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Properties

Label 16.0.28733701216...0625.3
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 5^{8}\cdot 7^{12}$
Root discriminant $21.94$
Ramified primes $3, 5, 7$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2^2\wr C_2$ (as 16T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 4, -18, -40, -43, 87, -68, 232, 68, 87, 43, -40, 18, 4, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 4*x^14 + 18*x^13 - 40*x^12 + 43*x^11 + 87*x^10 + 68*x^9 + 232*x^8 - 68*x^7 + 87*x^6 - 43*x^5 - 40*x^4 - 18*x^3 + 4*x^2 + 5*x + 1)
 
gp: K = bnfinit(x^16 - 5*x^15 + 4*x^14 + 18*x^13 - 40*x^12 + 43*x^11 + 87*x^10 + 68*x^9 + 232*x^8 - 68*x^7 + 87*x^6 - 43*x^5 - 40*x^4 - 18*x^3 + 4*x^2 + 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 4 x^{14} + 18 x^{13} - 40 x^{12} + 43 x^{11} + 87 x^{10} + 68 x^{9} + 232 x^{8} - 68 x^{7} + 87 x^{6} - 43 x^{5} - 40 x^{4} - 18 x^{3} + 4 x^{2} + 5 x + 1 \)

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:  \(2873370121635406640625=3^{12}\cdot 5^{8}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.94$
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:  $3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{4}$, $\frac{1}{20} a^{10} - \frac{1}{10} a^{9} - \frac{1}{10} a^{8} + \frac{1}{10} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{3}{10} a^{4} + \frac{1}{10} a^{3} + \frac{1}{10} a^{2} + \frac{2}{5} a - \frac{1}{20}$, $\frac{1}{20} a^{11} + \frac{1}{5} a^{9} - \frac{1}{10} a^{8} - \frac{1}{5} a^{6} + \frac{1}{10} a^{5} + \frac{3}{10} a^{3} - \frac{2}{5} a^{2} - \frac{1}{4} a - \frac{1}{10}$, $\frac{1}{100} a^{12} + \frac{1}{50} a^{11} - \frac{1}{100} a^{10} + \frac{3}{50} a^{9} - \frac{6}{25} a^{8} + \frac{3}{50} a^{7} - \frac{3}{50} a^{6} + \frac{1}{25} a^{5} - \frac{7}{50} a^{4} + \frac{11}{25} a^{3} + \frac{9}{100} a^{2} + \frac{7}{25} a + \frac{41}{100}$, $\frac{1}{500} a^{13} - \frac{1}{500} a^{12} - \frac{3}{125} a^{11} + \frac{9}{500} a^{10} - \frac{28}{125} a^{9} + \frac{22}{125} a^{8} - \frac{31}{125} a^{7} - \frac{27}{125} a^{6} + \frac{16}{125} a^{5} + \frac{9}{125} a^{4} + \frac{147}{500} a^{3} - \frac{59}{500} a^{2} - \frac{42}{125} a + \frac{87}{500}$, $\frac{1}{500} a^{14} + \frac{1}{250} a^{12} + \frac{1}{250} a^{11} + \frac{7}{500} a^{10} - \frac{17}{250} a^{9} - \frac{24}{125} a^{8} + \frac{27}{125} a^{7} - \frac{17}{250} a^{6} + \frac{11}{50} a^{5} - \frac{27}{500} a^{4} - \frac{38}{125} a^{3} - \frac{71}{250} a^{2} - \frac{9}{125} a + \frac{127}{500}$, $\frac{1}{5000} a^{15} - \frac{1}{2500} a^{14} - \frac{1}{2500} a^{13} + \frac{3}{1250} a^{12} - \frac{13}{625} a^{11} + \frac{31}{5000} a^{10} + \frac{39}{250} a^{9} + \frac{77}{1250} a^{8} - \frac{118}{625} a^{7} - \frac{813}{5000} a^{5} + \frac{59}{2500} a^{4} + \frac{607}{2500} a^{3} + \frac{181}{1250} a^{2} + \frac{97}{625} a - \frac{1467}{5000}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( \frac{8}{125} a^{15} + \frac{17}{125} a^{14} - \frac{967}{500} a^{13} + \frac{1311}{500} a^{12} + \frac{698}{125} a^{11} - \frac{6871}{500} a^{10} + \frac{5859}{250} a^{9} + \frac{2253}{50} a^{8} + \frac{2803}{50} a^{7} + \frac{29413}{250} a^{6} + \frac{913}{125} a^{5} + \frac{14857}{250} a^{4} - \frac{2567}{500} a^{3} - \frac{5509}{500} a^{2} - \frac{2193}{250} a - \frac{681}{500} \) (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:  \( 30068.4245215 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\wr C_2$ (as 16T46):

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 $C_2^2\wr C_2$
Character table for $C_2^2\wr C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{-35}) \), 4.0.4725.1, 4.0.189.1, 4.4.231525.1, 4.0.231525.1, 4.0.25725.1, \(\Q(\sqrt{-3}, \sqrt{-35})\), 4.0.25725.2, 8.0.5955980625.2, 8.0.1093955625.3, 8.0.53603825625.6

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 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.4.3.2$x^{4} - 7$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
7.4.3.2$x^{4} - 7$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
7.8.6.2$x^{8} - 49 x^{4} + 3969$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$