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Properties

Label 16.4.31342639780...813.11
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![29237, 153283, 26000, -185692, 85563, 127518, -185880, 140155, -64519, 13691, 1387, -1210, -17, 109, -15, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 15*x^14 + 109*x^13 - 17*x^12 - 1210*x^11 + 1387*x^10 + 13691*x^9 - 64519*x^8 + 140155*x^7 - 185880*x^6 + 127518*x^5 + 85563*x^4 - 185692*x^3 + 26000*x^2 + 153283*x + 29237)
 
gp: K = bnfinit(x^16 - 4*x^15 - 15*x^14 + 109*x^13 - 17*x^12 - 1210*x^11 + 1387*x^10 + 13691*x^9 - 64519*x^8 + 140155*x^7 - 185880*x^6 + 127518*x^5 + 85563*x^4 - 185692*x^3 + 26000*x^2 + 153283*x + 29237, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 15 x^{14} + 109 x^{13} - 17 x^{12} - 1210 x^{11} + 1387 x^{10} + 13691 x^{9} - 64519 x^{8} + 140155 x^{7} - 185880 x^{6} + 127518 x^{5} + 85563 x^{4} - 185692 x^{3} + 26000 x^{2} + 153283 x + 29237 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13} a^{12} - \frac{3}{13} a^{11} - \frac{2}{13} a^{10} - \frac{6}{13} a^{9} - \frac{3}{13} a^{8} + \frac{4}{13} a^{7} + \frac{4}{13} a^{6} + \frac{5}{13} a^{5} + \frac{4}{13} a^{4} - \frac{5}{13} a^{3} + \frac{1}{13} a^{2}$, $\frac{1}{13} a^{13} + \frac{2}{13} a^{11} + \frac{1}{13} a^{10} + \frac{5}{13} a^{9} - \frac{5}{13} a^{8} + \frac{3}{13} a^{7} + \frac{4}{13} a^{6} + \frac{6}{13} a^{5} - \frac{6}{13} a^{4} - \frac{1}{13} a^{3} + \frac{3}{13} a^{2}$, $\frac{1}{14986023} a^{14} - \frac{210829}{14986023} a^{13} + \frac{86873}{14986023} a^{12} - \frac{3987364}{14986023} a^{11} - \frac{180538}{1152771} a^{10} + \frac{2391741}{4995341} a^{9} + \frac{5786822}{14986023} a^{8} + \frac{569921}{1152771} a^{7} - \frac{1572495}{4995341} a^{6} - \frac{541219}{14986023} a^{5} + \frac{1140443}{4995341} a^{4} - \frac{6750563}{14986023} a^{3} - \frac{548979}{4995341} a^{2} + \frac{385984}{1152771} a + \frac{243914}{1152771}$, $\frac{1}{3035335967459282541470716423431633} a^{15} + \frac{64052295043658158831753049}{3035335967459282541470716423431633} a^{14} + \frac{982380782829910806114699360104}{233487382112252503190055109494741} a^{13} + \frac{28094010873259218675671171495663}{3035335967459282541470716423431633} a^{12} - \frac{1229144002370949817821437939371762}{3035335967459282541470716423431633} a^{11} - \frac{5990469048058137561137810262831}{1011778655819760847156905474477211} a^{10} + \frac{55285260741140485842751298423963}{3035335967459282541470716423431633} a^{9} + \frac{105755057538521263681362707458070}{3035335967459282541470716423431633} a^{8} + \frac{433311382849399867484234619765184}{1011778655819760847156905474477211} a^{7} + \frac{1376936887778410866734606182981550}{3035335967459282541470716423431633} a^{6} + \frac{435028032818317232661519763161219}{1011778655819760847156905474477211} a^{5} + \frac{1359307731759979321135426307886817}{3035335967459282541470716423431633} a^{4} - \frac{176743505126920559790990141435732}{1011778655819760847156905474477211} a^{3} + \frac{287397723573849078659537934279589}{3035335967459282541470716423431633} a^{2} - \frac{83799539881143806072750790183718}{233487382112252503190055109494741} a - \frac{19094278736309896762712569876612}{77829127370750834396685036498247}$

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:  \( 317898884.513 \) (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, 8.4.2929680361933.3

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 $16$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ 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$$[\ ]_{4}$
$53$53.4.3.4$x^{4} + 424$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.0.1$x^{4} - x + 18$$1$$4$$0$$C_4$$[\ ]^{4}$
53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$