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Properties

Label 16.0.44133664812...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{4}\cdot 29^{8}\cdot 109^{4}$
Root discriminant $26.02$
Ramified primes $5, 29, 109$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T969

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1205, -1095, 1892, -1861, 1843, -1854, 2134, -1912, 1526, -1223, 931, -598, 312, -126, 38, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 38*x^14 - 126*x^13 + 312*x^12 - 598*x^11 + 931*x^10 - 1223*x^9 + 1526*x^8 - 1912*x^7 + 2134*x^6 - 1854*x^5 + 1843*x^4 - 1861*x^3 + 1892*x^2 - 1095*x + 1205)
 
gp: K = bnfinit(x^16 - 8*x^15 + 38*x^14 - 126*x^13 + 312*x^12 - 598*x^11 + 931*x^10 - 1223*x^9 + 1526*x^8 - 1912*x^7 + 2134*x^6 - 1854*x^5 + 1843*x^4 - 1861*x^3 + 1892*x^2 - 1095*x + 1205, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 38 x^{14} - 126 x^{13} + 312 x^{12} - 598 x^{11} + 931 x^{10} - 1223 x^{9} + 1526 x^{8} - 1912 x^{7} + 2134 x^{6} - 1854 x^{5} + 1843 x^{4} - 1861 x^{3} + 1892 x^{2} - 1095 x + 1205 \)

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:  \(44133664812763327950625=5^{4}\cdot 29^{8}\cdot 109^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.02$
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:  $5, 29, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{4282971675} a^{14} - \frac{7}{4282971675} a^{13} + \frac{34293464}{1427657225} a^{12} - \frac{617282261}{4282971675} a^{11} + \frac{611274296}{1427657225} a^{10} + \frac{772277794}{4282971675} a^{9} + \frac{575258931}{1427657225} a^{8} - \frac{398330607}{1427657225} a^{7} + \frac{458236528}{4282971675} a^{6} - \frac{172815253}{856594335} a^{5} + \frac{439211559}{1427657225} a^{4} - \frac{130207039}{1427657225} a^{3} - \frac{953444977}{4282971675} a^{2} + \frac{27903454}{57106289} a + \frac{61473239}{856594335}$, $\frac{1}{6857037651675} a^{15} + \frac{793}{6857037651675} a^{14} + \frac{83314296389}{2285679217225} a^{13} - \frac{399433453486}{6857037651675} a^{12} + \frac{259540981446}{2285679217225} a^{11} - \frac{839396128002}{2285679217225} a^{10} + \frac{2312749480843}{6857037651675} a^{9} - \frac{2011259466796}{6857037651675} a^{8} + \frac{3236066392328}{6857037651675} a^{7} - \frac{31930542533}{1371407530335} a^{6} - \frac{1713573607648}{6857037651675} a^{5} - \frac{1675963493717}{6857037651675} a^{4} + \frac{1131338773123}{6857037651675} a^{3} + \frac{96806282990}{274281506067} a^{2} - \frac{47994746506}{1371407530335} a - \frac{36600652405}{91427168689}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 128228.18223 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T969:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 44 conjugacy class representatives for t16n969
Character table for t16n969 is not computed

Intermediate fields

\(\Q(\sqrt{29}) \), 4.4.91669.1, 8.4.210080139025.1, 8.4.42016027805.1, 8.0.42016027805.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
$29$29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$109$109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$