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Properties

Label 16.4.10003544168...3125.1
Degree $16$
Signature $[4, 6]$
Discriminant $5^{12}\cdot 31^{3}\cdot 41^{3}\cdot 1259^{3}$
Root discriminant $48.70$
Ramified primes $5, 31, 41, 1259$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1871

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-181939, -241270, 1183363, -1528675, 977156, -445185, 99934, 19695, -32711, 15945, -5443, 1200, -106, -40, 21, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 21*x^14 - 40*x^13 - 106*x^12 + 1200*x^11 - 5443*x^10 + 15945*x^9 - 32711*x^8 + 19695*x^7 + 99934*x^6 - 445185*x^5 + 977156*x^4 - 1528675*x^3 + 1183363*x^2 - 241270*x - 181939)
 
gp: K = bnfinit(x^16 - 5*x^15 + 21*x^14 - 40*x^13 - 106*x^12 + 1200*x^11 - 5443*x^10 + 15945*x^9 - 32711*x^8 + 19695*x^7 + 99934*x^6 - 445185*x^5 + 977156*x^4 - 1528675*x^3 + 1183363*x^2 - 241270*x - 181939, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 21 x^{14} - 40 x^{13} - 106 x^{12} + 1200 x^{11} - 5443 x^{10} + 15945 x^{9} - 32711 x^{8} + 19695 x^{7} + 99934 x^{6} - 445185 x^{5} + 977156 x^{4} - 1528675 x^{3} + 1183363 x^{2} - 241270 x - 181939 \)

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:  \(1000354416862195134033203125=5^{12}\cdot 31^{3}\cdot 41^{3}\cdot 1259^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.70$
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:  $5, 31, 41, 1259$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{379} a^{14} + \frac{123}{379} a^{13} + \frac{70}{379} a^{12} - \frac{35}{379} a^{11} + \frac{33}{379} a^{10} - \frac{107}{379} a^{9} - \frac{31}{379} a^{8} - \frac{135}{379} a^{7} - \frac{54}{379} a^{6} + \frac{112}{379} a^{5} - \frac{102}{379} a^{4} - \frac{68}{379} a^{3} + \frac{101}{379} a^{2} - \frac{130}{379} a - \frac{56}{379}$, $\frac{1}{6064716443697143171515151849979463082021819} a^{15} + \frac{6995545752596058116768492094862971788534}{6064716443697143171515151849979463082021819} a^{14} + \frac{2035530230548973691095681686588343070491444}{6064716443697143171515151849979463082021819} a^{13} - \frac{2845900982899548525348609919586978566198148}{6064716443697143171515151849979463082021819} a^{12} + \frac{1782830500450778743999272866663825163474677}{6064716443697143171515151849979463082021819} a^{11} - \frac{2833708152447850854270152367793178034419528}{6064716443697143171515151849979463082021819} a^{10} - \frac{2226790797682279069711987840021630377712226}{6064716443697143171515151849979463082021819} a^{9} + \frac{1768430730137554875026790419331799022956705}{6064716443697143171515151849979463082021819} a^{8} - \frac{2626472694918639032317466791495469155643224}{6064716443697143171515151849979463082021819} a^{7} + \frac{554373679669778966160437451813764274029915}{6064716443697143171515151849979463082021819} a^{6} + \frac{1038614620076446807527032207382502198965749}{6064716443697143171515151849979463082021819} a^{5} + \frac{1047283496293900632819959216920277786249152}{6064716443697143171515151849979463082021819} a^{4} - \frac{391828230651293605676910967442640386140720}{6064716443697143171515151849979463082021819} a^{3} + \frac{2331800798863995550106766872233836183843026}{6064716443697143171515151849979463082021819} a^{2} - \frac{371551253814651300042348630617723889876785}{6064716443697143171515151849979463082021819} a - \frac{2076064757839780925252613322757543526641271}{6064716443697143171515151849979463082021819}$

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:  $9$
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:  \( 2749998.42027 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1871:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 73728
The 104 conjugacy class representatives for t16n1871 are not computed
Character table for t16n1871 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 8.8.1000118125.1

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 $16$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ $16$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ R $16$ R $16$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
5Data not computed
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.3.0.1$x^{3} - x + 9$$1$$3$$0$$C_3$$[\ ]^{3}$
31.3.0.1$x^{3} - x + 9$$1$$3$$0$$C_3$$[\ ]^{3}$
31.4.3.1$x^{4} + 217$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$41$41.4.3.3$x^{4} + 246$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.8.0.1$x^{8} - x + 12$$1$$8$$0$$C_8$$[\ ]^{8}$
1259Data not computed