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Properties

Label 16.0.39770658357...1264.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 3^{14}\cdot 11^{2}$
Root discriminant $16.79$
Ramified primes $2, 3, 11$
Class number $1$
Class group Trivial
Galois group $C_2^3.C_2^4.C_2$ (as 16T608)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13, 56, 2, -192, 14, 340, -176, -276, 337, -24, -188, 140, -16, -36, 26, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 26*x^14 - 36*x^13 - 16*x^12 + 140*x^11 - 188*x^10 - 24*x^9 + 337*x^8 - 276*x^7 - 176*x^6 + 340*x^5 + 14*x^4 - 192*x^3 + 2*x^2 + 56*x + 13)
 
gp: K = bnfinit(x^16 - 8*x^15 + 26*x^14 - 36*x^13 - 16*x^12 + 140*x^11 - 188*x^10 - 24*x^9 + 337*x^8 - 276*x^7 - 176*x^6 + 340*x^5 + 14*x^4 - 192*x^3 + 2*x^2 + 56*x + 13, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 26 x^{14} - 36 x^{13} - 16 x^{12} + 140 x^{11} - 188 x^{10} - 24 x^{9} + 337 x^{8} - 276 x^{7} - 176 x^{6} + 340 x^{5} + 14 x^{4} - 192 x^{3} + 2 x^{2} + 56 x + 13 \)

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:  \(39770658357865611264=2^{36}\cdot 3^{14}\cdot 11^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.79$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13} a^{14} + \frac{5}{13} a^{13} - \frac{4}{13} a^{12} - \frac{4}{13} a^{11} - \frac{6}{13} a^{8} + \frac{2}{13} a^{7} - \frac{3}{13} a^{6} + \frac{2}{13} a^{5} + \frac{5}{13} a^{4} - \frac{6}{13} a^{3} - \frac{6}{13} a^{2} + \frac{1}{13} a$, $\frac{1}{77160083} a^{15} - \frac{1771901}{77160083} a^{14} - \frac{36669433}{77160083} a^{13} - \frac{15959170}{77160083} a^{12} - \frac{23926333}{77160083} a^{11} + \frac{2790618}{5935391} a^{10} + \frac{271334}{580151} a^{9} - \frac{26687272}{77160083} a^{8} - \frac{32235127}{77160083} a^{7} + \frac{37983485}{77160083} a^{6} + \frac{15341566}{77160083} a^{5} + \frac{14415404}{77160083} a^{4} - \frac{98834}{4061057} a^{3} - \frac{38012314}{77160083} a^{2} + \frac{1478506}{7014553} a - \frac{1983294}{5935391}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{42724590}{77160083} a^{15} - \frac{354081765}{77160083} a^{14} + \frac{93458460}{5935391} a^{13} - \frac{1901260190}{77160083} a^{12} - \frac{109739710}{77160083} a^{11} + \frac{463319096}{5935391} a^{10} - \frac{74493066}{580151} a^{9} + \frac{2060298950}{77160083} a^{8} + \frac{13809832898}{77160083} a^{7} - \frac{16275327468}{77160083} a^{6} - \frac{2270264208}{77160083} a^{5} + \frac{15394328289}{77160083} a^{4} - \frac{243082118}{4061057} a^{3} - \frac{6671694319}{77160083} a^{2} + \frac{227232424}{7014553} a + \frac{118241660}{5935391} \) (order $12$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 12278.7384077 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T608):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), 4.2.6912.1 x2, 4.0.1728.1 x2, \(\Q(\zeta_{12})\), 8.0.47775744.1

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
3Data not computed
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$