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Properties

Label 16.0.21542958950...1664.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 73^{6}\cdot 937^{4}$
Root discriminant $383.12$
Ramified primes $2, 3, 73, 937$
Class number $2146680000$ (GRH)
Class group $[2, 2, 2, 2, 2, 67083750]$ (GRH)
Galois group 16T1191

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4107750751677169, 0, 1157359870269768, 0, 97478917371936, 0, 2556746204384, 0, 31228142425, 0, 202880976, 0, 723214, 0, 1336, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 1336*x^14 + 723214*x^12 + 202880976*x^10 + 31228142425*x^8 + 2556746204384*x^6 + 97478917371936*x^4 + 1157359870269768*x^2 + 4107750751677169)
 
gp: K = bnfinit(x^16 + 1336*x^14 + 723214*x^12 + 202880976*x^10 + 31228142425*x^8 + 2556746204384*x^6 + 97478917371936*x^4 + 1157359870269768*x^2 + 4107750751677169, 1)
 

Normalized defining polynomial

\( x^{16} + 1336 x^{14} + 723214 x^{12} + 202880976 x^{10} + 31228142425 x^{8} + 2556746204384 x^{6} + 97478917371936 x^{4} + 1157359870269768 x^{2} + 4107750751677169 \)

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:  \(215429589500286246610837301474983006961664=2^{48}\cdot 3^{8}\cdot 73^{6}\cdot 937^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $383.12$
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:  $2, 3, 73, 937$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{937} a^{10} + \frac{399}{937} a^{8} - \frac{150}{937} a^{6} - \frac{138}{937} a^{4} + \frac{384}{937} a^{2}$, $\frac{1}{937} a^{11} + \frac{399}{937} a^{9} - \frac{150}{937} a^{7} - \frac{138}{937} a^{5} + \frac{384}{937} a^{3}$, $\frac{1}{64091737} a^{12} + \frac{1336}{64091737} a^{10} + \frac{723214}{64091737} a^{8} + \frac{10605765}{64091737} a^{6} + \frac{15466506}{64091737} a^{4} - \frac{20}{937} a^{2}$, $\frac{1}{64091737} a^{13} + \frac{1336}{64091737} a^{11} + \frac{723214}{64091737} a^{9} + \frac{10605765}{64091737} a^{7} + \frac{15466506}{64091737} a^{5} - \frac{20}{937} a^{3}$, $\frac{1}{77763187430904769466652694949312300875367} a^{14} + \frac{392234003233877632584865347351991}{77763187430904769466652694949312300875367} a^{12} + \frac{23707667907810780322999920713362948567}{77763187430904769466652694949312300875367} a^{10} - \frac{12751089094940710689037420418924439105318}{77763187430904769466652694949312300875367} a^{8} - \frac{21926292681204710686814392685847042808295}{77763187430904769466652694949312300875367} a^{6} + \frac{7706184873460900658172917367384487226}{82991662146109679260034893222318357391} a^{4} - \frac{48399741479450491011122178928509}{1213310655489096347422331445772991} a^{2} - \frac{53649703262230082639546094878}{1294888639796260776331196847143}$, $\frac{1}{77763187430904769466652694949312300875367} a^{15} + \frac{392234003233877632584865347351991}{77763187430904769466652694949312300875367} a^{13} + \frac{23707667907810780322999920713362948567}{77763187430904769466652694949312300875367} a^{11} - \frac{12751089094940710689037420418924439105318}{77763187430904769466652694949312300875367} a^{9} - \frac{21926292681204710686814392685847042808295}{77763187430904769466652694949312300875367} a^{7} + \frac{7706184873460900658172917367384487226}{82991662146109679260034893222318357391} a^{5} - \frac{48399741479450491011122178928509}{1213310655489096347422331445772991} a^{3} - \frac{53649703262230082639546094878}{1294888639796260776331196847143} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{67083750}$, which has order $2146680000$ (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:  \( 52830.2866296 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1191:

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 t16n1191 are not computed
Character table for t16n1191 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 4.4.10512.1, 8.8.28288548864.4

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
3Data not computed
$73$73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.8.6.1$x^{8} - 14527 x^{4} + 78021889$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
937Data not computed