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Properties

Label 16.4.45365592064...0000.3
Degree $16$
Signature $[4, 6]$
Discriminant $2^{36}\cdot 5^{8}\cdot 13^{2}$
Root discriminant $14.66$
Ramified primes $2, 5, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1701

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -4, 14, -24, 32, -84, 188, -216, 81, 92, -136, 64, 14, -40, 26, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 26*x^14 - 40*x^13 + 14*x^12 + 64*x^11 - 136*x^10 + 92*x^9 + 81*x^8 - 216*x^7 + 188*x^6 - 84*x^5 + 32*x^4 - 24*x^3 + 14*x^2 - 4*x - 1)
 
gp: K = bnfinit(x^16 - 8*x^15 + 26*x^14 - 40*x^13 + 14*x^12 + 64*x^11 - 136*x^10 + 92*x^9 + 81*x^8 - 216*x^7 + 188*x^6 - 84*x^5 + 32*x^4 - 24*x^3 + 14*x^2 - 4*x - 1, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 26 x^{14} - 40 x^{13} + 14 x^{12} + 64 x^{11} - 136 x^{10} + 92 x^{9} + 81 x^{8} - 216 x^{7} + 188 x^{6} - 84 x^{5} + 32 x^{4} - 24 x^{3} + 14 x^{2} - 4 x - 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4536559206400000000=2^{36}\cdot 5^{8}\cdot 13^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.66$
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:  $2, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} + \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{4} a^{5} + \frac{1}{8} a^{4} + \frac{1}{8} a^{3} + \frac{1}{8} a + \frac{1}{8}$, $\frac{1}{475976} a^{15} + \frac{4419}{118994} a^{14} + \frac{44669}{475976} a^{13} - \frac{15113}{475976} a^{12} - \frac{29187}{118994} a^{11} - \frac{85165}{475976} a^{10} + \frac{108559}{475976} a^{9} - \frac{22937}{118994} a^{8} + \frac{95065}{237988} a^{7} - \frac{214251}{475976} a^{6} - \frac{105033}{475976} a^{5} + \frac{105839}{237988} a^{4} - \frac{58965}{475976} a^{3} - \frac{52159}{475976} a^{2} + \frac{45185}{118994} a + \frac{86813}{475976}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $9$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1303.44598694 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1701:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 8192
The 119 conjugacy class representatives for t16n1701 are not computed
Character table for t16n1701 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.400.1, 8.4.163840000.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.8.18.11$x^{8} + 2 x^{4} + 12 x^{2} + 4$$4$$2$$18$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 2, 3, 7/2, 7/2]^{2}$
2.8.18.11$x^{8} + 2 x^{4} + 12 x^{2} + 4$$4$$2$$18$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 2, 3, 7/2, 7/2]^{2}$
5Data not computed
$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}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$