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Properties

Label 16.4.17627096310...0000.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{20}\cdot 5^{8}\cdot 19^{3}\cdot 89^{4}$
Root discriminant $28.37$
Ramified primes $2, 5, 19, 89$
Class number $1$
Class group Trivial
Galois group 16T1567

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19, 0, 0, 0, 331, 0, 0, 0, -100, 0, 0, 0, -7, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^12 - 100*x^8 + 331*x^4 + 19)
 
gp: K = bnfinit(x^16 - 7*x^12 - 100*x^8 + 331*x^4 + 19, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{12} - 100 x^{8} + 331 x^{4} + 19 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(176270963105382400000000=2^{20}\cdot 5^{8}\cdot 19^{3}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.37$
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:  $2, 5, 19, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{6} a^{8} - \frac{1}{2} a^{6} - \frac{1}{3} a^{4} - \frac{1}{2} a^{2} - \frac{1}{6}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{7} - \frac{1}{3} a^{5} - \frac{1}{2} a^{3} - \frac{1}{6} a$, $\frac{1}{6} a^{10} + \frac{1}{6} a^{6} - \frac{1}{2} a^{4} + \frac{1}{3} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{11} + \frac{1}{6} a^{7} - \frac{1}{2} a^{5} + \frac{1}{3} a^{3} - \frac{1}{2} a$, $\frac{1}{5982} a^{12} - \frac{184}{2991} a^{8} - \frac{425}{2991} a^{4} + \frac{35}{1994}$, $\frac{1}{5982} a^{13} - \frac{184}{2991} a^{9} - \frac{425}{2991} a^{5} + \frac{35}{1994} a$, $\frac{1}{11964} a^{14} - \frac{1}{11964} a^{13} - \frac{1}{12} a^{11} - \frac{92}{2991} a^{10} - \frac{629}{11964} a^{9} - \frac{1}{12} a^{8} + \frac{1}{6} a^{7} + \frac{2141}{11964} a^{6} + \frac{1945}{3988} a^{5} - \frac{1}{3} a^{4} - \frac{5}{12} a^{3} - \frac{481}{1994} a^{2} + \frac{3883}{11964} a + \frac{1}{12}$, $\frac{1}{11964} a^{15} - \frac{1}{11964} a^{13} - \frac{1}{11964} a^{12} + \frac{629}{11964} a^{11} - \frac{1}{12} a^{10} + \frac{92}{2991} a^{9} - \frac{629}{11964} a^{8} + \frac{49}{3988} a^{7} + \frac{1}{6} a^{6} + \frac{3841}{11964} a^{5} + \frac{1945}{3988} a^{4} + \frac{2099}{11964} a^{3} - \frac{5}{12} a^{2} - \frac{258}{997} a + \frac{3883}{11964}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 542854.955237 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1567:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 106 conjugacy class representatives for t16n1567 are not computed
Character table for t16n1567 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2225.1, 8.4.1504990000.1

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R $16$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.8.16.49$x^{8} + 14 x^{4} + 12 x^{2} + 8 x + 4$$4$$2$$16$$Z_8 : Z_8^\times$$[2, 2, 3, 3]^{2}$
5Data not computed
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$89$89.8.0.1$x^{8} - x + 62$$1$$8$$0$$C_8$$[\ ]^{8}$
89.8.4.1$x^{8} + 427734 x^{4} - 704969 x^{2} + 45739093689$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$