↑

Properties

Label 16.0.28204278669...5744.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 3^{12}\cdot 13^{6}$
Root discriminant $33.74$
Ramified primes $2, 3, 13$
Class number $24$ (GRH)
Class group $[2, 2, 6]$ (GRH)
Galois group $C_2\times SD_{16}$ (as 16T48)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1369, 0, 452, 0, 2476, 0, 1892, 0, 1714, 0, 740, 0, 172, 0, 20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 20*x^14 + 172*x^12 + 740*x^10 + 1714*x^8 + 1892*x^6 + 2476*x^4 + 452*x^2 + 1369)
 
gp: K = bnfinit(x^16 + 20*x^14 + 172*x^12 + 740*x^10 + 1714*x^8 + 1892*x^6 + 2476*x^4 + 452*x^2 + 1369, 1)
 

Normalized defining polynomial

\( x^{16} + 20 x^{14} + 172 x^{12} + 740 x^{10} + 1714 x^{8} + 1892 x^{6} + 2476 x^{4} + 452 x^{2} + 1369 \)

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:  \(2820427866999756888735744=2^{40}\cdot 3^{12}\cdot 13^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.74$
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:  $2, 3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{12} a^{8} - \frac{1}{6} a^{6} - \frac{1}{6} a^{2} + \frac{1}{12}$, $\frac{1}{12} a^{9} - \frac{1}{6} a^{7} - \frac{1}{6} a^{3} + \frac{1}{12} a$, $\frac{1}{12} a^{10} - \frac{1}{3} a^{6} - \frac{1}{6} a^{4} - \frac{1}{4} a^{2} + \frac{1}{6}$, $\frac{1}{12} a^{11} - \frac{1}{3} a^{7} - \frac{1}{6} a^{5} - \frac{1}{4} a^{3} + \frac{1}{6} a$, $\frac{1}{24} a^{12} - \frac{1}{24} a^{8} - \frac{1}{2} a^{7} - \frac{1}{3} a^{6} + \frac{3}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} - \frac{3}{8}$, $\frac{1}{24} a^{13} - \frac{1}{24} a^{9} - \frac{1}{3} a^{7} + \frac{3}{8} a^{5} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} - \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{1136337144} a^{14} + \frac{64093}{378779048} a^{12} + \frac{36411253}{1136337144} a^{10} - \frac{4628115}{378779048} a^{8} - \frac{1}{2} a^{7} + \frac{176176983}{378779048} a^{6} - \frac{1}{2} a^{5} - \frac{34091753}{1136337144} a^{4} - \frac{1}{2} a^{3} + \frac{514477613}{1136337144} a^{2} - \frac{1}{2} a - \frac{90953119}{378779048}$, $\frac{1}{42044474328} a^{15} - \frac{59168203}{3503706194} a^{13} - \frac{1478704939}{42044474328} a^{11} - \frac{18742681}{568168572} a^{9} - \frac{2475276353}{14014824776} a^{7} - \frac{1}{2} a^{6} + \frac{1991903909}{10511118582} a^{5} - \frac{2038051909}{14014824776} a^{3} - \frac{1}{2} a^{2} - \frac{400581917}{7007412388} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{6}$, which has order $24$ (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:  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:  \( 22694.8390411 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times SD_{16}$ (as 16T48):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2\times SD_{16}$
Character table for $C_2\times SD_{16}$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), 4.4.7488.1, 4.4.29952.1, \(\Q(\sqrt{2}, \sqrt{3})\), 8.0.104963309568.1, 8.0.419853238272.1, 8.8.3588489216.1

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

Sibling fields

Galois closure: data not computed
Degree 16 sibling: data not computed

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$