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Properties

Label 16.8.38974026016...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{16}\cdot 5^{14}\cdot 29^{2}\cdot 41^{5}$
Root discriminant $39.76$
Ramified primes $2, 5, 29, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1782

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![80, 0, -880, 1280, 760, -2220, 1550, -420, -449, 692, -442, 124, 40, -56, 28, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 40*x^12 + 124*x^11 - 442*x^10 + 692*x^9 - 449*x^8 - 420*x^7 + 1550*x^6 - 2220*x^5 + 760*x^4 + 1280*x^3 - 880*x^2 + 80)
 
gp: K = bnfinit(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 40*x^12 + 124*x^11 - 442*x^10 + 692*x^9 - 449*x^8 - 420*x^7 + 1550*x^6 - 2220*x^5 + 760*x^4 + 1280*x^3 - 880*x^2 + 80, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 28 x^{14} - 56 x^{13} + 40 x^{12} + 124 x^{11} - 442 x^{10} + 692 x^{9} - 449 x^{8} - 420 x^{7} + 1550 x^{6} - 2220 x^{5} + 760 x^{4} + 1280 x^{3} - 880 x^{2} + 80 \)

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:  \(38974026016400000000000000=2^{16}\cdot 5^{14}\cdot 29^{2}\cdot 41^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.76$
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:  $2, 5, 29, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{8} + \frac{1}{8} a^{6}$, $\frac{1}{24} a^{13} + \frac{1}{24} a^{12} + \frac{1}{24} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{24} a^{7} - \frac{1}{24} a^{6} - \frac{1}{12} a^{5} - \frac{1}{12} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{4312152} a^{14} - \frac{7}{4312152} a^{13} + \frac{48437}{1078038} a^{12} - \frac{84359}{4312152} a^{11} - \frac{37799}{359346} a^{10} - \frac{343}{110568} a^{9} + \frac{119071}{2156076} a^{8} + \frac{356107}{4312152} a^{7} - \frac{31593}{479128} a^{6} - \frac{176935}{1078038} a^{5} - \frac{245761}{2156076} a^{4} - \frac{91721}{539019} a^{3} - \frac{488269}{1078038} a^{2} - \frac{47297}{539019} a - \frac{78062}{539019}$, $\frac{1}{245792664} a^{15} + \frac{7}{81930888} a^{14} - \frac{345467}{245792664} a^{13} - \frac{2531957}{81930888} a^{12} + \frac{8503759}{245792664} a^{11} + \frac{21623}{535496} a^{10} + \frac{30587669}{245792664} a^{9} - \frac{4486213}{81930888} a^{8} + \frac{4982005}{61448166} a^{7} + \frac{19921267}{122896332} a^{6} - \frac{6978167}{40965444} a^{5} + \frac{5669221}{40965444} a^{4} + \frac{1806337}{4726782} a^{3} - \frac{5805025}{30724083} a^{2} - \frac{10026682}{30724083} a + \frac{10211701}{30724083}$

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:  \( 13947781.9872 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1782:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16384
The 130 conjugacy class representatives for t16n1782 are not computed
Character table for t16n1782 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.5125.1, 8.6.974980000000.4

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

Sibling fields

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.8.7$x^{8} + 2 x^{6} + 4 x^{5} + 16$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
2.8.8.7$x^{8} + 2 x^{6} + 4 x^{5} + 16$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
$41$41.4.3.4$x^{4} + 8856$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.8.0.1$x^{8} - x + 12$$1$$8$$0$$C_8$$[\ ]^{8}$