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Properties

Label 16.8.35332779248...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{8}\cdot 5^{6}\cdot 13^{8}\cdot 101^{8}$
Root discriminant $93.70$
Ramified primes $2, 5, 13, 101$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1275

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![400, 0, -21340, 0, -8239, 0, 16878, 0, 23711, 0, -5144, 0, 127, 0, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 6*x^14 + 127*x^12 - 5144*x^10 + 23711*x^8 + 16878*x^6 - 8239*x^4 - 21340*x^2 + 400)
 
gp: K = bnfinit(x^16 + 6*x^14 + 127*x^12 - 5144*x^10 + 23711*x^8 + 16878*x^6 - 8239*x^4 - 21340*x^2 + 400, 1)
 

Normalized defining polynomial

\( x^{16} + 6 x^{14} + 127 x^{12} - 5144 x^{10} + 23711 x^{8} + 16878 x^{6} - 8239 x^{4} - 21340 x^{2} + 400 \)

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:  \(35332779248867141512750084000000=2^{8}\cdot 5^{6}\cdot 13^{8}\cdot 101^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $93.70$
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:  $2, 5, 13, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{3}$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{4}$, $\frac{1}{16} a^{9} - \frac{1}{16} a^{7} - \frac{1}{16} a^{5} + \frac{1}{16} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{80} a^{10} + \frac{3}{80} a^{8} + \frac{1}{80} a^{6} + \frac{17}{80} a^{4} - \frac{1}{4} a^{3} - \frac{11}{40} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{160} a^{11} - \frac{1}{160} a^{10} + \frac{3}{160} a^{9} + \frac{7}{160} a^{8} + \frac{1}{160} a^{7} + \frac{9}{160} a^{6} - \frac{3}{160} a^{5} + \frac{33}{160} a^{4} + \frac{19}{80} a^{3} - \frac{3}{10} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{13120} a^{12} - \frac{19}{3280} a^{10} - \frac{373}{6560} a^{8} + \frac{43}{820} a^{6} - \frac{11}{2624} a^{4} - \frac{1}{4} a^{3} + \frac{987}{3280} a^{2} - \frac{1}{4} a + \frac{79}{164}$, $\frac{1}{13120} a^{13} + \frac{3}{6560} a^{11} - \frac{1}{160} a^{10} + \frac{1}{41} a^{9} + \frac{7}{160} a^{8} - \frac{5}{1312} a^{7} + \frac{9}{160} a^{6} - \frac{1121}{13120} a^{5} - \frac{7}{160} a^{4} + \frac{331}{3280} a^{3} - \frac{1}{20} a^{2} + \frac{19}{82} a$, $\frac{1}{19759612160} a^{14} - \frac{549223}{19759612160} a^{12} - \frac{8110603}{9879806080} a^{10} + \frac{10824941}{210208640} a^{8} + \frac{320469617}{19759612160} a^{6} + \frac{4718989009}{19759612160} a^{4} - \frac{1748564103}{4939903040} a^{2} - \frac{98617895}{246995152}$, $\frac{1}{98798060800} a^{15} - \frac{3561359}{98798060800} a^{13} - \frac{78895799}{49399030400} a^{11} - \frac{1}{160} a^{10} + \frac{9767489}{1051043200} a^{9} + \frac{7}{160} a^{8} + \frac{4052506121}{98798060800} a^{7} + \frac{9}{160} a^{6} - \frac{7588598687}{98798060800} a^{5} - \frac{7}{160} a^{4} - \frac{1140112631}{24699515200} a^{3} - \frac{1}{20} a^{2} + \frac{589655181}{1234975760} a$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 22552817244.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1275:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1275
Character table for t16n1275 is not computed

Intermediate fields

\(\Q(\sqrt{101}) \), 4.4.663065.1, 8.4.371508639120125.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$101$101.8.4.1$x^{8} + 244824 x^{4} - 1030301 x^{2} + 14984697744$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
101.8.4.1$x^{8} + 244824 x^{4} - 1030301 x^{2} + 14984697744$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$