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Properties

Label 16.8.33910015523...9856.4
Degree $16$
Signature $[8, 4]$
Discriminant $2^{58}\cdot 7^{6}$
Root discriminant $25.59$
Ramified primes $2, 7$
Class number $1$
Class group Trivial
Galois group $C_2^4.C_2^3.C_2$ (as 16T646)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-17, 40, 24, -176, 280, -48, -456, 392, 182, -280, -56, 192, -80, -16, 24, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 24*x^14 - 16*x^13 - 80*x^12 + 192*x^11 - 56*x^10 - 280*x^9 + 182*x^8 + 392*x^7 - 456*x^6 - 48*x^5 + 280*x^4 - 176*x^3 + 24*x^2 + 40*x - 17)
 
gp: K = bnfinit(x^16 - 8*x^15 + 24*x^14 - 16*x^13 - 80*x^12 + 192*x^11 - 56*x^10 - 280*x^9 + 182*x^8 + 392*x^7 - 456*x^6 - 48*x^5 + 280*x^4 - 176*x^3 + 24*x^2 + 40*x - 17, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 24 x^{14} - 16 x^{13} - 80 x^{12} + 192 x^{11} - 56 x^{10} - 280 x^{9} + 182 x^{8} + 392 x^{7} - 456 x^{6} - 48 x^{5} + 280 x^{4} - 176 x^{3} + 24 x^{2} + 40 x - 17 \)

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:  \(33910015523872734969856=2^{58}\cdot 7^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.59$
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:  $2, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{17} a^{14} + \frac{5}{17} a^{13} - \frac{4}{17} a^{12} - \frac{6}{17} a^{11} - \frac{7}{17} a^{10} - \frac{4}{17} a^{9} - \frac{1}{17} a^{8} - \frac{6}{17} a^{7} - \frac{7}{17} a^{6} - \frac{8}{17} a^{5} + \frac{6}{17} a^{4} - \frac{8}{17} a^{3} - \frac{8}{17} a^{2} + \frac{5}{17} a$, $\frac{1}{52147100433751} a^{15} - \frac{480616077517}{52147100433751} a^{14} + \frac{14182798791099}{52147100433751} a^{13} - \frac{12329879429927}{52147100433751} a^{12} + \frac{14180887922428}{52147100433751} a^{11} + \frac{3011848180611}{52147100433751} a^{10} - \frac{25380576794274}{52147100433751} a^{9} + \frac{10560794396881}{52147100433751} a^{8} + \frac{418001284734}{1682164530121} a^{7} - \frac{2733649868400}{52147100433751} a^{6} - \frac{20051826325159}{52147100433751} a^{5} - \frac{17727165509015}{52147100433751} a^{4} - \frac{23499278026438}{52147100433751} a^{3} - \frac{23299964601283}{52147100433751} a^{2} - \frac{3254836677653}{52147100433751} a - \frac{266661914710}{3067476496103}$

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:  $11$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 403981.183446 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T646):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.14336.1, \(\Q(\zeta_{16})^+\), 4.4.7168.1, 8.8.3288334336.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$