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Properties

Label 16.0.32942089830...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{14}\cdot 19^{2}\cdot 59^{2}$
Root discriminant $39.34$
Ramified primes $2, 5, 19, 59$
Class number $80$ (GRH)
Class group $[2, 40]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T554)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6400, 0, 6400, 0, 11600, 0, 7400, 0, 5245, 0, 2130, 0, 390, 0, 30, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 30*x^14 + 390*x^12 + 2130*x^10 + 5245*x^8 + 7400*x^6 + 11600*x^4 + 6400*x^2 + 6400)
 
gp: K = bnfinit(x^16 + 30*x^14 + 390*x^12 + 2130*x^10 + 5245*x^8 + 7400*x^6 + 11600*x^4 + 6400*x^2 + 6400, 1)
 

Normalized defining polynomial

\( x^{16} + 30 x^{14} + 390 x^{12} + 2130 x^{10} + 5245 x^{8} + 7400 x^{6} + 11600 x^{4} + 6400 x^{2} + 6400 \)

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:  \(32942089830400000000000000=2^{32}\cdot 5^{14}\cdot 19^{2}\cdot 59^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.34$
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:  $2, 5, 19, 59$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{10} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{10} a^{9} - \frac{1}{2} a$, $\frac{1}{20} a^{10} - \frac{1}{4} a^{2}$, $\frac{1}{40} a^{11} - \frac{1}{20} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{80} a^{12} - \frac{1}{40} a^{10} - \frac{1}{40} a^{8} + \frac{1}{8} a^{6} + \frac{1}{16} a^{4}$, $\frac{1}{160} a^{13} - \frac{1}{80} a^{11} + \frac{3}{80} a^{9} - \frac{3}{16} a^{7} - \frac{7}{32} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{53928658880} a^{14} - \frac{32360035}{5392865888} a^{12} - \frac{666331593}{26964329440} a^{10} - \frac{388379899}{26964329440} a^{8} + \frac{2495786529}{10785731776} a^{6} + \frac{730232371}{2696432944} a^{4} - \frac{128033815}{337054118} a^{2} - \frac{65594529}{168527059}$, $\frac{1}{107857317760} a^{15} - \frac{32360035}{10785731776} a^{13} - \frac{666331593}{53928658880} a^{11} - \frac{388379899}{53928658880} a^{9} + \frac{2495786529}{21571463552} a^{7} + \frac{730232371}{5392865888} a^{5} + \frac{209020303}{674108236} a^{3} + \frac{51466265}{168527059} a$

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

Class group and class number

$C_{2}\times C_{40}$, which has order $80$ (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:  \( 20680.5664472 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T554):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 40 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.8.5120000000.1, 8.0.5739520000000.2, 8.0.4484000000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ 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
$2$2.8.16.35$x^{8} + 6 x^{6} + 4 x^{5} + 10 x^{4} + 4 x^{2} + 12$$4$$2$$16$$C_8:C_2$$[2, 3, 3]^{2}$
2.8.16.35$x^{8} + 6 x^{6} + 4 x^{5} + 10 x^{4} + 4 x^{2} + 12$$4$$2$$16$$C_8:C_2$$[2, 3, 3]^{2}$
5Data not computed
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$59$$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$