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Properties

Label 16.0.18288680905...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{4}\cdot 13^{8}\cdot 17^{4}$
Root discriminant $43.79$
Ramified primes $2, 5, 13, 17$
Class number $40$ (GRH)
Class group $[2, 2, 10]$ (GRH)
Galois group $C_4^2:C_2^2$ (as 16T117)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4624, 2720, 7792, 3152, 3608, 3384, 5092, -6004, 6057, -3424, 1672, -684, 297, -116, 38, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 38*x^14 - 116*x^13 + 297*x^12 - 684*x^11 + 1672*x^10 - 3424*x^9 + 6057*x^8 - 6004*x^7 + 5092*x^6 + 3384*x^5 + 3608*x^4 + 3152*x^3 + 7792*x^2 + 2720*x + 4624)
 
gp: K = bnfinit(x^16 - 8*x^15 + 38*x^14 - 116*x^13 + 297*x^12 - 684*x^11 + 1672*x^10 - 3424*x^9 + 6057*x^8 - 6004*x^7 + 5092*x^6 + 3384*x^5 + 3608*x^4 + 3152*x^3 + 7792*x^2 + 2720*x + 4624, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 38 x^{14} - 116 x^{13} + 297 x^{12} - 684 x^{11} + 1672 x^{10} - 3424 x^{9} + 6057 x^{8} - 6004 x^{7} + 5092 x^{6} + 3384 x^{5} + 3608 x^{4} + 3152 x^{3} + 7792 x^{2} + 2720 x + 4624 \)

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:  \(182886809054238170152960000=2^{32}\cdot 5^{4}\cdot 13^{8}\cdot 17^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.79$
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:  $2, 5, 13, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{24} a^{12} + \frac{1}{12} a^{11} - \frac{1}{4} a^{10} + \frac{1}{6} a^{9} - \frac{1}{8} a^{8} - \frac{1}{12} a^{7} + \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{11}{24} a^{4} + \frac{1}{4} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{6}$, $\frac{1}{24} a^{13} + \frac{1}{12} a^{11} + \frac{1}{6} a^{10} - \frac{11}{24} a^{9} + \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{3} a^{6} - \frac{11}{24} a^{5} + \frac{1}{6} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{48} a^{14} + \frac{1}{48} a^{10} + \frac{5}{12} a^{9} + \frac{1}{24} a^{8} - \frac{1}{12} a^{7} + \frac{5}{48} a^{6} - \frac{5}{12} a^{5} - \frac{5}{24} a^{4} + \frac{1}{12} a^{3} + \frac{1}{4} a^{2} + \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{362236063585095154655198112} a^{15} - \frac{47936557175543247618585}{7546584658022815721983294} a^{14} + \frac{214728323253253475901236}{11319876987034223582974941} a^{13} - \frac{334195433943003916320575}{30186338632091262887933176} a^{12} + \frac{13310459614694867619152819}{120745354528365051551732704} a^{11} + \frac{4611524939282314324344311}{30186338632091262887933176} a^{10} - \frac{15832893947849592950851085}{60372677264182525775866352} a^{9} - \frac{3928448898413323285994055}{15093169316045631443966588} a^{8} - \frac{53613195469821184093513715}{362236063585095154655198112} a^{7} - \frac{44332561014620245531454575}{90559015896273788663799528} a^{6} + \frac{28654613619008923182636073}{60372677264182525775866352} a^{5} - \frac{3169700253477654583258309}{22639753974068447165949882} a^{4} + \frac{4743437697907353783243457}{90559015896273788663799528} a^{3} + \frac{1216476266696499805576537}{3773292329011407860991647} a^{2} + \frac{19364751242645511340193219}{45279507948136894331899764} a + \frac{88739482292262700864730}{665875116884366093116173}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{10}$, which has order $40$ (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:  \( -1 \) (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:  \( 65218.9169673 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2:C_2^2$ (as 16T117):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $C_4^2:C_2^2$
Character table for $C_4^2:C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{26}) \), 4.4.183872.3, \(\Q(\sqrt{2}, \sqrt{13})\), 4.4.183872.2, 8.8.33808912384.1, 8.0.46794342400.1, 8.0.13523564953600.5

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
$5$5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$