↑

Properties

Label 16.0.60250206922...6033.1
Degree $16$
Signature $[0, 8]$
Discriminant $17^{7}\cdot 59^{8}$
Root discriminant $26.53$
Ramified primes $17, 59$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T289)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 36, -272, 734, -908, 1005, -814, 910, -924, 490, -74, -9, -25, 24, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 24*x^14 - 25*x^13 - 9*x^12 - 74*x^11 + 490*x^10 - 924*x^9 + 910*x^8 - 814*x^7 + 1005*x^6 - 908*x^5 + 734*x^4 - 272*x^3 + 36*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^16 - 8*x^15 + 24*x^14 - 25*x^13 - 9*x^12 - 74*x^11 + 490*x^10 - 924*x^9 + 910*x^8 - 814*x^7 + 1005*x^6 - 908*x^5 + 734*x^4 - 272*x^3 + 36*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 24 x^{14} - 25 x^{13} - 9 x^{12} - 74 x^{11} + 490 x^{10} - 924 x^{9} + 910 x^{8} - 814 x^{7} + 1005 x^{6} - 908 x^{5} + 734 x^{4} - 272 x^{3} + 36 x^{2} - 4 x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(60250206922566378206033=17^{7}\cdot 59^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.53$
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:  $17, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{1019} a^{14} - \frac{2}{1019} a^{13} + \frac{325}{1019} a^{12} + \frac{280}{1019} a^{11} + \frac{477}{1019} a^{10} - \frac{263}{1019} a^{9} + \frac{458}{1019} a^{8} + \frac{6}{1019} a^{7} - \frac{398}{1019} a^{6} - \frac{305}{1019} a^{5} - \frac{62}{1019} a^{4} + \frac{60}{1019} a^{3} + \frac{30}{1019} a^{2} + \frac{346}{1019} a + \frac{293}{1019}$, $\frac{1}{3492590552773607} a^{15} + \frac{1315319739394}{3492590552773607} a^{14} - \frac{206329665987913}{3492590552773607} a^{13} + \frac{1140526717703332}{3492590552773607} a^{12} - \frac{1397966047920851}{3492590552773607} a^{11} + \frac{1558422656241668}{3492590552773607} a^{10} + \frac{840323457676811}{3492590552773607} a^{9} + \frac{256135950772982}{3492590552773607} a^{8} - \frac{1304236619817653}{3492590552773607} a^{7} + \frac{1202654534319808}{3492590552773607} a^{6} + \frac{1085687046667601}{3492590552773607} a^{5} - \frac{460162184816735}{3492590552773607} a^{4} + \frac{568087736122834}{3492590552773607} a^{3} + \frac{1262931032717910}{3492590552773607} a^{2} + \frac{535602153384338}{3492590552773607} a + \frac{1013830850973587}{3492590552773607}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $7$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 9007.32885109 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T289):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-59}) \), 4.0.59177.1, 8.0.59532594593.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ 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
$17$17.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
17.8.7.6$x^{8} + 37179$$8$$1$$7$$C_8$$[\ ]_{8}$
59Data not computed