# Properties

 Label 16.0.25051689769...4656.1 Degree $16$ Signature $[0, 8]$ Discriminant $2^{16}\cdot 3^{8}\cdot 17^{12}$ Root discriminant $29.00$ Ramified primes $2, 3, 17$ Class number $20$ (GRH) Class group $[20]$ (GRH) Galois group $C_4\times C_2^2$ (as 16T2)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -13, 0, 129, 0, -494, 0, 1430, 0, -494, 0, 129, 0, -13, 0, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 13*x^14 + 129*x^12 - 494*x^10 + 1430*x^8 - 494*x^6 + 129*x^4 - 13*x^2 + 1)

gp: K = bnfinit(x^16 - 13*x^14 + 129*x^12 - 494*x^10 + 1430*x^8 - 494*x^6 + 129*x^4 - 13*x^2 + 1, 1)

## Normalizeddefining polynomial

$$x^{16} - 13 x^{14} + 129 x^{12} - 494 x^{10} + 1430 x^{8} - 494 x^{6} + 129 x^{4} - 13 x^{2} + 1$$

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: $$250516897691366976454656=2^{16}\cdot 3^{8}\cdot 17^{12}$$ magma: Discriminant(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $29.00$ 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]~ $|\Gal(K/\Q)|$: $16$ This field is Galois and abelian over $\Q$. Conductor: $$204=2^{2}\cdot 3\cdot 17$$ Dirichlet character group: $\lbrace$$\chi_{204}(1,·), \chi_{204}(67,·), \chi_{204}(137,·), \chi_{204}(203,·), \chi_{204}(13,·), \chi_{204}(149,·), \chi_{204}(89,·), \chi_{204}(157,·), \chi_{204}(35,·), \chi_{204}(101,·), \chi_{204}(103,·), \chi_{204}(169,·), \chi_{204}(47,·), \chi_{204}(115,·), \chi_{204}(55,·), \chi_{204}(191,·)$$\rbrace$ This is a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{8} a^{6} - \frac{3}{8}$, $\frac{1}{8} a^{7} - \frac{3}{8} a$, $\frac{1}{8} a^{8} - \frac{3}{8} a^{2}$, $\frac{1}{8} a^{9} - \frac{3}{8} a^{3}$, $\frac{1}{8} a^{10} - \frac{3}{8} a^{4}$, $\frac{1}{8} a^{11} - \frac{3}{8} a^{5}$, $\frac{1}{1664} a^{12} - \frac{3}{64} a^{6} + \frac{545}{1664}$, $\frac{1}{1664} a^{13} - \frac{3}{64} a^{7} + \frac{545}{1664} a$, $\frac{1}{1664} a^{14} - \frac{3}{64} a^{8} + \frac{545}{1664} a^{2}$, $\frac{1}{1664} a^{15} - \frac{3}{64} a^{9} + \frac{545}{1664} a^{3}$

magma: IntegralBasis(K);

sage: K.integral_basis()

gp: K.zk

## Class group and class number

$C_{20}$, which has order $20$ (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: $$-\frac{733}{1664} a^{15} + \frac{4817}{832} a^{13} - \frac{461}{8} a^{11} + \frac{14431}{64} a^{9} - \frac{21071}{32} a^{7} + \frac{2383}{8} a^{5} - \frac{98925}{1664} a^{3} + \frac{4985}{832} a$$ (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: $$165082.730127$$ (assuming GRH) magma: Regulator(K);  sage: K.regulator()  gp: K.reg

## Galois group

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

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{16}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2} 2.4.4.1x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2} 2.4.4.1x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4} 3.8.4.1x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2} 17.8.6.1x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$