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Properties

Label 16.0.71594983564...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 11^{2}\cdot 101^{6}\cdot 1511^{2}$
Root discriminant $63.60$
Ramified primes $5, 11, 101, 1511$
Class number $2240$ (GRH)
Class group $[2, 2, 560]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T707)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3248561, -234427, 3038941, -373248, 1117415, -202907, 238958, -56734, 38308, -10281, 4718, -1276, 554, -104, 32, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 32*x^14 - 104*x^13 + 554*x^12 - 1276*x^11 + 4718*x^10 - 10281*x^9 + 38308*x^8 - 56734*x^7 + 238958*x^6 - 202907*x^5 + 1117415*x^4 - 373248*x^3 + 3038941*x^2 - 234427*x + 3248561)
 
gp: K = bnfinit(x^16 - 6*x^15 + 32*x^14 - 104*x^13 + 554*x^12 - 1276*x^11 + 4718*x^10 - 10281*x^9 + 38308*x^8 - 56734*x^7 + 238958*x^6 - 202907*x^5 + 1117415*x^4 - 373248*x^3 + 3038941*x^2 - 234427*x + 3248561, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 32 x^{14} - 104 x^{13} + 554 x^{12} - 1276 x^{11} + 4718 x^{10} - 10281 x^{9} + 38308 x^{8} - 56734 x^{7} + 238958 x^{6} - 202907 x^{5} + 1117415 x^{4} - 373248 x^{3} + 3038941 x^{2} - 234427 x + 3248561 \)

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:  \(71594983564208250058837890625=5^{12}\cdot 11^{2}\cdot 101^{6}\cdot 1511^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.60$
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:  $5, 11, 101, 1511$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{11} + \frac{1}{5} a^{8} - \frac{1}{5} a^{6} + \frac{1}{5} a^{4} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{11} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{2} - \frac{1}{5}$, $\frac{1}{25} a^{14} - \frac{2}{25} a^{12} + \frac{9}{25} a^{11} + \frac{1}{25} a^{10} - \frac{1}{25} a^{9} + \frac{3}{25} a^{8} - \frac{4}{25} a^{7} + \frac{7}{25} a^{6} - \frac{6}{25} a^{5} + \frac{9}{25} a^{4} + \frac{1}{25} a^{3} - \frac{2}{5} a^{2} + \frac{8}{25} a + \frac{4}{25}$, $\frac{1}{13427886722779751930029877261537663033352625} a^{15} + \frac{202659949074887202943899064579183666044882}{13427886722779751930029877261537663033352625} a^{14} + \frac{162059010470247168421676012931975406386543}{13427886722779751930029877261537663033352625} a^{13} - \frac{136324479687018677157211409078274520465769}{2685577344555950386005975452307532606670525} a^{12} + \frac{1679232072805578923194933243618195445086104}{13427886722779751930029877261537663033352625} a^{11} - \frac{2542488097777763230008292127775712210428344}{13427886722779751930029877261537663033352625} a^{10} + \frac{3013131728859213793938964961377245957191666}{13427886722779751930029877261537663033352625} a^{9} + \frac{2273417719821533208377810310794541741055032}{13427886722779751930029877261537663033352625} a^{8} + \frac{1691073568358823969852290678113469318905659}{13427886722779751930029877261537663033352625} a^{7} - \frac{768631797397674096881766136536974751530922}{13427886722779751930029877261537663033352625} a^{6} - \frac{998974321127584935727797344573357488081888}{13427886722779751930029877261537663033352625} a^{5} + \frac{6209443392006399850483579329780642515129854}{13427886722779751930029877261537663033352625} a^{4} - \frac{3527298913298476977545323977499881511482278}{13427886722779751930029877261537663033352625} a^{3} + \frac{16818456547000675786008542121089422436887}{123191621309905981009448415243464798471125} a^{2} + \frac{66525559621872786932489514026715133618464}{2685577344555950386005975452307532606670525} a - \frac{6672376897361714671484059795321227964866407}{13427886722779751930029877261537663033352625}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{560}$, which has order $2240$ (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:  \( 38120.6275869 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T707):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 31 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, 8.8.16098453125.1, 8.0.53514477878125.1, 8.0.529846315625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$101$101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
1511Data not computed