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Properties

Label 16.4.41943040000...0000.2
Degree $16$
Signature $[4, 6]$
Discriminant $2^{40}\cdot 5^{18}$
Root discriminant $34.59$
Ramified primes $2, 5$
Class number $1$
Class group Trivial
Galois group $A_{16}$ (as 16T1953)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, -20, 0, 100, -160, -4, 336, -460, 290, -60, -16, 44, -40, 20, 0, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 20*x^13 - 40*x^12 + 44*x^11 - 16*x^10 - 60*x^9 + 290*x^8 - 460*x^7 + 336*x^6 - 4*x^5 - 160*x^4 + 100*x^3 - 20*x + 5)
 
gp: K = bnfinit(x^16 - 4*x^15 + 20*x^13 - 40*x^12 + 44*x^11 - 16*x^10 - 60*x^9 + 290*x^8 - 460*x^7 + 336*x^6 - 4*x^5 - 160*x^4 + 100*x^3 - 20*x + 5, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 20 x^{13} - 40 x^{12} + 44 x^{11} - 16 x^{10} - 60 x^{9} + 290 x^{8} - 460 x^{7} + 336 x^{6} - 4 x^{5} - 160 x^{4} + 100 x^{3} - 20 x + 5 \)

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:  \(4194304000000000000000000=2^{40}\cdot 5^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.59$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{8} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{8} a^{2} + \frac{3}{8}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2} + \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{8} - \frac{1}{8} a^{4} + \frac{1}{8}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{12} + \frac{1}{16} a^{9} - \frac{1}{16} a^{8} + \frac{3}{16} a^{5} - \frac{3}{16} a^{4} - \frac{5}{16} a + \frac{5}{16}$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{12} - \frac{1}{16} a^{10} + \frac{1}{16} a^{8} - \frac{1}{16} a^{6} + \frac{1}{16} a^{4} + \frac{1}{16} a^{2} - \frac{1}{16}$, $\frac{1}{89744} a^{15} - \frac{1039}{44872} a^{14} + \frac{515}{22436} a^{13} - \frac{3971}{89744} a^{12} - \frac{3807}{89744} a^{11} - \frac{855}{44872} a^{10} + \frac{409}{22436} a^{9} + \frac{321}{89744} a^{8} + \frac{2007}{89744} a^{7} - \frac{6159}{44872} a^{6} - \frac{421}{5609} a^{5} + \frac{9659}{89744} a^{4} - \frac{36841}{89744} a^{3} + \frac{6877}{44872} a^{2} + \frac{2202}{5609} a - \frac{2745}{89744}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2751187.82391 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_{16}$ (as 16T1953):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 10461394944000
The 123 conjugacy class representatives for $A_{16}$ are not computed
Character table for $A_{16}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.13.0.1}{13} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ $15{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.9.0.1}{9} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
2Data not computed
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.5.8.8$x^{5} + 15 x^{4} + 5$$5$$1$$8$$F_5$$[2]^{4}$
5.10.10.14$x^{10} + 10 x^{6} + 10 x^{5} + 100 x^{2} + 50 x + 25$$5$$2$$10$$(C_5^2 : C_4) : C_2$$[5/4, 5/4]_{4}^{2}$