Properties

Label 16.0.60446290980...3088.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{79}$
Root discriminant $30.64$
Ramified prime $2$
Class number $17$ (GRH)
Class group $[17]$ (GRH)
Galois Group $C_{16}$ (as 16T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, 64, 0, 336, 0, 672, 0, 660, 0, 352, 0, 104, 0, 16, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 16*x^14 + 104*x^12 + 352*x^10 + 660*x^8 + 672*x^6 + 336*x^4 + 64*x^2 + 2)
gp: K = bnfinit(x^16 + 16*x^14 + 104*x^12 + 352*x^10 + 660*x^8 + 672*x^6 + 336*x^4 + 64*x^2 + 2, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut +\mathstrut 16 x^{14} \) \(\mathstrut +\mathstrut 104 x^{12} \) \(\mathstrut +\mathstrut 352 x^{10} \) \(\mathstrut +\mathstrut 660 x^{8} \) \(\mathstrut +\mathstrut 672 x^{6} \) \(\mathstrut +\mathstrut 336 x^{4} \) \(\mathstrut +\mathstrut 64 x^{2} \) \(\mathstrut +\mathstrut 2 \)

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:  \(604462909807314587353088=2^{79}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $30.64$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is Galois and abelian over $\Q$.
Conductor:  \(64=2^{6}\)
Dirichlet character group:    $\lbrace$$\chi_{64}(1,·)$, $\chi_{64}(3,·)$, $\chi_{64}(9,·)$, $\chi_{64}(11,·)$, $\chi_{64}(17,·)$, $\chi_{64}(19,·)$, $\chi_{64}(25,·)$, $\chi_{64}(27,·)$, $\chi_{64}(33,·)$, $\chi_{64}(35,·)$, $\chi_{64}(41,·)$, $\chi_{64}(43,·)$, $\chi_{64}(49,·)$, $\chi_{64}(51,·)$, $\chi_{64}(57,·)$, $\chi_{64}(59,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$

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

Class group and class number

$C_{17}$, which has order $17$ (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:  \( a^{8} + 8 a^{6} + 20 a^{4} + 16 a^{2} + 1 \),  \( a^{4} + 4 a^{2} + 1 \),  \( a^{8} + 8 a^{6} + 21 a^{4} + 20 a^{2} + 5 \),  \( a^{6} + 6 a^{4} + 9 a^{2} + 1 \),  \( a^{14} + 14 a^{12} + 77 a^{10} + 210 a^{8} + 294 a^{6} + 196 a^{4} + 49 a^{2} + 3 \),  \( a^{6} + 7 a^{4} + 14 a^{2} + 7 \),  \( a^{2} + 1 \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 15753.9498624 \) (assuming GRH)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_{16}$ (as 16T1):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A cyclic group of order 16
The 16 conjugacy class representatives for $C_{16}$
Character table for $C_{16}$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{32})^+\)

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 $16$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ $16$ $16$

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

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.2e6.16t1.1c1$1$ $ 2^{6}$ $x^{16} + 16 x^{14} + 104 x^{12} + 352 x^{10} + 660 x^{8} + 672 x^{6} + 336 x^{4} + 64 x^{2} + 2$ $C_{16}$ (as 16T1) $0$ $-1$
* 1.2e5.8t1.1c1$1$ $ 2^{5}$ $x^{8} - 8 x^{6} + 20 x^{4} - 16 x^{2} + 2$ $C_8$ (as 8T1) $0$ $1$
* 1.2e6.16t1.1c2$1$ $ 2^{6}$ $x^{16} + 16 x^{14} + 104 x^{12} + 352 x^{10} + 660 x^{8} + 672 x^{6} + 336 x^{4} + 64 x^{2} + 2$ $C_{16}$ (as 16T1) $0$ $-1$
* 1.2e4.4t1.1c1$1$ $ 2^{4}$ $x^{4} - 4 x^{2} + 2$ $C_4$ (as 4T1) $0$ $1$
* 1.2e6.16t1.1c3$1$ $ 2^{6}$ $x^{16} + 16 x^{14} + 104 x^{12} + 352 x^{10} + 660 x^{8} + 672 x^{6} + 336 x^{4} + 64 x^{2} + 2$ $C_{16}$ (as 16T1) $0$ $-1$
* 1.2e5.8t1.1c2$1$ $ 2^{5}$ $x^{8} - 8 x^{6} + 20 x^{4} - 16 x^{2} + 2$ $C_8$ (as 8T1) $0$ $1$
* 1.2e6.16t1.1c4$1$ $ 2^{6}$ $x^{16} + 16 x^{14} + 104 x^{12} + 352 x^{10} + 660 x^{8} + 672 x^{6} + 336 x^{4} + 64 x^{2} + 2$ $C_{16}$ (as 16T1) $0$ $-1$
* 1.2e3.2t1.1c1$1$ $ 2^{3}$ $x^{2} - 2$ $C_2$ (as 2T1) $1$ $1$
* 1.2e6.16t1.1c5$1$ $ 2^{6}$ $x^{16} + 16 x^{14} + 104 x^{12} + 352 x^{10} + 660 x^{8} + 672 x^{6} + 336 x^{4} + 64 x^{2} + 2$ $C_{16}$ (as 16T1) $0$ $-1$
* 1.2e5.8t1.1c3$1$ $ 2^{5}$ $x^{8} - 8 x^{6} + 20 x^{4} - 16 x^{2} + 2$ $C_8$ (as 8T1) $0$ $1$
* 1.2e6.16t1.1c6$1$ $ 2^{6}$ $x^{16} + 16 x^{14} + 104 x^{12} + 352 x^{10} + 660 x^{8} + 672 x^{6} + 336 x^{4} + 64 x^{2} + 2$ $C_{16}$ (as 16T1) $0$ $-1$
* 1.2e4.4t1.1c2$1$ $ 2^{4}$ $x^{4} - 4 x^{2} + 2$ $C_4$ (as 4T1) $0$ $1$
* 1.2e6.16t1.1c7$1$ $ 2^{6}$ $x^{16} + 16 x^{14} + 104 x^{12} + 352 x^{10} + 660 x^{8} + 672 x^{6} + 336 x^{4} + 64 x^{2} + 2$ $C_{16}$ (as 16T1) $0$ $-1$
* 1.2e5.8t1.1c4$1$ $ 2^{5}$ $x^{8} - 8 x^{6} + 20 x^{4} - 16 x^{2} + 2$ $C_8$ (as 8T1) $0$ $1$
* 1.2e6.16t1.1c8$1$ $ 2^{6}$ $x^{16} + 16 x^{14} + 104 x^{12} + 352 x^{10} + 660 x^{8} + 672 x^{6} + 336 x^{4} + 64 x^{2} + 2$ $C_{16}$ (as 16T1) $0$ $-1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.