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Properties

Label 16.0.71338966047...7817.1
Degree $16$
Signature $[0, 8]$
Discriminant $71^{8}\cdot 73^{7}$
Root discriminant $55.06$
Ramified primes $71, 73$
Class number $7$ (GRH)
Class group $[7]$ (GRH)
Galois group $D_{16}$ (as 16T56)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![107, -29, -326, -1167, 1030, 2845, 2662, -5011, -2482, 1930, 944, -165, -27, -21, 4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 4*x^14 - 21*x^13 - 27*x^12 - 165*x^11 + 944*x^10 + 1930*x^9 - 2482*x^8 - 5011*x^7 + 2662*x^6 + 2845*x^5 + 1030*x^4 - 1167*x^3 - 326*x^2 - 29*x + 107)
 
gp: K = bnfinit(x^16 - 2*x^15 + 4*x^14 - 21*x^13 - 27*x^12 - 165*x^11 + 944*x^10 + 1930*x^9 - 2482*x^8 - 5011*x^7 + 2662*x^6 + 2845*x^5 + 1030*x^4 - 1167*x^3 - 326*x^2 - 29*x + 107, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 4 x^{14} - 21 x^{13} - 27 x^{12} - 165 x^{11} + 944 x^{10} + 1930 x^{9} - 2482 x^{8} - 5011 x^{7} + 2662 x^{6} + 2845 x^{5} + 1030 x^{4} - 1167 x^{3} - 326 x^{2} - 29 x + 107 \)

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:  \(7133896604786078388958797817=71^{8}\cdot 73^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.06$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{2}{5} a^{10} + \frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{11} - \frac{2}{5} a^{10} + \frac{1}{5} a^{9} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{16025} a^{14} - \frac{1444}{16025} a^{13} + \frac{1474}{16025} a^{12} - \frac{6222}{16025} a^{11} + \frac{166}{641} a^{10} - \frac{3924}{16025} a^{9} + \frac{5327}{16025} a^{8} + \frac{7393}{16025} a^{7} + \frac{5256}{16025} a^{6} + \frac{2658}{16025} a^{5} - \frac{3642}{16025} a^{4} - \frac{253}{3205} a^{3} + \frac{1336}{16025} a^{2} - \frac{6884}{16025} a - \frac{7656}{16025}$, $\frac{1}{81234622373830448891525} a^{15} + \frac{915439001032016644}{81234622373830448891525} a^{14} + \frac{7961857077742920760772}{81234622373830448891525} a^{13} - \frac{484764835576020954926}{16246924474766089778305} a^{12} + \frac{37638312604919922732299}{81234622373830448891525} a^{11} - \frac{7706731773215374101949}{81234622373830448891525} a^{10} + \frac{1185510987451042406337}{16246924474766089778305} a^{9} - \frac{18699982319820464835551}{81234622373830448891525} a^{8} + \frac{4573507926580991008422}{16246924474766089778305} a^{7} + \frac{35505850860515899786441}{81234622373830448891525} a^{6} - \frac{34825293376917993601288}{81234622373830448891525} a^{5} + \frac{22167310012632206963234}{81234622373830448891525} a^{4} - \frac{30902548703042641551564}{81234622373830448891525} a^{3} - \frac{32829256445567489156416}{81234622373830448891525} a^{2} - \frac{11517375215974931114763}{81234622373830448891525} a + \frac{38044319630428093152622}{81234622373830448891525}$

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

Class group and class number

$C_{7}$, which has order $7$ (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:  \( 3343233.44259 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{16}$ (as 16T56):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $D_{16}$
Character table for $D_{16}$

Intermediate fields

\(\Q(\sqrt{-71}) \), 4.0.367993.1, 8.0.9885575907577.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $16$ $16$ $16$

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$71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$73$73.2.1.2$x^{2} + 365$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.4.2.1$x^{4} + 1533 x^{2} + 644809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
73.4.2.1$x^{4} + 1533 x^{2} + 644809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
73.4.2.1$x^{4} + 1533 x^{2} + 644809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$