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Properties

Label 16.0.82409021384...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{12}\cdot 3^{8}\cdot 5^{8}\cdot 151^{5}$
Root discriminant $31.24$
Ramified primes $2, 3, 5, 151$
Class number $2$
Class group $[2]$
Galois group 16T1862

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![151, 0, 604, 0, 1963, 0, 3064, 0, 2104, 0, 763, 0, 160, 0, 19, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 19*x^14 + 160*x^12 + 763*x^10 + 2104*x^8 + 3064*x^6 + 1963*x^4 + 604*x^2 + 151)
 
gp: K = bnfinit(x^16 + 19*x^14 + 160*x^12 + 763*x^10 + 2104*x^8 + 3064*x^6 + 1963*x^4 + 604*x^2 + 151, 1)
 

Normalized defining polynomial

\( x^{16} + 19 x^{14} + 160 x^{12} + 763 x^{10} + 2104 x^{8} + 3064 x^{6} + 1963 x^{4} + 604 x^{2} + 151 \)

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:  \(824090213843697600000000=2^{12}\cdot 3^{8}\cdot 5^{8}\cdot 151^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.24$
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:  $2, 3, 5, 151$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{6} - \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{6} - \frac{1}{9} a^{2} + \frac{1}{9}$, $\frac{1}{9} a^{9} - \frac{1}{9} a^{7} - \frac{1}{9} a^{3} + \frac{1}{9} a$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{6} - \frac{1}{9} a^{4} + \frac{1}{9}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{7} - \frac{1}{9} a^{5} + \frac{1}{9} a$, $\frac{1}{27} a^{12} - \frac{2}{27} a^{6} + \frac{1}{27}$, $\frac{1}{27} a^{13} - \frac{2}{27} a^{7} + \frac{1}{27} a$, $\frac{1}{39582} a^{14} - \frac{1}{54} a^{13} - \frac{223}{13194} a^{12} - \frac{697}{13194} a^{10} - \frac{1}{18} a^{9} + \frac{1225}{39582} a^{8} - \frac{2}{27} a^{7} + \frac{1973}{13194} a^{6} - \frac{1}{6} a^{5} + \frac{68}{6597} a^{4} + \frac{7}{18} a^{3} + \frac{9062}{19791} a^{2} + \frac{23}{54} a + \frac{3539}{13194}$, $\frac{1}{39582} a^{15} + \frac{32}{19791} a^{13} - \frac{1}{54} a^{12} - \frac{697}{13194} a^{11} - \frac{487}{19791} a^{9} - \frac{1}{18} a^{8} + \frac{55}{39582} a^{7} - \frac{2}{27} a^{6} - \frac{2063}{13194} a^{5} - \frac{1}{6} a^{4} - \frac{6065}{39582} a^{3} + \frac{7}{18} a^{2} - \frac{5320}{19791} a + \frac{23}{54}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 112840.665534 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1862:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 73728
The 152 conjugacy class representatives for t16n1862 are not computed
Character table for t16n1862 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 8.4.1154300625.1

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 R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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}$
2.12.12.21$x^{12} + 44 x^{10} + 45 x^{8} - 48 x^{6} + 59 x^{4} - 60 x^{2} + 23$$2$$6$$12$12T134$[2, 2, 2, 2, 2, 2]^{6}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$151$$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
151.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
151.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
151.4.0.1$x^{4} - x + 6$$1$$4$$0$$C_4$$[\ ]^{4}$
151.6.5.5$x^{6} + 18875$$6$$1$$5$$C_6$$[\ ]_{6}$