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Properties

Label 16.0.27851617895...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{12}\cdot 11^{4}\cdot 29^{4}\cdot 41^{2}$
Root discriminant $44.96$
Ramified primes $2, 5, 11, 29, 41$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1086

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![24611521, 0, -22264968, 0, 3095180, 0, 1493272, 0, 93094, 0, -5272, 0, -500, 0, 8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 8*x^14 - 500*x^12 - 5272*x^10 + 93094*x^8 + 1493272*x^6 + 3095180*x^4 - 22264968*x^2 + 24611521)
 
gp: K = bnfinit(x^16 + 8*x^14 - 500*x^12 - 5272*x^10 + 93094*x^8 + 1493272*x^6 + 3095180*x^4 - 22264968*x^2 + 24611521, 1)
 

Normalized defining polynomial

\( x^{16} + 8 x^{14} - 500 x^{12} - 5272 x^{10} + 93094 x^{8} + 1493272 x^{6} + 3095180 x^{4} - 22264968 x^{2} + 24611521 \)

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:  \(278516178950416000000000000=2^{16}\cdot 5^{12}\cdot 11^{4}\cdot 29^{4}\cdot 41^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.96$
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:  $2, 5, 11, 29, 41$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4}$, $\frac{1}{4} a^{5} - \frac{1}{4} a$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{8} a^{2} + \frac{1}{8}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{8} a^{3} + \frac{1}{8} a$, $\frac{1}{32} a^{8} + \frac{1}{16} a^{4} + \frac{13}{32}$, $\frac{1}{32} a^{9} + \frac{1}{16} a^{5} + \frac{13}{32} a$, $\frac{1}{704} a^{10} - \frac{3}{704} a^{8} - \frac{19}{352} a^{6} - \frac{7}{352} a^{4} - \frac{43}{704} a^{2} - \frac{29}{64}$, $\frac{1}{704} a^{11} - \frac{3}{704} a^{9} - \frac{19}{352} a^{7} - \frac{7}{352} a^{5} - \frac{43}{704} a^{3} - \frac{29}{64} a$, $\frac{1}{15488} a^{12} + \frac{1}{1936} a^{10} + \frac{105}{15488} a^{8} - \frac{27}{968} a^{6} - \frac{197}{15488} a^{4} + \frac{7}{176} a^{2} + \frac{51}{128}$, $\frac{1}{15488} a^{13} + \frac{1}{1936} a^{11} + \frac{105}{15488} a^{9} - \frac{27}{968} a^{7} - \frac{197}{15488} a^{5} + \frac{7}{176} a^{3} + \frac{51}{128} a$, $\frac{1}{212897443227952384} a^{14} - \frac{6396755274891}{212897443227952384} a^{12} + \frac{55476400907065}{212897443227952384} a^{10} - \frac{941529959629603}{212897443227952384} a^{8} + \frac{4314959168089307}{212897443227952384} a^{6} - \frac{1189751762424571}{19354313020722944} a^{4} + \frac{309489467796099}{1759483001883904} a^{2} - \frac{378207820059}{3901292687104}$, $\frac{1}{212897443227952384} a^{15} - \frac{6396755274891}{212897443227952384} a^{13} + \frac{55476400907065}{212897443227952384} a^{11} - \frac{941529959629603}{212897443227952384} a^{9} + \frac{4314959168089307}{212897443227952384} a^{7} - \frac{1189751762424571}{19354313020722944} a^{5} + \frac{309489467796099}{1759483001883904} a^{3} - \frac{378207820059}{3901292687104} a$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{336165195}{1298155141633856} a^{14} + \frac{4369160349}{2596310283267712} a^{12} - \frac{21852329857}{162269392704232} a^{10} - \frac{2724457308305}{2596310283267712} a^{8} + \frac{32754831138659}{1298155141633856} a^{6} + \frac{74689190258889}{236028207569792} a^{4} + \frac{2767285589229}{5364277444768} a^{2} - \frac{3749078526325}{1950646343552} \) (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:  \( 4080033.34438 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1086:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 97 conjugacy class representatives for t16n1086 are not computed
Character table for t16n1086 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, \(\Q(\zeta_{5})\), 4.0.3625.1, 8.0.13140625.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.8.3$x^{8} + 2 x^{7} + 2 x^{6} + 16$$2$$4$$8$$C_2^3: C_4$$[2, 2, 2]^{4}$
2.8.8.3$x^{8} + 2 x^{7} + 2 x^{6} + 16$$2$$4$$8$$C_2^3: C_4$$[2, 2, 2]^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$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}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
41Data not computed