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Properties

Label 16.0.11020639142...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{12}\cdot 101^{5}$
Root discriminant $56.58$
Ramified primes $2, 5, 101$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 16T1354

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25757525, 0, 18361800, 0, 4394510, 0, 317140, 0, -13764, 0, -564, 0, 161, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 + 161*x^12 - 564*x^10 - 13764*x^8 + 317140*x^6 + 4394510*x^4 + 18361800*x^2 + 25757525)
 
gp: K = bnfinit(x^16 - 4*x^14 + 161*x^12 - 564*x^10 - 13764*x^8 + 317140*x^6 + 4394510*x^4 + 18361800*x^2 + 25757525, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{14} + 161 x^{12} - 564 x^{10} - 13764 x^{8} + 317140 x^{6} + 4394510 x^{4} + 18361800 x^{2} + 25757525 \)

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:  \(11020639142936576000000000000=2^{32}\cdot 5^{12}\cdot 101^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.58$
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:  $2, 5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5555} a^{12} - \frac{1014}{5555} a^{10} + \frac{1171}{5555} a^{8} + \frac{1961}{5555} a^{6} + \frac{376}{5555} a^{4} + \frac{2}{11} a^{2} + \frac{1}{11}$, $\frac{1}{27775} a^{13} - \frac{6569}{27775} a^{11} - \frac{4384}{27775} a^{9} + \frac{7516}{27775} a^{7} - \frac{10734}{27775} a^{5} + \frac{24}{55} a^{3} + \frac{1}{55} a$, $\frac{1}{6870728404386874492025} a^{14} - \frac{401264196023915459}{6870728404386874492025} a^{12} - \frac{2406316721571159778449}{6870728404386874492025} a^{10} - \frac{1614540601072697854849}{6870728404386874492025} a^{8} + \frac{1690695531541203308601}{6870728404386874492025} a^{6} - \frac{6382407932272547059}{13605402780964107905} a^{4} - \frac{3111520908834236044}{13605402780964107905} a^{2} - \frac{2753632497912811}{26941391645473481}$, $\frac{1}{6870728404386874492025} a^{15} + \frac{93477723283870283}{6870728404386874492025} a^{13} + \frac{110404728625715470398}{624611673126079499275} a^{11} + \frac{3087239229068843944248}{6870728404386874492025} a^{9} - \frac{1461552607328353546552}{6870728404386874492025} a^{7} - \frac{16464825378817167598}{68027013904820539525} a^{5} - \frac{4843117626411486141}{13605402780964107905} a^{3} - \frac{8869727644932513}{134706958227367405} a$

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

Class group and class number

$C_{4}$, 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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{34729589236196}{9943167010690122275} a^{14} + \frac{286628080442664}{9943167010690122275} a^{12} - \frac{6757871836852196}{9943167010690122275} a^{10} + \frac{47490828620534204}{9943167010690122275} a^{8} + \frac{292715274730160604}{9943167010690122275} a^{6} - \frac{24555394376862891}{19689439625128955} a^{4} - \frac{196512854843273836}{19689439625128955} a^{2} - \frac{760809260663693}{38988989356691} \) (order $10$)
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:  \( 46569140.1677 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1354:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 59 conjugacy class representatives for t16n1354 are not computed
Character table for t16n1354 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.404000000.2

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 $16$ R $16$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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$5.4.3.2$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}$
101Data not computed