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Properties

Label 16.4.71411348623...0000.5
Degree $16$
Signature $[4, 6]$
Discriminant $2^{16}\cdot 5^{9}\cdot 11^{7}\cdot 31^{5}$
Root discriminant $41.29$
Ramified primes $2, 5, 11, 31$
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![16, -416, -40, 680, -336, 24, -292, -140, 89, 70, 48, 26, -11, 10, 0, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 10*x^13 - 11*x^12 + 26*x^11 + 48*x^10 + 70*x^9 + 89*x^8 - 140*x^7 - 292*x^6 + 24*x^5 - 336*x^4 + 680*x^3 - 40*x^2 - 416*x + 16)
 
gp: K = bnfinit(x^16 - 4*x^15 + 10*x^13 - 11*x^12 + 26*x^11 + 48*x^10 + 70*x^9 + 89*x^8 - 140*x^7 - 292*x^6 + 24*x^5 - 336*x^4 + 680*x^3 - 40*x^2 - 416*x + 16, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 10 x^{13} - 11 x^{12} + 26 x^{11} + 48 x^{10} + 70 x^{9} + 89 x^{8} - 140 x^{7} - 292 x^{6} + 24 x^{5} - 336 x^{4} + 680 x^{3} - 40 x^{2} - 416 x + 16 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(71411348623593088000000000=2^{16}\cdot 5^{9}\cdot 11^{7}\cdot 31^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.29$
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:  $2, 5, 11, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{10} a^{10} - \frac{1}{5} a^{8} - \frac{1}{10} a^{7} + \frac{1}{10} a^{6} - \frac{3}{10} a^{5} + \frac{2}{5} a^{4} - \frac{1}{10} a^{3} - \frac{3}{10} a^{2} + \frac{2}{5}$, $\frac{1}{10} a^{11} - \frac{1}{5} a^{9} - \frac{1}{10} a^{8} + \frac{1}{10} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{3}{10} a^{3} - \frac{1}{2} a^{2} + \frac{2}{5} a$, $\frac{1}{20} a^{12} + \frac{1}{5} a^{9} - \frac{3}{20} a^{8} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{4} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5}$, $\frac{1}{100} a^{13} - \frac{1}{50} a^{12} + \frac{1}{50} a^{11} + \frac{3}{20} a^{9} + \frac{11}{50} a^{8} + \frac{1}{50} a^{7} - \frac{6}{25} a^{6} - \frac{31}{100} a^{5} - \frac{1}{2} a^{4} + \frac{2}{5} a^{3} + \frac{4}{25} a^{2} - \frac{1}{25} a - \frac{3}{25}$, $\frac{1}{3800} a^{14} + \frac{1}{475} a^{13} + \frac{1}{1900} a^{12} - \frac{11}{380} a^{11} - \frac{3}{152} a^{10} - \frac{169}{1900} a^{9} - \frac{189}{1900} a^{8} + \frac{353}{1900} a^{7} + \frac{849}{3800} a^{6} - \frac{1}{2} a^{5} + \frac{31}{380} a^{4} - \frac{261}{950} a^{3} - \frac{241}{950} a^{2} - \frac{11}{25} a - \frac{8}{95}$, $\frac{1}{8915752876600} a^{15} - \frac{125250847}{1114469109575} a^{14} - \frac{10955444437}{2228938219150} a^{13} + \frac{83813840249}{4457876438300} a^{12} + \frac{378079554089}{8915752876600} a^{11} - \frac{68897273329}{4457876438300} a^{10} - \frac{328417988009}{2228938219150} a^{9} - \frac{810358736629}{4457876438300} a^{8} - \frac{1076983387511}{8915752876600} a^{7} - \frac{528200974971}{2228938219150} a^{6} - \frac{502638135783}{2228938219150} a^{5} + \frac{568045200039}{2228938219150} a^{4} + \frac{118189821219}{1114469109575} a^{3} + \frac{417688105919}{2228938219150} a^{2} - \frac{73029375031}{222893821915} a - \frac{263692349313}{1114469109575}$

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:  $9$
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:  \( 15535260.0958 \) (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

\(\Q(\sqrt{5}) \), 4.4.8525.1, 8.4.204654560000.2

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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$2.8.8.9$x^{8} + 6 x^{6} + 4 x^{5} + 16$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
2.8.8.9$x^{8} + 6 x^{6} + 4 x^{5} + 16$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
$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}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
31Data not computed