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Properties

Label 16.8.23948260540...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{32}\cdot 3^{8}\cdot 5^{12}\cdot 59^{2}$
Root discriminant $38.57$
Ramified primes $2, 3, 5, 59$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1102

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3481, 0, 5664, 0, -2015, 0, -3954, 0, 1094, 0, 366, 0, -70, 0, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^14 - 70*x^12 + 366*x^10 + 1094*x^8 - 3954*x^6 - 2015*x^4 + 5664*x^2 + 3481)
 
gp: K = bnfinit(x^16 - 6*x^14 - 70*x^12 + 366*x^10 + 1094*x^8 - 3954*x^6 - 2015*x^4 + 5664*x^2 + 3481, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{14} - 70 x^{12} + 366 x^{10} + 1094 x^{8} - 3954 x^{6} - 2015 x^{4} + 5664 x^{2} + 3481 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(23948260540416000000000000=2^{32}\cdot 3^{8}\cdot 5^{12}\cdot 59^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.57$
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:  $2, 3, 5, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13977008424979} a^{14} - \frac{1282747367501}{13977008424979} a^{12} - \frac{1165946838273}{13977008424979} a^{10} - \frac{2538307273667}{13977008424979} a^{8} + \frac{1849998801247}{13977008424979} a^{6} - \frac{689075057282}{13977008424979} a^{4} - \frac{4016781491107}{13977008424979} a^{2} + \frac{38939400372}{236898447881}$, $\frac{1}{13977008424979} a^{15} - \frac{1282747367501}{13977008424979} a^{13} - \frac{1165946838273}{13977008424979} a^{11} - \frac{2538307273667}{13977008424979} a^{9} + \frac{1849998801247}{13977008424979} a^{7} - \frac{689075057282}{13977008424979} a^{5} - \frac{4016781491107}{13977008424979} a^{3} + \frac{38939400372}{236898447881} a$

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:  $11$
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:  \( 3814098.90077 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1102:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 64 conjugacy class representatives for t16n1102 are not computed
Character table for t16n1102 is not computed

Intermediate fields

\(\Q(\sqrt{15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{3}) \), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{20})^+\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\zeta_{60})^+\)

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ R

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
3Data not computed
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
59Data not computed