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Properties

Label 16.16.4253672347...3664.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{66}\cdot 7^{8}$
Root discriminant $46.16$
Ramified primes $2, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\wr C_4$ (as 16T172)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 448, 4480, 5200, -10752, -18576, 4296, 18800, 4340, -6816, -3008, 848, 556, -32, -40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 40*x^14 - 32*x^13 + 556*x^12 + 848*x^11 - 3008*x^10 - 6816*x^9 + 4340*x^8 + 18800*x^7 + 4296*x^6 - 18576*x^5 - 10752*x^4 + 5200*x^3 + 4480*x^2 + 448*x - 2)
 
gp: K = bnfinit(x^16 - 40*x^14 - 32*x^13 + 556*x^12 + 848*x^11 - 3008*x^10 - 6816*x^9 + 4340*x^8 + 18800*x^7 + 4296*x^6 - 18576*x^5 - 10752*x^4 + 5200*x^3 + 4480*x^2 + 448*x - 2, 1)
 

Normalized defining polynomial

\( x^{16} - 40 x^{14} - 32 x^{13} + 556 x^{12} + 848 x^{11} - 3008 x^{10} - 6816 x^{9} + 4340 x^{8} + 18800 x^{7} + 4296 x^{6} - 18576 x^{5} - 10752 x^{4} + 5200 x^{3} + 4480 x^{2} + 448 x - 2 \)

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:  \(425367234731459587461873664=2^{66}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.16$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{163036600387703329} a^{15} - \frac{78393395516795896}{163036600387703329} a^{14} - \frac{36694151373743711}{163036600387703329} a^{13} - \frac{16078355181006446}{163036600387703329} a^{12} - \frac{79993930097010407}{163036600387703329} a^{11} - \frac{59890340475942273}{163036600387703329} a^{10} - \frac{30072213634890844}{163036600387703329} a^{9} - \frac{10068744857993688}{163036600387703329} a^{8} - \frac{11283710982880572}{163036600387703329} a^{7} + \frac{63168165544602820}{163036600387703329} a^{6} - \frac{44533277187882432}{163036600387703329} a^{5} + \frac{76264388574715396}{163036600387703329} a^{4} - \frac{69463294653641289}{163036600387703329} a^{3} + \frac{72365807573552656}{163036600387703329} a^{2} + \frac{57131068262455499}{163036600387703329} a + \frac{19630842770083133}{163036600387703329}$

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:  $15$
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:  \( 226771933.203 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T172):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 4.4.14336.1, 4.4.7168.1, 8.8.5156108238848.4, 8.8.5156108238848.3, 8.8.3288334336.1

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

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.6.2$x^{8} - 49 x^{4} + 3969$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$