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Properties

Label 16.0.28196723068...5089.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 73^{10}$
Root discriminant $25.30$
Ramified primes $3, 73$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4\wr C_2$ (as 16T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![268, -1600, 2469, 36, 6427, -3542, 5100, -5082, 3384, -2304, 1404, -578, 280, -78, 27, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 27*x^14 - 78*x^13 + 280*x^12 - 578*x^11 + 1404*x^10 - 2304*x^9 + 3384*x^8 - 5082*x^7 + 5100*x^6 - 3542*x^5 + 6427*x^4 + 36*x^3 + 2469*x^2 - 1600*x + 268)
 
gp: K = bnfinit(x^16 - 4*x^15 + 27*x^14 - 78*x^13 + 280*x^12 - 578*x^11 + 1404*x^10 - 2304*x^9 + 3384*x^8 - 5082*x^7 + 5100*x^6 - 3542*x^5 + 6427*x^4 + 36*x^3 + 2469*x^2 - 1600*x + 268, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 27 x^{14} - 78 x^{13} + 280 x^{12} - 578 x^{11} + 1404 x^{10} - 2304 x^{9} + 3384 x^{8} - 5082 x^{7} + 5100 x^{6} - 3542 x^{5} + 6427 x^{4} + 36 x^{3} + 2469 x^{2} - 1600 x + 268 \)

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:  \(28196723068685041735089=3^{8}\cdot 73^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.30$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{16} a^{11} + \frac{1}{16} a^{8} - \frac{1}{4} a^{6} + \frac{3}{16} a^{5} + \frac{3}{16} a^{2} + \frac{1}{4}$, $\frac{1}{32} a^{12} - \frac{1}{32} a^{11} - \frac{1}{16} a^{10} + \frac{1}{32} a^{9} + \frac{1}{32} a^{8} - \frac{1}{8} a^{7} + \frac{7}{32} a^{6} + \frac{5}{32} a^{5} - \frac{3}{16} a^{4} - \frac{5}{32} a^{3} - \frac{5}{32} a^{2} + \frac{1}{8} a + \frac{1}{8}$, $\frac{1}{64} a^{13} + \frac{1}{64} a^{11} + \frac{3}{64} a^{10} + \frac{1}{32} a^{9} - \frac{3}{64} a^{8} + \frac{3}{64} a^{7} + \frac{3}{16} a^{6} - \frac{5}{64} a^{5} - \frac{15}{64} a^{4} - \frac{5}{32} a^{3} - \frac{17}{64} a^{2} - \frac{1}{8} a + \frac{1}{16}$, $\frac{1}{64} a^{14} - \frac{1}{64} a^{12} + \frac{1}{64} a^{11} - \frac{1}{32} a^{10} + \frac{3}{64} a^{9} - \frac{3}{64} a^{8} + \frac{1}{16} a^{7} - \frac{3}{64} a^{6} - \frac{5}{64} a^{5} + \frac{5}{32} a^{4} - \frac{15}{64} a^{3} + \frac{11}{32} a^{2} - \frac{5}{16} a + \frac{1}{8}$, $\frac{1}{23398297185480249344} a^{15} - \frac{158480774445969737}{23398297185480249344} a^{14} + \frac{1311114411674297}{2924787148185031168} a^{13} + \frac{55781018441955429}{11699148592740124672} a^{12} - \frac{166187204515055021}{11699148592740124672} a^{11} - \frac{2449217733925719}{91399598380782224} a^{10} - \frac{23903961953256033}{5849574296370062336} a^{9} + \frac{107834262482084325}{5849574296370062336} a^{8} - \frac{108746415567967339}{5849574296370062336} a^{7} - \frac{1720076609426508863}{11699148592740124672} a^{6} + \frac{652067558997697521}{11699148592740124672} a^{5} - \frac{26026721287363143}{365598393523128896} a^{4} - \frac{8421215496220242981}{23398297185480249344} a^{3} + \frac{3565921100530634525}{23398297185480249344} a^{2} - \frac{1691721319714703499}{5849574296370062336} a - \frac{185566276888154649}{5849574296370062336}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{1602690543}{993771917312} a^{15} + \frac{5615017511}{993771917312} a^{14} - \frac{5032347911}{124221489664} a^{13} + \frac{52419930421}{496885958656} a^{12} - \frac{197225627325}{496885958656} a^{11} + \frac{5748155779}{7763843104} a^{10} - \frac{474902863985}{248442979328} a^{9} + \frac{717905793173}{248442979328} a^{8} - \frac{1065623117819}{248442979328} a^{7} + \frac{3368238515281}{496885958656} a^{6} - \frac{3085337478591}{496885958656} a^{5} + \frac{65593966495}{15527686208} a^{4} - \frac{10497194745973}{993771917312} a^{3} - \frac{2598294039187}{993771917312} a^{2} - \frac{1382385421115}{248442979328} a + \frac{537934268375}{248442979328} \) (order $6$)
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:  \( 686254.798714 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T42):

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 $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{73}) \), \(\Q(\sqrt{-219}) \), 4.0.657.1 x2, 4.2.15987.1 x2, \(\Q(\sqrt{-3}, \sqrt{73})\), 8.0.167918799033.1, 8.0.31510377.1, 8.0.2300257521.1

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 8 siblings: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$73$73.8.6.2$x^{8} + 1533 x^{4} + 644809$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
73.8.4.1$x^{8} + 138554 x^{4} - 389017 x^{2} + 4799302729$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$