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Properties

Label 16.8.43149301773...5625.4
Degree $16$
Signature $[8, 4]$
Discriminant $5^{8}\cdot 101^{10}$
Root discriminant $40.01$
Ramified primes $5, 101$
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![25, 0, -60065, 0, 33196, 0, -9692, 0, 4554, 0, -361, 0, 89, 0, -22, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 22*x^14 + 89*x^12 - 361*x^10 + 4554*x^8 - 9692*x^6 + 33196*x^4 - 60065*x^2 + 25)
 
gp: K = bnfinit(x^16 - 22*x^14 + 89*x^12 - 361*x^10 + 4554*x^8 - 9692*x^6 + 33196*x^4 - 60065*x^2 + 25, 1)
 

Normalized defining polynomial

\( x^{16} - 22 x^{14} + 89 x^{12} - 361 x^{10} + 4554 x^{8} - 9692 x^{6} + 33196 x^{4} - 60065 x^{2} + 25 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(43149301773875176172265625=5^{8}\cdot 101^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.01$
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:  $5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{10} - \frac{1}{2} a^{7} + \frac{3}{10} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} + \frac{1}{10} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{11} + \frac{3}{10} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{10} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{12} - \frac{1}{5} a^{8} + \frac{1}{10} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{50} a^{13} - \frac{1}{50} a^{11} + \frac{3}{50} a^{9} + \frac{11}{25} a^{7} - \frac{1}{2} a^{6} - \frac{2}{25} a^{5} + \frac{9}{50} a^{3} + \frac{2}{5} a - \frac{1}{2}$, $\frac{1}{25463580466776650} a^{14} - \frac{468101337070361}{25463580466776650} a^{12} - \frac{1265506530061187}{25463580466776650} a^{10} + \frac{698937158958821}{12731790233388325} a^{8} - \frac{1}{2} a^{7} + \frac{3309839822519088}{12731790233388325} a^{6} - \frac{883021918011041}{2314870951525150} a^{4} - \frac{665470307877967}{2546358046677665} a^{2} - \frac{1}{2} a + \frac{19619138519617}{509271609335533}$, $\frac{1}{25463580466776650} a^{15} + \frac{20585136132586}{12731790233388325} a^{13} + \frac{154315981456189}{5092716093355330} a^{11} + \frac{2925689145924241}{25463580466776650} a^{9} - \frac{851276323753}{25463580466776650} a^{7} - \frac{1068211594133053}{2314870951525150} a^{5} - \frac{1}{2} a^{4} + \frac{237549725958896}{12731790233388325} a^{3} - \frac{313080224139363}{5092716093355330} a - \frac{1}{2}$

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:  $11$
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:  \( 28231704.2902 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{101}) \), \(\Q(\sqrt{505}) \), 4.4.2525.1 x2, 4.4.51005.1 x2, \(\Q(\sqrt{5}, \sqrt{101})\), 8.4.6568812813125.5, 8.4.643938125.1, 8.8.65037750625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
101Data not computed