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Properties

Label 16.0.73146118344...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{12}\cdot 17^{8}$
Root discriminant $55.15$
Ramified primes $2, 5, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2\times D_4):C_4$ (as 16T120)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2560000, 0, 2880000, 0, 1464000, 0, 434000, 0, 83300, 0, 10400, 0, 830, 0, 40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 40*x^14 + 830*x^12 + 10400*x^10 + 83300*x^8 + 434000*x^6 + 1464000*x^4 + 2880000*x^2 + 2560000)
 
gp: K = bnfinit(x^16 + 40*x^14 + 830*x^12 + 10400*x^10 + 83300*x^8 + 434000*x^6 + 1464000*x^4 + 2880000*x^2 + 2560000, 1)
 

Normalized defining polynomial

\( x^{16} + 40 x^{14} + 830 x^{12} + 10400 x^{10} + 83300 x^{8} + 434000 x^{6} + 1464000 x^{4} + 2880000 x^{2} + 2560000 \)

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:  \(7314611834454016000000000000=2^{32}\cdot 5^{12}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.15$
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:  $2, 5, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{10} a^{4}$, $\frac{1}{10} a^{5}$, $\frac{1}{10} a^{6}$, $\frac{1}{10} a^{7}$, $\frac{1}{100} a^{8}$, $\frac{1}{200} a^{9} - \frac{1}{20} a^{5} - \frac{1}{2} a$, $\frac{1}{400} a^{10} - \frac{1}{40} a^{6} + \frac{1}{4} a^{2}$, $\frac{1}{800} a^{11} + \frac{3}{80} a^{7} + \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{8000} a^{12} + \frac{3}{800} a^{8} - \frac{1}{20} a^{6} + \frac{1}{80} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{16000} a^{13} + \frac{3}{1600} a^{9} - \frac{1}{40} a^{7} + \frac{1}{160} a^{5} - \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{2530112000} a^{14} - \frac{17071}{316264000} a^{12} - \frac{202877}{253011200} a^{10} - \frac{14037}{15813200} a^{8} + \frac{16917}{1946240} a^{6} - \frac{20687}{1265056} a^{4} - \frac{110235}{316264} a^{2} + \frac{9110}{39533}$, $\frac{1}{5060224000} a^{15} - \frac{17071}{632528000} a^{13} - \frac{202877}{506022400} a^{11} - \frac{14037}{31626400} a^{9} - \frac{177707}{3892480} a^{7} - \frac{20687}{2530112} a^{5} - \frac{110235}{632528} a^{3} + \frac{4555}{39533} a$

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:  $7$
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:  \( 15294705.252 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times D_4):C_4$ (as 16T120):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $(C_2\times D_4):C_4$
Character table for $(C_2\times D_4):C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{17}) \), 4.0.57800.1, \(\Q(\sqrt{5}, \sqrt{17})\), 4.0.2312.1, 8.0.3340840000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.4.6.4$x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.8.22.5$x^{8} + 10 x^{4} + 16 x + 36$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
5Data not computed
$17$17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$