↑

Properties

Label 16.4.89007517432...5625.1
Degree $16$
Signature $[4, 6]$
Discriminant $5^{8}\cdot 29^{10}\cdot 41\cdot 1321$
Root discriminant $36.25$
Ramified primes $5, 29, 41, 1321$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1722

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1222489, -130350, -252667, 194942, -133249, -44174, -5874, -21989, 1931, -165, -312, 390, -139, -17, 6, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 6*x^14 - 17*x^13 - 139*x^12 + 390*x^11 - 312*x^10 - 165*x^9 + 1931*x^8 - 21989*x^7 - 5874*x^6 - 44174*x^5 - 133249*x^4 + 194942*x^3 - 252667*x^2 - 130350*x + 1222489)
 
gp: K = bnfinit(x^16 - 2*x^15 + 6*x^14 - 17*x^13 - 139*x^12 + 390*x^11 - 312*x^10 - 165*x^9 + 1931*x^8 - 21989*x^7 - 5874*x^6 - 44174*x^5 - 133249*x^4 + 194942*x^3 - 252667*x^2 - 130350*x + 1222489, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 6 x^{14} - 17 x^{13} - 139 x^{12} + 390 x^{11} - 312 x^{10} - 165 x^{9} + 1931 x^{8} - 21989 x^{7} - 5874 x^{6} - 44174 x^{5} - 133249 x^{4} + 194942 x^{3} - 252667 x^{2} - 130350 x + 1222489 \)

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:  \(8900751743270385297265625=5^{8}\cdot 29^{10}\cdot 41\cdot 1321\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.25$
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:  $5, 29, 41, 1321$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7} a^{14} + \frac{2}{7} a^{13} + \frac{2}{7} a^{12} + \frac{2}{7} a^{11} - \frac{1}{7} a^{10} - \frac{2}{7} a^{9} - \frac{1}{7} a^{7} + \frac{2}{7} a^{6} - \frac{3}{7} a^{5} - \frac{2}{7} a^{4} + \frac{3}{7} a^{3} - \frac{3}{7} a^{2} - \frac{1}{7}$, $\frac{1}{21221719216746303793764389722225547128229611} a^{15} + \frac{161327772839647267364373761500523437993430}{21221719216746303793764389722225547128229611} a^{14} + \frac{3788742435537070224992541281060188782594013}{21221719216746303793764389722225547128229611} a^{13} - \frac{10417599212950650397670282312234995755327613}{21221719216746303793764389722225547128229611} a^{12} - \frac{622668997176544497783947629577969562989532}{21221719216746303793764389722225547128229611} a^{11} + \frac{2032293580168223240086533308662728614539318}{21221719216746303793764389722225547128229611} a^{10} - \frac{1150258144400513678717795845572933215862927}{3031674173820900541966341388889363875461373} a^{9} - \frac{2300627522345506903637385381413554470191654}{21221719216746303793764389722225547128229611} a^{8} + \frac{2547285948753274565661937955773536521124974}{21221719216746303793764389722225547128229611} a^{7} + \frac{7097947868508166336371211217812738925737446}{21221719216746303793764389722225547128229611} a^{6} + \frac{2391006038301111860415429188658289869312108}{21221719216746303793764389722225547128229611} a^{5} + \frac{8510087834682354583977445819464357648916306}{21221719216746303793764389722225547128229611} a^{4} + \frac{127023108223939271024501671461998888211743}{21221719216746303793764389722225547128229611} a^{3} + \frac{286182628252597268843423404908194575971062}{3031674173820900541966341388889363875461373} a^{2} + \frac{8382027319287866365864460172361834541009661}{21221719216746303793764389722225547128229611} a - \frac{137935135208982313946722384382095188993268}{3031674173820900541966341388889363875461373}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 543729.174338 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1722:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 8192
The 104 conjugacy class representatives for t16n1722 are not computed
Character table for t16n1722 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.4.15243125.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.8.4.2$x^{8} - 24389 x^{2} + 13438339$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
1321Data not computed