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Properties

Label 16.6.81968949183...2631.1
Degree $16$
Signature $[6, 5]$
Discriminant $-\,71^{3}\cdot 73^{12}$
Root discriminant $55.54$
Ramified primes $71, 73$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1251

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2116, 5474, -1475, -16095, 2139, 16319, -3952, -7391, 2682, 1504, -862, -39, 110, -21, 9, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 9*x^14 - 21*x^13 + 110*x^12 - 39*x^11 - 862*x^10 + 1504*x^9 + 2682*x^8 - 7391*x^7 - 3952*x^6 + 16319*x^5 + 2139*x^4 - 16095*x^3 - 1475*x^2 + 5474*x + 2116)
 
gp: K = bnfinit(x^16 - 7*x^15 + 9*x^14 - 21*x^13 + 110*x^12 - 39*x^11 - 862*x^10 + 1504*x^9 + 2682*x^8 - 7391*x^7 - 3952*x^6 + 16319*x^5 + 2139*x^4 - 16095*x^3 - 1475*x^2 + 5474*x + 2116, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 9 x^{14} - 21 x^{13} + 110 x^{12} - 39 x^{11} - 862 x^{10} + 1504 x^{9} + 2682 x^{8} - 7391 x^{7} - 3952 x^{6} + 16319 x^{5} + 2139 x^{4} - 16095 x^{3} - 1475 x^{2} + 5474 x + 2116 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-8196894918367374985699192631=-\,71^{3}\cdot 73^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.54$
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:  $71, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{2} a^{7} + \frac{3}{8} a^{6} + \frac{3}{8} a^{5} + \frac{3}{8} a^{4} + \frac{1}{8} a^{3} + \frac{3}{8} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{16} a^{13} + \frac{1}{8} a^{11} + \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{16} a^{8} + \frac{3}{16} a^{7} + \frac{1}{8} a^{6} - \frac{3}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{5}{16} a^{2} - \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{128} a^{14} + \frac{3}{128} a^{13} + \frac{1}{64} a^{12} + \frac{1}{8} a^{11} + \frac{1}{32} a^{10} - \frac{15}{128} a^{9} - \frac{1}{16} a^{8} - \frac{5}{128} a^{7} + \frac{3}{8} a^{6} - \frac{3}{64} a^{5} + \frac{9}{32} a^{4} + \frac{5}{128} a^{3} + \frac{25}{128} a^{2} + \frac{17}{64} a - \frac{3}{32}$, $\frac{1}{628242515557741760449024} a^{15} + \frac{73946749820961043291}{19632578611179430014032} a^{14} - \frac{7809922167999210873255}{628242515557741760449024} a^{13} - \frac{19602202997448197816091}{314121257778870880224512} a^{12} + \frac{36483237100210551655301}{157060628889435440112256} a^{11} + \frac{110107308058666955051749}{628242515557741760449024} a^{10} + \frac{111220207227516629203333}{628242515557741760449024} a^{9} + \frac{99250299885931871651347}{628242515557741760449024} a^{8} + \frac{147780362960323545220255}{628242515557741760449024} a^{7} + \frac{23070731321995969773653}{314121257778870880224512} a^{6} + \frac{57483875285563693583995}{314121257778870880224512} a^{5} + \frac{123859203347834635842969}{628242515557741760449024} a^{4} - \frac{329503961659242556685}{13657445990385690444544} a^{3} + \frac{46079875756183973100023}{628242515557741760449024} a^{2} + \frac{145113424898583566621799}{314121257778870880224512} a + \frac{129501861435339182271}{6828722995192845222272}$

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:  $10$
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:  \( 793198315.417 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1251:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1251
Character table for t16n1251 is not computed

Intermediate fields

\(\Q(\sqrt{73}) \), 4.4.389017.1, 8.6.10744730066519.1

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
Arithmetically equvalently 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 ${\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ $16$ $16$

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
$71$$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$73$73.8.6.1$x^{8} - 14527 x^{4} + 78021889$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
73.8.6.1$x^{8} - 14527 x^{4} + 78021889$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$