Properties

Label 16.0.11161211905...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{12}\cdot 17^{8}$
Root discriminant $27.57$
Ramified primes $2, 5, 17$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![65536, 0, -36864, 0, 16640, 0, -7056, 0, 2929, 0, -441, 0, 65, 0, -9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 9*x^14 + 65*x^12 - 441*x^10 + 2929*x^8 - 7056*x^6 + 16640*x^4 - 36864*x^2 + 65536)
 
gp: K = bnfinit(x^16 - 9*x^14 + 65*x^12 - 441*x^10 + 2929*x^8 - 7056*x^6 + 16640*x^4 - 36864*x^2 + 65536, 1)
 

Normalized defining polynomial

\( x^{16} - 9 x^{14} + 65 x^{12} - 441 x^{10} + 2929 x^{8} - 7056 x^{6} + 16640 x^{4} - 36864 x^{2} + 65536 \)

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:  \(111612119056000000000000=2^{16}\cdot 5^{12}\cdot 17^{8}\)
magma: Discriminant(K);
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.57$
magma: Abs(Discriminant(K))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5, 17$
magma: PrimeDivisors(Discriminant(K));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(340=2^{2}\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{340}(1,·)$, $\chi_{340}(67,·)$, $\chi_{340}(69,·)$, $\chi_{340}(137,·)$, $\chi_{340}(203,·)$, $\chi_{340}(271,·)$, $\chi_{340}(273,·)$, $\chi_{340}(339,·)$, $\chi_{340}(33,·)$, $\chi_{340}(101,·)$, $\chi_{340}(103,·)$, $\chi_{340}(169,·)$, $\chi_{340}(171,·)$, $\chi_{340}(237,·)$, $\chi_{340}(239,·)$, $\chi_{340}(307,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{46864} a^{10} + \frac{7}{16} a^{8} + \frac{1}{16} a^{6} + \frac{7}{16} a^{4} + \frac{1}{16} a^{2} - \frac{441}{2929}$, $\frac{1}{187456} a^{11} + \frac{7}{64} a^{9} - \frac{15}{64} a^{7} + \frac{23}{64} a^{5} - \frac{31}{64} a^{3} + \frac{5417}{11716} a$, $\frac{1}{749824} a^{12} + \frac{7}{749824} a^{10} + \frac{65}{256} a^{8} + \frac{71}{256} a^{6} + \frac{113}{256} a^{4} + \frac{311}{5858} a^{2} - \frac{376}{2929}$, $\frac{1}{2999296} a^{13} + \frac{7}{2999296} a^{11} + \frac{65}{1024} a^{9} - \frac{441}{1024} a^{7} - \frac{143}{1024} a^{5} + \frac{311}{23432} a^{3} - \frac{94}{2929} a$, $\frac{1}{11997184} a^{14} + \frac{7}{11997184} a^{12} - \frac{79}{11997184} a^{10} + \frac{1607}{4096} a^{8} + \frac{881}{4096} a^{6} + \frac{311}{93728} a^{4} - \frac{47}{5858} a^{2} + \frac{56}{2929}$, $\frac{1}{47988736} a^{15} + \frac{7}{47988736} a^{13} - \frac{79}{47988736} a^{11} + \frac{1607}{16384} a^{9} + \frac{881}{16384} a^{7} + \frac{94039}{374912} a^{5} - \frac{5905}{23432} a^{3} + \frac{2985}{11716} a$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( -\frac{1629}{47988736} a^{15} + \frac{11765}{47988736} a^{13} - \frac{79821}{47988736} a^{11} + \frac{181}{16384} a^{9} - \frac{1165}{16384} a^{7} + \frac{11765}{187456} a^{5} - \frac{1629}{11716} a^{3} + \frac{724}{2929} a \) (order $20$)
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:  \( 114932.409132 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

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 $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{-85}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-17}) \), \(\Q(i, \sqrt{85})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{17})\), \(\Q(\sqrt{5}, \sqrt{17})\), \(\Q(\sqrt{-5}, \sqrt{-17})\), \(\Q(\sqrt{5}, \sqrt{-17})\), \(\Q(\sqrt{-5}, \sqrt{17})\), 4.0.36125.1, 4.4.578000.1, \(\Q(\zeta_{5})\), \(\Q(\zeta_{20})^+\), 8.0.13363360000.1, 8.0.334084000000.2, \(\Q(\zeta_{20})\), 8.0.1305015625.1, 8.8.334084000000.1, 8.0.334084000000.1, 8.0.334084000000.3

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$17$17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$