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Properties

Label 16.4.31342639780...8813.8
Degree $16$
Signature $[4, 6]$
Discriminant $13^{11}\cdot 53^{10}$
Root discriminant $69.74$
Ramified primes $13, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4:D_4.D_4$ (as 16T681)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![909, -981, -3038, 6137, -3444, 237, 1339, 643, -3520, 774, 1667, -987, -19, 118, -11, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 11*x^14 + 118*x^13 - 19*x^12 - 987*x^11 + 1667*x^10 + 774*x^9 - 3520*x^8 + 643*x^7 + 1339*x^6 + 237*x^5 - 3444*x^4 + 6137*x^3 - 3038*x^2 - 981*x + 909)
 
gp: K = bnfinit(x^16 - 6*x^15 - 11*x^14 + 118*x^13 - 19*x^12 - 987*x^11 + 1667*x^10 + 774*x^9 - 3520*x^8 + 643*x^7 + 1339*x^6 + 237*x^5 - 3444*x^4 + 6137*x^3 - 3038*x^2 - 981*x + 909, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 11 x^{14} + 118 x^{13} - 19 x^{12} - 987 x^{11} + 1667 x^{10} + 774 x^{9} - 3520 x^{8} + 643 x^{7} + 1339 x^{6} + 237 x^{5} - 3444 x^{4} + 6137 x^{3} - 3038 x^{2} - 981 x + 909 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(313426397802392026311505288813=13^{11}\cdot 53^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.74$
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:  $13, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{39} a^{14} - \frac{16}{39} a^{13} - \frac{19}{39} a^{12} - \frac{7}{39} a^{11} + \frac{2}{13} a^{10} + \frac{4}{13} a^{9} - \frac{7}{39} a^{8} - \frac{2}{39} a^{7} + \frac{16}{39} a^{6} + \frac{16}{39} a^{4} - \frac{1}{39} a^{3} + \frac{1}{39} a^{2} + \frac{16}{39} a - \frac{4}{13}$, $\frac{1}{25253023283485619806766584743} a^{15} - \frac{59832213006489828101469550}{8417674427828539935588861581} a^{14} - \frac{5729006501780099149400836043}{25253023283485619806766584743} a^{13} + \frac{283754216464408361298344767}{1942540252575816908212814211} a^{12} - \frac{5159664147598038200820259537}{25253023283485619806766584743} a^{11} - \frac{2791059060604764356297105950}{8417674427828539935588861581} a^{10} - \frac{12570442752207668233487616562}{25253023283485619806766584743} a^{9} - \frac{184265926775943905171076255}{8417674427828539935588861581} a^{8} + \frac{6264436632387415820287536185}{25253023283485619806766584743} a^{7} - \frac{3280074718546322597960816351}{25253023283485619806766584743} a^{6} + \frac{8875696991117487766777290778}{25253023283485619806766584743} a^{5} - \frac{2892780827004403193381222253}{8417674427828539935588861581} a^{4} + \frac{79411935719031227970701274}{647513417525272302737604737} a^{3} - \frac{398850672906025022208845458}{25253023283485619806766584743} a^{2} + \frac{4207695532012186759252934269}{25253023283485619806766584743} a + \frac{3400876215383643447590124267}{8417674427828539935588861581}$

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:  $9$
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:  \( 1831702069.86 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:D_4.D_4$ (as 16T681):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 19 conjugacy class representatives for $C_4:D_4.D_4$
Character table for $C_4:D_4.D_4$

Intermediate fields

\(\Q(\sqrt{13}) \), 4.4.8957.1, 8.4.2929680361933.5

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 $16$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
$53$53.4.0.1$x^{4} - x + 18$$1$$4$$0$$C_4$$[\ ]^{4}$
53.4.3.3$x^{4} + 106$$4$$1$$3$$C_4$$[\ ]_{4}$
53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$