.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{8}\cdot 17^{8}$
Root discriminant $28.57$
Ramified primes $2, 3, 17$
Class number $16$ (GRH)
Class group $[4, 4]$ (GRH)
Galois group $C_2^4$ (as 16T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![65536, 0, 0, 0, -12544, 0, 0, 0, 2145, 0, 0, 0, -49, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 49*x^12 + 2145*x^8 - 12544*x^4 + 65536)
 
gp: K = bnfinit(x^16 - 49*x^12 + 2145*x^8 - 12544*x^4 + 65536, 1)
 

Normalized defining polynomial

\( x^{16} - 49 x^{12} + 2145 x^{8} - 12544 x^{4} + 65536 \)

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:  \(196571825135013064605696=2^{32}\cdot 3^{8}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.57$
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:  $2, 3, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(408=2^{3}\cdot 3\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{408}(1,·)$, $\chi_{408}(67,·)$, $\chi_{408}(137,·)$, $\chi_{408}(203,·)$, $\chi_{408}(205,·)$, $\chi_{408}(271,·)$, $\chi_{408}(341,·)$, $\chi_{408}(407,·)$, $\chi_{408}(35,·)$, $\chi_{408}(101,·)$, $\chi_{408}(103,·)$, $\chi_{408}(169,·)$, $\chi_{408}(239,·)$, $\chi_{408}(305,·)$, $\chi_{408}(307,·)$, $\chi_{408}(373,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{9} a^{8} - \frac{2}{9} a^{4} + \frac{4}{9}$, $\frac{1}{36} a^{9} + \frac{7}{36} a^{5} + \frac{13}{36} a$, $\frac{1}{144} a^{10} - \frac{65}{144} a^{6} + \frac{49}{144} a^{2}$, $\frac{1}{576} a^{11} + \frac{79}{576} a^{7} - \frac{95}{576} a^{3}$, $\frac{1}{4942080} a^{12} + \frac{47}{2304} a^{8} + \frac{257}{2304} a^{4} - \frac{4339}{19305}$, $\frac{1}{19768320} a^{13} + \frac{47}{9216} a^{9} + \frac{2561}{9216} a^{5} - \frac{4339}{77220} a$, $\frac{1}{79073280} a^{14} + \frac{47}{36864} a^{10} - \frac{6655}{36864} a^{6} - \frac{81559}{308880} a^{2}$, $\frac{1}{316293120} a^{15} + \frac{47}{147456} a^{11} - \frac{43519}{147456} a^{7} - \frac{390439}{1235520} a^{3}$

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

Class group and class number

$C_{4}\times C_{4}$, which has order $16$ (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:  \( \frac{1}{77220} a^{13} + \frac{49661}{77220} a \) (order $24$)
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:  \( 148181.522325 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4$ (as 16T3):

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 $C_2^4$
Character table for $C_2^4$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{102}) \), \(\Q(\sqrt{-102}) \), \(\Q(\sqrt{-51}) \), \(\Q(\sqrt{51}) \), \(\Q(\sqrt{-34}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-17}) \), \(\Q(\zeta_{8})\), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(i, \sqrt{102})\), \(\Q(i, \sqrt{51})\), \(\Q(i, \sqrt{34})\), \(\Q(i, \sqrt{17})\), \(\Q(\sqrt{-2}, \sqrt{-51})\), \(\Q(\sqrt{-2}, \sqrt{51})\), \(\Q(\sqrt{-2}, \sqrt{17})\), \(\Q(\sqrt{-2}, \sqrt{-17})\), \(\Q(\sqrt{2}, \sqrt{51})\), \(\Q(\sqrt{2}, \sqrt{-51})\), \(\Q(\sqrt{2}, \sqrt{-17})\), \(\Q(\sqrt{2}, \sqrt{17})\), \(\Q(\sqrt{-3}, \sqrt{-34})\), \(\Q(\sqrt{-3}, \sqrt{34})\), \(\Q(\sqrt{-3}, \sqrt{17})\), \(\Q(\sqrt{-3}, \sqrt{-17})\), \(\Q(\sqrt{3}, \sqrt{34})\), \(\Q(\sqrt{3}, \sqrt{-34})\), \(\Q(\sqrt{3}, \sqrt{-17})\), \(\Q(\sqrt{3}, \sqrt{17})\), \(\Q(\sqrt{6}, \sqrt{17})\), \(\Q(\sqrt{6}, \sqrt{-17})\), \(\Q(\sqrt{6}, \sqrt{-34})\), \(\Q(\sqrt{6}, \sqrt{34})\), \(\Q(\sqrt{-6}, \sqrt{-17})\), \(\Q(\sqrt{-6}, \sqrt{17})\), \(\Q(\sqrt{-6}, \sqrt{34})\), \(\Q(\sqrt{-6}, \sqrt{-34})\), \(\Q(\zeta_{24})\), 8.0.443364212736.8, 8.0.5473632256.1, 8.0.443364212736.9, 8.0.1731891456.1, 8.0.443364212736.3, 8.0.443364212736.2, 8.0.27710263296.5, 8.0.443364212736.7, 8.0.443364212736.1, 8.0.443364212736.4, 8.0.443364212736.6, 8.0.27710263296.1, 8.8.443364212736.1, 8.0.443364212736.5

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 R ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$