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Properties

Label 16.0.20723941878...0625.6
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 13^{8}\cdot 101^{4}$
Root discriminant $38.22$
Ramified primes $5, 13, 101$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^4:C_2^2.C_2$ (as 16T317)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5281, 36288, 109982, 187062, 188102, 104525, 20608, -7864, -3755, 154, 328, -140, -28, 13, 7, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 7*x^14 + 13*x^13 - 28*x^12 - 140*x^11 + 328*x^10 + 154*x^9 - 3755*x^8 - 7864*x^7 + 20608*x^6 + 104525*x^5 + 188102*x^4 + 187062*x^3 + 109982*x^2 + 36288*x + 5281)
 
gp: K = bnfinit(x^16 - 3*x^15 + 7*x^14 + 13*x^13 - 28*x^12 - 140*x^11 + 328*x^10 + 154*x^9 - 3755*x^8 - 7864*x^7 + 20608*x^6 + 104525*x^5 + 188102*x^4 + 187062*x^3 + 109982*x^2 + 36288*x + 5281, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 7 x^{14} + 13 x^{13} - 28 x^{12} - 140 x^{11} + 328 x^{10} + 154 x^{9} - 3755 x^{8} - 7864 x^{7} + 20608 x^{6} + 104525 x^{5} + 188102 x^{4} + 187062 x^{3} + 109982 x^{2} + 36288 x + 5281 \)

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:  \(20723941878730254150390625=5^{12}\cdot 13^{8}\cdot 101^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.22$
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:  $5, 13, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{31} a^{14} - \frac{14}{31} a^{13} + \frac{14}{31} a^{12} - \frac{5}{31} a^{11} + \frac{15}{31} a^{10} - \frac{4}{31} a^{9} - \frac{4}{31} a^{8} + \frac{11}{31} a^{7} - \frac{2}{31} a^{6} - \frac{4}{31} a^{5} - \frac{10}{31} a^{4} + \frac{9}{31} a^{3} + \frac{1}{31} a^{2} + \frac{7}{31} a - \frac{13}{31}$, $\frac{1}{79644628792077303036733106602661} a^{15} + \frac{1270067775192385911337808591580}{79644628792077303036733106602661} a^{14} + \frac{19681502718697579268278183223764}{79644628792077303036733106602661} a^{13} - \frac{15921635111546929059003202409105}{79644628792077303036733106602661} a^{12} - \frac{37998397829770394826021410538762}{79644628792077303036733106602661} a^{11} + \frac{36916728279098275852497728078115}{79644628792077303036733106602661} a^{10} - \frac{21437360474335417421295623735222}{79644628792077303036733106602661} a^{9} + \frac{12028842586782507431354153034000}{79644628792077303036733106602661} a^{8} + \frac{24230039829116827085672607329533}{79644628792077303036733106602661} a^{7} + \frac{24157668549665245550486399073196}{79644628792077303036733106602661} a^{6} + \frac{18035566000616715892138435055155}{79644628792077303036733106602661} a^{5} + \frac{39682978420177712067822187935005}{79644628792077303036733106602661} a^{4} - \frac{32276421573800012593241281943138}{79644628792077303036733106602661} a^{3} - \frac{210385764916848961457000376506}{79644628792077303036733106602661} a^{2} - \frac{17583604126391284456250808327724}{79644628792077303036733106602661} a - \frac{19999474945355853990878159759649}{79644628792077303036733106602661}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{115764245889213379625044588674}{79644628792077303036733106602661} a^{15} + \frac{518052317477584858884914437464}{79644628792077303036733106602661} a^{14} - \frac{1538535597844523381243449926538}{79644628792077303036733106602661} a^{13} + \frac{552747219579813896965570962676}{79644628792077303036733106602661} a^{12} + \frac{3141149980897545187089654938900}{79644628792077303036733106602661} a^{11} + \frac{10699731848551427490046840609559}{79644628792077303036733106602661} a^{10} - \frac{54438368029964617549261197916518}{79644628792077303036733106602661} a^{9} + \frac{60729804279186129222630076375630}{79644628792077303036733106602661} a^{8} + \frac{366290156530421313529886147955812}{79644628792077303036733106602661} a^{7} + \frac{325923531602794301597162343931913}{79644628792077303036733106602661} a^{6} - \frac{2950165658294442958741244254792601}{79644628792077303036733106602661} a^{5} - \frac{7685673652663423988976979327870560}{79644628792077303036733106602661} a^{4} - \frac{9420892754402244508247732252185881}{79644628792077303036733106602661} a^{3} - \frac{6740235884903272862806530777822544}{79644628792077303036733106602661} a^{2} - \frac{2990187221457856929909114269922678}{79644628792077303036733106602661} a - \frac{669762634775413466625752360550348}{79644628792077303036733106602661} \) (order $10$)
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:  \( 747944.552124 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:C_2^2.C_2$ (as 16T317):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $C_2^4:C_2^2.C_2$
Character table for $C_2^4:C_2^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{65}) \), \(\Q(\zeta_{5})\), 4.0.21125.1, \(\Q(\sqrt{5}, \sqrt{13})\), 8.0.446265625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
101Data not computed