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Properties

Label 16.8.20837857125...1201.1
Degree $16$
Signature $[8, 4]$
Discriminant $31^{4}\cdot 41^{12}$
Root discriminant $38.23$
Ramified primes $31, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T158)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-529, -1889, 766, 7377, 9198, -3190, -21047, 2662, 7638, -2455, 260, 44, -177, 76, 0, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 76*x^13 - 177*x^12 + 44*x^11 + 260*x^10 - 2455*x^9 + 7638*x^8 + 2662*x^7 - 21047*x^6 - 3190*x^5 + 9198*x^4 + 7377*x^3 + 766*x^2 - 1889*x - 529)
 
gp: K = bnfinit(x^16 - 6*x^15 + 76*x^13 - 177*x^12 + 44*x^11 + 260*x^10 - 2455*x^9 + 7638*x^8 + 2662*x^7 - 21047*x^6 - 3190*x^5 + 9198*x^4 + 7377*x^3 + 766*x^2 - 1889*x - 529, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 76 x^{13} - 177 x^{12} + 44 x^{11} + 260 x^{10} - 2455 x^{9} + 7638 x^{8} + 2662 x^{7} - 21047 x^{6} - 3190 x^{5} + 9198 x^{4} + 7377 x^{3} + 766 x^{2} - 1889 x - 529 \)

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:  \(20837857125684480535711201=31^{4}\cdot 41^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.23$
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:  $31, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{5} a^{10} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{11} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{50} a^{12} + \frac{1}{25} a^{11} + \frac{3}{50} a^{9} - \frac{2}{25} a^{8} - \frac{12}{25} a^{7} + \frac{9}{50} a^{6} + \frac{4}{25} a^{5} + \frac{9}{25} a^{4} + \frac{3}{10} a^{3} - \frac{7}{25} a^{2} + \frac{9}{25} a + \frac{9}{50}$, $\frac{1}{50} a^{13} - \frac{2}{25} a^{11} + \frac{3}{50} a^{10} + \frac{2}{25} a^{8} - \frac{3}{50} a^{7} + \frac{1}{5} a^{6} - \frac{9}{25} a^{5} + \frac{9}{50} a^{4} - \frac{2}{25} a^{3} - \frac{7}{25} a^{2} - \frac{7}{50} a + \frac{1}{25}$, $\frac{1}{1150} a^{14} - \frac{3}{1150} a^{13} + \frac{7}{1150} a^{12} - \frac{43}{1150} a^{11} - \frac{79}{1150} a^{10} - \frac{53}{1150} a^{9} - \frac{59}{1150} a^{8} + \frac{109}{230} a^{7} - \frac{319}{1150} a^{6} + \frac{351}{1150} a^{5} - \frac{143}{1150} a^{4} - \frac{487}{1150} a^{3} + \frac{251}{1150} a^{2} - \frac{99}{1150} a - \frac{9}{50}$, $\frac{1}{256848765710109606080150} a^{15} - \frac{5903871218980059901}{128424382855054803040075} a^{14} + \frac{1712068875725070867031}{256848765710109606080150} a^{13} + \frac{2031863447381653947093}{256848765710109606080150} a^{12} + \frac{5059564070484539232109}{128424382855054803040075} a^{11} + \frac{17469474802215179409519}{256848765710109606080150} a^{10} - \frac{715389037696563925271}{10273950628404384243206} a^{9} - \frac{11154985839161288723296}{128424382855054803040075} a^{8} - \frac{1641925910682621206511}{256848765710109606080150} a^{7} + \frac{74970333839332615986603}{256848765710109606080150} a^{6} + \frac{49870674912613249604472}{128424382855054803040075} a^{5} - \frac{7535842141807850409343}{51369753142021921216030} a^{4} + \frac{78027207103656959573681}{256848765710109606080150} a^{3} + \frac{33355202036413582033589}{128424382855054803040075} a^{2} + \frac{89346699400673205507387}{256848765710109606080150} a - \frac{582825764509320748112}{5583668819784991436525}$

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:  $11$
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:  \( 4240818.47303 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T158):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.2.52111.1, 4.4.68921.1, 4.2.2136551.1, 8.4.111337809161.1 x2, 8.6.147253231471.1 x2, 8.4.4564850175601.2

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

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$31$31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
$41$41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$