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Properties

Label 16.8.15998368041...0000.3
Degree $16$
Signature $[8, 4]$
Discriminant $2^{16}\cdot 5^{12}\cdot 29^{6}\cdot 41^{2}$
Root discriminant $37.61$
Ramified primes $2, 5, 29, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2.C_2^5.C_2$ (as 16T610)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![18901, -8112, -54815, 8210, 40322, -8234, -9790, 2280, 944, 938, -793, -138, 168, -52, 13, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 13*x^14 - 52*x^13 + 168*x^12 - 138*x^11 - 793*x^10 + 938*x^9 + 944*x^8 + 2280*x^7 - 9790*x^6 - 8234*x^5 + 40322*x^4 + 8210*x^3 - 54815*x^2 - 8112*x + 18901)
 
gp: K = bnfinit(x^16 - 2*x^15 + 13*x^14 - 52*x^13 + 168*x^12 - 138*x^11 - 793*x^10 + 938*x^9 + 944*x^8 + 2280*x^7 - 9790*x^6 - 8234*x^5 + 40322*x^4 + 8210*x^3 - 54815*x^2 - 8112*x + 18901, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 13 x^{14} - 52 x^{13} + 168 x^{12} - 138 x^{11} - 793 x^{10} + 938 x^{9} + 944 x^{8} + 2280 x^{7} - 9790 x^{6} - 8234 x^{5} + 40322 x^{4} + 8210 x^{3} - 54815 x^{2} - 8112 x + 18901 \)

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:  \(15998368041616000000000000=2^{16}\cdot 5^{12}\cdot 29^{6}\cdot 41^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.61$
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:  $2, 5, 29, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{9591962703186975580226880705424061} a^{15} - \frac{1057530552837421504378563528674808}{9591962703186975580226880705424061} a^{14} + \frac{87061555010300375128005545338234}{504840142272998714748783195022319} a^{13} + \frac{3228300258054611642009488310779717}{9591962703186975580226880705424061} a^{12} - \frac{2361669709327592674924402754510158}{9591962703186975580226880705424061} a^{11} - \frac{3392219667583362240703609528257490}{9591962703186975580226880705424061} a^{10} + \frac{33740582402310352860227122928078}{121417249407430070635783300068659} a^{9} + \frac{1816106344505147826311287107162837}{9591962703186975580226880705424061} a^{8} - \frac{3994347695059071082012415491583477}{9591962703186975580226880705424061} a^{7} + \frac{3665531348319039601119456767664893}{9591962703186975580226880705424061} a^{6} - \frac{754372496717341805790986405360009}{9591962703186975580226880705424061} a^{5} - \frac{2316482920349035544103111645974077}{9591962703186975580226880705424061} a^{4} + \frac{19190387335990880374268822995627}{504840142272998714748783195022319} a^{3} + \frac{226550930615033796526968676944506}{9591962703186975580226880705424061} a^{2} - \frac{1165859281976313993585754217900921}{9591962703186975580226880705424061} a + \frac{1320665693023776572879081062288332}{9591962703186975580226880705424061}$

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:  $11$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3448695.75807 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^5.C_2$ (as 16T610):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 40 conjugacy class representatives for $C_2^2.C_2^5.C_2$
Character table for $C_2^2.C_2^5.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.58000.1, \(\Q(\zeta_{20})^+\), 4.4.725.1, 8.8.3364000000.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
5Data not computed
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$