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Properties

Label 16.8.21939893472...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{12}\cdot 5^{8}\cdot 13^{8}\cdot 41^{2}$
Root discriminant $21.57$
Ramified primes $2, 5, 13, 41$
Class number $1$
Class group Trivial
Galois group 16T1123

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-16, 112, -44, -332, 447, -46, -270, 249, -181, 40, 143, -124, -5, 39, -9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 9*x^14 + 39*x^13 - 5*x^12 - 124*x^11 + 143*x^10 + 40*x^9 - 181*x^8 + 249*x^7 - 270*x^6 - 46*x^5 + 447*x^4 - 332*x^3 - 44*x^2 + 112*x - 16)
 
gp: K = bnfinit(x^16 - 3*x^15 - 9*x^14 + 39*x^13 - 5*x^12 - 124*x^11 + 143*x^10 + 40*x^9 - 181*x^8 + 249*x^7 - 270*x^6 - 46*x^5 + 447*x^4 - 332*x^3 - 44*x^2 + 112*x - 16, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 9 x^{14} + 39 x^{13} - 5 x^{12} - 124 x^{11} + 143 x^{10} + 40 x^{9} - 181 x^{8} + 249 x^{7} - 270 x^{6} - 46 x^{5} + 447 x^{4} - 332 x^{3} - 44 x^{2} + 112 x - 16 \)

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:  \(2193989347201600000000=2^{12}\cdot 5^{8}\cdot 13^{8}\cdot 41^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.57$
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:  $2, 5, 13, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{10} a^{12} - \frac{1}{5} a^{11} - \frac{1}{10} a^{10} + \frac{1}{10} a^{9} - \frac{2}{5} a^{8} - \frac{3}{10} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{3}{10} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{2} a - \frac{2}{5}$, $\frac{1}{10} a^{13} - \frac{1}{10} a^{10} - \frac{1}{5} a^{9} - \frac{1}{10} a^{8} + \frac{1}{10} a^{7} - \frac{1}{10} a^{5} - \frac{1}{5} a^{4} - \frac{3}{10} a^{3} + \frac{3}{10} a^{2} + \frac{1}{10} a + \frac{1}{5}$, $\frac{1}{400} a^{14} + \frac{3}{80} a^{13} - \frac{7}{400} a^{12} + \frac{13}{400} a^{11} - \frac{19}{80} a^{10} + \frac{41}{200} a^{9} + \frac{159}{400} a^{8} - \frac{57}{200} a^{7} + \frac{7}{80} a^{6} + \frac{151}{400} a^{5} - \frac{33}{100} a^{4} + \frac{19}{40} a^{3} + \frac{3}{400} a^{2} - \frac{79}{200} a + \frac{17}{100}$, $\frac{1}{7450688800} a^{15} + \frac{4700911}{7450688800} a^{14} - \frac{276408487}{7450688800} a^{13} - \frac{153352859}{7450688800} a^{12} - \frac{1209164647}{7450688800} a^{11} - \frac{405102859}{3725344400} a^{10} + \frac{705417671}{7450688800} a^{9} + \frac{344030387}{745068880} a^{8} + \frac{1829436371}{7450688800} a^{7} - \frac{735480089}{7450688800} a^{6} + \frac{738422971}{1862672200} a^{5} + \frac{1638686579}{3725344400} a^{4} + \frac{2406612203}{7450688800} a^{3} - \frac{198792163}{745068880} a^{2} - \frac{83366111}{372534440} a + \frac{85644559}{232834025}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 42895.2433063 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1123:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 55 conjugacy class representatives for t16n1123 are not computed
Character table for t16n1123 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{5}, \sqrt{13})\), 8.8.1142440000.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ 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$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13Data not computed
$41$41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.2.2$x^{4} - 41 x^{2} + 20172$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$