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Properties

Label 16.0.98856461372...8129.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{4}\cdot 73^{14}$
Root discriminant $56.19$
Ramified primes $3, 73$
Class number $89$ (GRH)
Class group $[89]$ (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2916352, -1929216, 2335744, -1005056, 642304, -163440, 70900, -4508, -4787, 2471, -1544, 243, -70, -11, 12, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 12*x^14 - 11*x^13 - 70*x^12 + 243*x^11 - 1544*x^10 + 2471*x^9 - 4787*x^8 - 4508*x^7 + 70900*x^6 - 163440*x^5 + 642304*x^4 - 1005056*x^3 + 2335744*x^2 - 1929216*x + 2916352)
 
gp: K = bnfinit(x^16 - 3*x^15 + 12*x^14 - 11*x^13 - 70*x^12 + 243*x^11 - 1544*x^10 + 2471*x^9 - 4787*x^8 - 4508*x^7 + 70900*x^6 - 163440*x^5 + 642304*x^4 - 1005056*x^3 + 2335744*x^2 - 1929216*x + 2916352, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 12 x^{14} - 11 x^{13} - 70 x^{12} + 243 x^{11} - 1544 x^{10} + 2471 x^{9} - 4787 x^{8} - 4508 x^{7} + 70900 x^{6} - 163440 x^{5} + 642304 x^{4} - 1005056 x^{3} + 2335744 x^{2} - 1929216 x + 2916352 \)

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:  \(9885646137219473682569328129=3^{4}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.19$
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:  $3, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{1}{4} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{9} - \frac{1}{32} a^{8} - \frac{1}{32} a^{7} - \frac{1}{8} a^{6} + \frac{7}{32} a^{5} + \frac{7}{32} a^{4} - \frac{7}{32} a^{3} - \frac{1}{16} a^{2}$, $\frac{1}{64} a^{10} + \frac{1}{32} a^{8} - \frac{1}{64} a^{7} - \frac{1}{64} a^{6} + \frac{3}{32} a^{5} + \frac{1}{16} a^{4} + \frac{3}{64} a^{3} - \frac{3}{32} a^{2} - \frac{1}{8} a$, $\frac{1}{64} a^{11} + \frac{1}{64} a^{8} + \frac{1}{64} a^{7} - \frac{1}{32} a^{6} + \frac{3}{32} a^{5} - \frac{11}{64} a^{4} + \frac{3}{8} a^{3} + \frac{3}{16} a^{2} - \frac{1}{2} a$, $\frac{1}{512} a^{12} - \frac{1}{512} a^{11} + \frac{1}{256} a^{10} + \frac{1}{512} a^{9} - \frac{1}{128} a^{8} - \frac{13}{512} a^{7} + \frac{7}{256} a^{6} - \frac{117}{512} a^{5} + \frac{99}{512} a^{4} + \frac{49}{256} a^{3} - \frac{7}{32} a^{2} - \frac{3}{16} a - \frac{1}{2}$, $\frac{1}{2048} a^{13} + \frac{1}{2048} a^{12} + \frac{5}{2048} a^{10} + \frac{7}{1024} a^{9} + \frac{91}{2048} a^{8} + \frac{9}{512} a^{7} - \frac{153}{2048} a^{6} + \frac{425}{2048} a^{5} + \frac{11}{256} a^{4} + \frac{25}{512} a^{3} - \frac{13}{64} a^{2} - \frac{13}{32} a + \frac{1}{4}$, $\frac{1}{16384} a^{14} - \frac{3}{16384} a^{13} - \frac{1}{4096} a^{12} - \frac{27}{16384} a^{11} + \frac{29}{8192} a^{10} - \frac{221}{16384} a^{9} - \frac{61}{2048} a^{8} - \frac{393}{16384} a^{7} - \frac{499}{16384} a^{6} + \frac{157}{4096} a^{5} - \frac{39}{4096} a^{4} - \frac{487}{1024} a^{3} - \frac{91}{256} a^{2} + \frac{11}{64} a + \frac{3}{8}$, $\frac{1}{54037607137642816737378304} a^{15} + \frac{617698300050367146401}{54037607137642816737378304} a^{14} + \frac{218201469347490383043}{3377350446102676046086144} a^{13} - \frac{35365474357879631890763}{54037607137642816737378304} a^{12} - \frac{124769604878387787025321}{27018803568821408368689152} a^{11} + \frac{341472417704455413720683}{54037607137642816737378304} a^{10} - \frac{69765281456459990033823}{13509401784410704184344576} a^{9} + \frac{2616045898466926313508407}{54037607137642816737378304} a^{8} - \frac{3145050367868533730879607}{54037607137642816737378304} a^{7} - \frac{503463046599869172311195}{6754700892205352092172288} a^{6} + \frac{1430949206801200086697637}{13509401784410704184344576} a^{5} - \frac{200305362748514101117725}{1688675223051338023043072} a^{4} + \frac{157094488115271330084213}{422168805762834505760768} a^{3} + \frac{18029183134094901701001}{52771100720354313220096} a^{2} + \frac{18649321579944850677037}{52771100720354313220096} a + \frac{895296424488629957391}{6596387590044289152512}$

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

Class group and class number

$C_{89}$, which has order $89$ (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:  \( 103600962.089 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_8$ (as 16T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{73}) \), 4.2.15987.1, 4.4.389017.1, 4.2.1167051.1, 8.4.99426586671873.1, 8.0.11047398519097.1, 8.4.1362008036601.1

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

Sibling fields

Galois closure: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
73Data not computed