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Properties

Label 16.16.1979931599...0625.1
Degree $16$
Signature $[16, 0]$
Discriminant $3^{14}\cdot 5^{14}\cdot 7^{14}$
Root discriminant $58.69$
Ramified primes $3, 5, 7$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_8.C_4$ (as 16T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-149609, -364563, 122987, 995106, 679966, -388458, -533260, -20247, 142864, 28449, -17815, -4431, 1246, 273, -52, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 52*x^14 + 273*x^13 + 1246*x^12 - 4431*x^11 - 17815*x^10 + 28449*x^9 + 142864*x^8 - 20247*x^7 - 533260*x^6 - 388458*x^5 + 679966*x^4 + 995106*x^3 + 122987*x^2 - 364563*x - 149609)
 
gp: K = bnfinit(x^16 - 6*x^15 - 52*x^14 + 273*x^13 + 1246*x^12 - 4431*x^11 - 17815*x^10 + 28449*x^9 + 142864*x^8 - 20247*x^7 - 533260*x^6 - 388458*x^5 + 679966*x^4 + 995106*x^3 + 122987*x^2 - 364563*x - 149609, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 52 x^{14} + 273 x^{13} + 1246 x^{12} - 4431 x^{11} - 17815 x^{10} + 28449 x^{9} + 142864 x^{8} - 20247 x^{7} - 533260 x^{6} - 388458 x^{5} + 679966 x^{4} + 995106 x^{3} + 122987 x^{2} - 364563 x - 149609 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(19799315994393973883056640625=3^{14}\cdot 5^{14}\cdot 7^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.69$
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:  $3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{67240258043064487839116998} a^{15} - \frac{138050944090479543769333}{755508517337803234147382} a^{14} + \frac{12608490157950335831523477}{67240258043064487839116998} a^{13} + \frac{7127983261074364386413351}{67240258043064487839116998} a^{12} + \frac{18674014431012753439945353}{67240258043064487839116998} a^{11} - \frac{13631224877202636964408878}{33620129021532243919558499} a^{10} - \frac{26088817667450622410711433}{67240258043064487839116998} a^{9} - \frac{16375557980944689587823803}{67240258043064487839116998} a^{8} + \frac{4625897345956610776859905}{33620129021532243919558499} a^{7} + \frac{16114650527134662484171209}{33620129021532243919558499} a^{6} - \frac{12814985603010745264260629}{33620129021532243919558499} a^{5} - \frac{5853886073807070214714658}{33620129021532243919558499} a^{4} + \frac{27565513827582465373459155}{67240258043064487839116998} a^{3} + \frac{8617356236714497005462649}{33620129021532243919558499} a^{2} + \frac{25967703681957717459954787}{67240258043064487839116998} a - \frac{258404985556968393190087}{755508517337803234147382}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $15$
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:  \( 710707456.916 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8.C_4$ (as 16T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_8.C_4$
Character table for $C_8.C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{5}, \sqrt{21})\), 8.8.1340095640625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R R R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
3Data not computed
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
7Data not computed