Properties

Label 16.0.60516574977...0000.7
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 7^{8}$
Root discriminant $30.65$
Ramified primes $2, 3, 5, 7$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (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![14641, 0, -33759, 0, 63200, 0, -29161, 0, 9219, 0, -1741, 0, 240, 0, -19, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 19*x^14 + 240*x^12 - 1741*x^10 + 9219*x^8 - 29161*x^6 + 63200*x^4 - 33759*x^2 + 14641)
 
gp: K = bnfinit(x^16 - 19*x^14 + 240*x^12 - 1741*x^10 + 9219*x^8 - 29161*x^6 + 63200*x^4 - 33759*x^2 + 14641, 1)
 

Normalized defining polynomial

\( x^{16} - 19 x^{14} + 240 x^{12} - 1741 x^{10} + 9219 x^{8} - 29161 x^{6} + 63200 x^{4} - 33759 x^{2} + 14641 \)

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:  \(605165749776000000000000=2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 7^{8}\)
magma: Discriminant(K);
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.65$
magma: Abs(Discriminant(K))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(K));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(420=2^{2}\cdot 3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{420}(1,·)$, $\chi_{420}(83,·)$, $\chi_{420}(71,·)$, $\chi_{420}(13,·)$, $\chi_{420}(211,·)$, $\chi_{420}(281,·)$, $\chi_{420}(29,·)$, $\chi_{420}(223,·)$, $\chi_{420}(97,·)$, $\chi_{420}(293,·)$, $\chi_{420}(167,·)$, $\chi_{420}(169,·)$, $\chi_{420}(239,·)$, $\chi_{420}(307,·)$, $\chi_{420}(377,·)$, $\chi_{420}(379,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{11} a^{9} + \frac{1}{11} a^{7} + \frac{3}{11} a^{5} - \frac{2}{11} a^{3} + \frac{4}{11} a$, $\frac{1}{11} a^{10} + \frac{1}{11} a^{8} + \frac{3}{11} a^{6} - \frac{2}{11} a^{4} + \frac{4}{11} a^{2}$, $\frac{1}{11} a^{11} + \frac{2}{11} a^{7} - \frac{5}{11} a^{5} - \frac{5}{11} a^{3} - \frac{4}{11} a$, $\frac{1}{11} a^{12} + \frac{2}{11} a^{8} - \frac{5}{11} a^{6} - \frac{5}{11} a^{4} - \frac{4}{11} a^{2}$, $\frac{1}{11} a^{13} + \frac{4}{11} a^{7} + \frac{3}{11} a$, $\frac{1}{11137467154979} a^{14} - \frac{399578710000}{11137467154979} a^{12} + \frac{124755454976}{11137467154979} a^{10} + \frac{4072809174926}{11137467154979} a^{8} - \frac{3879007553519}{11137467154979} a^{6} + \frac{494941678642}{1012497014089} a^{4} + \frac{3589762146041}{11137467154979} a^{2} - \frac{10630726872}{92045183099}$, $\frac{1}{122512138704769} a^{15} + \frac{2637912332267}{122512138704769} a^{13} + \frac{124755454976}{122512138704769} a^{11} - \frac{989675895519}{122512138704769} a^{9} - \frac{7928995609875}{122512138704769} a^{7} - \frac{356657010010}{1012497014089} a^{5} + \frac{2577265131952}{122512138704769} a^{3} - \frac{470856642367}{1012497014089} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  \( -\frac{1913329}{92045183099} a^{15} + \frac{346706790}{1012497014089} a^{13} - \frac{4150184540}{1012497014089} a^{11} + \frac{25395094000}{1012497014089} a^{9} - \frac{115409043000}{1012497014089} a^{7} + \frac{202993944499}{1012497014089} a^{5} - \frac{9911044220}{92045183099} a^{3} - \frac{1313812737840}{1012497014089} a \) (order $12$)
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:  \( 76884.6053994 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{15}) \), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{15})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{-5})\), 4.4.882000.1, 4.0.55125.1, 4.0.98000.1, 4.4.6125.1, 8.0.12960000.1, 8.0.777924000000.8, 8.0.9604000000.1, 8.0.777924000000.9, 8.0.3038765625.1, 8.8.777924000000.1, 8.0.777924000000.5

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 R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{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}$
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$