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Properties

Label 16.4.84546860538...1104.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{24}\cdot 41^{6}\cdot 103^{2}$
Root discriminant $20.32$
Ramified primes $2, 41, 103$
Class number $1$
Class group Trivial
Galois group 16T1429

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 34, 61, -62, -33, 174, -118, 48, 29, -58, 44, -60, 48, -28, 12, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 12*x^14 - 28*x^13 + 48*x^12 - 60*x^11 + 44*x^10 - 58*x^9 + 29*x^8 + 48*x^7 - 118*x^6 + 174*x^5 - 33*x^4 - 62*x^3 + 61*x^2 + 34*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 12*x^14 - 28*x^13 + 48*x^12 - 60*x^11 + 44*x^10 - 58*x^9 + 29*x^8 + 48*x^7 - 118*x^6 + 174*x^5 - 33*x^4 - 62*x^3 + 61*x^2 + 34*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 12 x^{14} - 28 x^{13} + 48 x^{12} - 60 x^{11} + 44 x^{10} - 58 x^{9} + 29 x^{8} + 48 x^{7} - 118 x^{6} + 174 x^{5} - 33 x^{4} - 62 x^{3} + 61 x^{2} + 34 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(845468605385858351104=2^{24}\cdot 41^{6}\cdot 103^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.32$
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:  $2, 41, 103$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{276612073806577061} a^{15} - \frac{71865498128768070}{276612073806577061} a^{14} + \frac{45036386821072251}{276612073806577061} a^{13} + \frac{124257871216710638}{276612073806577061} a^{12} + \frac{55746909766587092}{276612073806577061} a^{11} - \frac{20837289313662365}{276612073806577061} a^{10} + \frac{109579829887863599}{276612073806577061} a^{9} + \frac{37933131760850279}{276612073806577061} a^{8} - \frac{123164794305106984}{276612073806577061} a^{7} + \frac{36654312634872231}{276612073806577061} a^{6} + \frac{115304984310528388}{276612073806577061} a^{5} - \frac{18533762650855764}{276612073806577061} a^{4} + \frac{114617633862785756}{276612073806577061} a^{3} - \frac{58313798114587216}{276612073806577061} a^{2} - \frac{130707206185995025}{276612073806577061} a - \frac{121263821889172357}{276612073806577061}$

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:  $9$
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:  \( 11772.8783317 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1429:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 77 conjugacy class representatives for t16n1429 are not computed
Character table for t16n1429 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.2624.1, 8.8.282300416.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$41$41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.3.4$x^{4} + 8856$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.4$x^{4} + 8856$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
$103$$\Q_{103}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{103}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{103}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{103}$$x + 2$$1$$1$$0$Trivial$[\ ]$
103.2.1.1$x^{2} - 103$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.1$x^{2} - 103$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
103.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$