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Properties

Label 16.8.54566491401...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 5^{8}\cdot 13^{6}\cdot 29^{7}$
Root discriminant $72.20$
Ramified primes $2, 5, 13, 29$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1558

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4709861, 0, -14046266, 0, 567385, 0, 417716, 0, 21474, 0, -4274, 0, -340, 0, 4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 4*x^14 - 340*x^12 - 4274*x^10 + 21474*x^8 + 417716*x^6 + 567385*x^4 - 14046266*x^2 + 4709861)
 
gp: K = bnfinit(x^16 + 4*x^14 - 340*x^12 - 4274*x^10 + 21474*x^8 + 417716*x^6 + 567385*x^4 - 14046266*x^2 + 4709861, 1)
 

Normalized defining polynomial

\( x^{16} + 4 x^{14} - 340 x^{12} - 4274 x^{10} + 21474 x^{8} + 417716 x^{6} + 567385 x^{4} - 14046266 x^{2} + 4709861 \)

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:  \(545664914012032080281600000000=2^{24}\cdot 5^{8}\cdot 13^{6}\cdot 29^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.20$
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:  $2, 5, 13, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{10} a^{8} + \frac{1}{5} a^{6} - \frac{1}{5} a^{4} - \frac{3}{10} a^{2} + \frac{1}{10}$, $\frac{1}{10} a^{9} + \frac{1}{5} a^{7} - \frac{1}{5} a^{5} - \frac{3}{10} a^{3} + \frac{1}{10} a$, $\frac{1}{10} a^{10} + \frac{2}{5} a^{6} + \frac{1}{10} a^{4} - \frac{3}{10} a^{2} - \frac{1}{5}$, $\frac{1}{10} a^{11} + \frac{2}{5} a^{7} + \frac{1}{10} a^{5} - \frac{3}{10} a^{3} - \frac{1}{5} a$, $\frac{1}{511420} a^{12} + \frac{4681}{127855} a^{10} - \frac{1}{20} a^{9} + \frac{1493}{511420} a^{8} - \frac{1}{10} a^{7} + \frac{12933}{102284} a^{6} + \frac{1}{10} a^{5} + \frac{12283}{511420} a^{4} - \frac{7}{20} a^{3} - \frac{1331}{5620} a^{2} + \frac{9}{20} a - \frac{10603}{39340}$, $\frac{1}{511420} a^{13} + \frac{4681}{127855} a^{11} - \frac{1}{20} a^{10} + \frac{1493}{511420} a^{9} + \frac{12933}{102284} a^{7} + \frac{3}{10} a^{6} + \frac{12283}{511420} a^{5} + \frac{9}{20} a^{4} - \frac{1331}{5620} a^{3} + \frac{3}{20} a^{2} - \frac{10603}{39340} a + \frac{1}{10}$, $\frac{1}{1493610484303557620} a^{14} + \frac{883994076319}{1493610484303557620} a^{12} - \frac{1}{20} a^{11} - \frac{28983547153015227}{1493610484303557620} a^{10} - \frac{1}{20} a^{9} + \frac{1312340426577637}{67891385650161710} a^{8} + \frac{1}{5} a^{7} + \frac{11506205728698689}{106686463164539830} a^{6} - \frac{9}{20} a^{5} + \frac{15661756385810529}{373402621075889405} a^{4} - \frac{1}{5} a^{3} - \frac{13919580703972793}{28723278544299185} a^{2} - \frac{9}{20} a + \frac{6167635024788793}{114893114177196740}$, $\frac{1}{46301925013410286220} a^{15} + \frac{35930190571651}{46301925013410286220} a^{13} - \frac{2210638484151178337}{46301925013410286220} a^{11} - \frac{1}{20} a^{10} + \frac{635801513046333}{64757937081692708} a^{9} - \frac{1}{20} a^{8} - \frac{3164529178723376634}{11575481253352571555} a^{7} + \frac{1}{5} a^{6} - \frac{26081184156504494}{11575481253352571555} a^{5} - \frac{9}{20} a^{4} - \frac{100715605828767403}{3561686539493098940} a^{3} - \frac{1}{5} a^{2} + \frac{240397070582851486}{890421634873274735} a - \frac{9}{20}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 277838154.403 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1558:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 94 conjugacy class representatives for t16n1558 are not computed
Character table for t16n1558 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.4.659478560000.1

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 R $16$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R $16$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.8.6.3$x^{8} + 65 x^{4} + 1352$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
13.8.0.1$x^{8} + 4 x^{2} - x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
$29$29.8.7.3$x^{8} + 58$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
29.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$