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Properties

Label 16.0.60404790201...6336.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{8}\cdot 11^{8}$
Root discriminant $22.98$
Ramified primes $2, 3, 11$
Class number $4$ (GRH)
Class group $[4]$ (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![6561, 0, 0, 0, -567, 0, 0, 0, -32, 0, 0, 0, -7, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^12 - 32*x^8 - 567*x^4 + 6561)
 
gp: K = bnfinit(x^16 - 7*x^12 - 32*x^8 - 567*x^4 + 6561, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{12} - 32 x^{8} - 567 x^{4} + 6561 \)

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:  \(6040479020157644046336=2^{32}\cdot 3^{8}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.98$
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:  $2, 3, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(264=2^{3}\cdot 3\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{264}(1,·)$, $\chi_{264}(67,·)$, $\chi_{264}(133,·)$, $\chi_{264}(65,·)$, $\chi_{264}(43,·)$, $\chi_{264}(131,·)$, $\chi_{264}(23,·)$, $\chi_{264}(89,·)$, $\chi_{264}(155,·)$, $\chi_{264}(221,·)$, $\chi_{264}(199,·)$, $\chi_{264}(197,·)$, $\chi_{264}(263,·)$, $\chi_{264}(109,·)$, $\chi_{264}(175,·)$, $\chi_{264}(241,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{1}{5} a^{4} + \frac{1}{5}$, $\frac{1}{15} a^{9} - \frac{1}{15} a^{5} + \frac{1}{15} a$, $\frac{1}{45} a^{10} - \frac{16}{45} a^{6} - \frac{14}{45} a^{2}$, $\frac{1}{135} a^{11} - \frac{61}{135} a^{7} - \frac{59}{135} a^{3}$, $\frac{1}{12960} a^{12} - \frac{23}{405} a^{8} - \frac{82}{405} a^{4} + \frac{5}{32}$, $\frac{1}{38880} a^{13} - \frac{23}{1215} a^{9} - \frac{487}{1215} a^{5} + \frac{37}{96} a$, $\frac{1}{116640} a^{14} - \frac{23}{3645} a^{10} + \frac{728}{3645} a^{6} - \frac{59}{288} a^{2}$, $\frac{1}{349920} a^{15} - \frac{23}{10935} a^{11} - \frac{2917}{10935} a^{7} + \frac{229}{864} a^{3}$

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:  \( \frac{91}{349920} a^{15} + \frac{13}{10935} a^{11} - \frac{253}{10935} a^{7} - \frac{39}{160} a^{3} \) (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:  \( 113529.247833 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-33}) \), \(\Q(\sqrt{66}) \), \(\Q(\sqrt{-66}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-22}) \), \(\Q(\sqrt{22}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(i, \sqrt{33})\), \(\Q(i, \sqrt{66})\), \(\Q(\zeta_{8})\), \(\Q(\sqrt{2}, \sqrt{33})\), \(\Q(\sqrt{-2}, \sqrt{33})\), \(\Q(\sqrt{-2}, \sqrt{-33})\), \(\Q(\sqrt{2}, \sqrt{-33})\), \(\Q(i, \sqrt{6})\), \(\Q(i, \sqrt{22})\), \(\Q(i, \sqrt{11})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{-6}, \sqrt{-22})\), \(\Q(\sqrt{6}, \sqrt{22})\), \(\Q(\sqrt{-3}, \sqrt{-11})\), \(\Q(\sqrt{3}, \sqrt{11})\), \(\Q(\sqrt{-6}, \sqrt{22})\), \(\Q(\sqrt{6}, \sqrt{-22})\), \(\Q(\sqrt{3}, \sqrt{-11})\), \(\Q(\sqrt{-3}, \sqrt{11})\), \(\Q(\sqrt{-6}, \sqrt{-11})\), \(\Q(\sqrt{6}, \sqrt{11})\), \(\Q(\sqrt{-3}, \sqrt{-22})\), \(\Q(\sqrt{3}, \sqrt{22})\), \(\Q(\sqrt{-6}, \sqrt{11})\), \(\Q(\sqrt{6}, \sqrt{-11})\), \(\Q(\sqrt{3}, \sqrt{-22})\), \(\Q(\sqrt{-3}, \sqrt{22})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{-11})\), \(\Q(\sqrt{2}, \sqrt{11})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{11})\), \(\Q(\sqrt{-2}, \sqrt{-11})\), 8.0.77720518656.8, 8.0.77720518656.1, 8.0.303595776.1, 8.0.77720518656.3, 8.0.77720518656.9, \(\Q(\zeta_{24})\), 8.0.959512576.1, 8.0.4857532416.1, 8.8.77720518656.1, 8.0.77720518656.4, 8.0.4857532416.2, 8.0.77720518656.2, 8.0.77720518656.6, 8.0.77720518656.7, 8.0.77720518656.5

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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}$
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$