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Properties

Label 16.8.20492018426...4768.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 41^{6}\cdot 137^{3}$
Root discriminant $28.64$
Ramified primes $2, 41, 137$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1722

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-73, 538, -789, -526, 701, -226, 336, -46, 162, -332, -261, 146, 69, 4, -13, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 13*x^14 + 4*x^13 + 69*x^12 + 146*x^11 - 261*x^10 - 332*x^9 + 162*x^8 - 46*x^7 + 336*x^6 - 226*x^5 + 701*x^4 - 526*x^3 - 789*x^2 + 538*x - 73)
 
gp: K = bnfinit(x^16 - 2*x^15 - 13*x^14 + 4*x^13 + 69*x^12 + 146*x^11 - 261*x^10 - 332*x^9 + 162*x^8 - 46*x^7 + 336*x^6 - 226*x^5 + 701*x^4 - 526*x^3 - 789*x^2 + 538*x - 73, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 13 x^{14} + 4 x^{13} + 69 x^{12} + 146 x^{11} - 261 x^{10} - 332 x^{9} + 162 x^{8} - 46 x^{7} + 336 x^{6} - 226 x^{5} + 701 x^{4} - 526 x^{3} - 789 x^{2} + 538 x - 73 \)

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:  \(204920184264750968864768=2^{24}\cdot 41^{6}\cdot 137^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.64$
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:  $2, 41, 137$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{125871296763087276395866909} a^{15} - \frac{31160939651868606848843438}{125871296763087276395866909} a^{14} + \frac{19947370286776206435586903}{125871296763087276395866909} a^{13} - \frac{18595690957353388354551472}{125871296763087276395866909} a^{12} + \frac{47645241269152675878585777}{125871296763087276395866909} a^{11} - \frac{22051238026296927921852459}{125871296763087276395866909} a^{10} - \frac{22717207482335667999810526}{125871296763087276395866909} a^{9} - \frac{48680597206448283523278957}{125871296763087276395866909} a^{8} - \frac{33033111371450250168767836}{125871296763087276395866909} a^{7} + \frac{49647024990875375003155503}{125871296763087276395866909} a^{6} + \frac{50428476449500441246577066}{125871296763087276395866909} a^{5} - \frac{6414463688960313380224716}{125871296763087276395866909} a^{4} + \frac{50877950904842071594566886}{125871296763087276395866909} a^{3} + \frac{31313770011723448336462846}{125871296763087276395866909} a^{2} + \frac{33872680349112771327200445}{125871296763087276395866909} a + \frac{19181365381496335347437460}{125871296763087276395866909}$

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:  \( 348629.168814 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1722:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 8192
The 104 conjugacy class representatives for t16n1722 are not computed
Character table for t16n1722 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 $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$41$41.8.0.1$x^{8} - x + 12$$1$$8$$0$$C_8$$[\ ]^{8}$
41.8.6.2$x^{8} + 943 x^{4} + 242064$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
137Data not computed