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Properties

Label 16.0.64790048254...0625.6
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 101^{6}$
Root discriminant $23.08$
Ramified primes $5, 101$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1263

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![42191, -30745, -66171, 39985, 54852, -25420, -27618, 9110, 9205, -2005, -2007, 245, 287, -15, -24, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 24*x^14 - 15*x^13 + 287*x^12 + 245*x^11 - 2007*x^10 - 2005*x^9 + 9205*x^8 + 9110*x^7 - 27618*x^6 - 25420*x^5 + 54852*x^4 + 39985*x^3 - 66171*x^2 - 30745*x + 42191)
 
gp: K = bnfinit(x^16 - 24*x^14 - 15*x^13 + 287*x^12 + 245*x^11 - 2007*x^10 - 2005*x^9 + 9205*x^8 + 9110*x^7 - 27618*x^6 - 25420*x^5 + 54852*x^4 + 39985*x^3 - 66171*x^2 - 30745*x + 42191, 1)
 

Normalized defining polynomial

\( x^{16} - 24 x^{14} - 15 x^{13} + 287 x^{12} + 245 x^{11} - 2007 x^{10} - 2005 x^{9} + 9205 x^{8} + 9110 x^{7} - 27618 x^{6} - 25420 x^{5} + 54852 x^{4} + 39985 x^{3} - 66171 x^{2} - 30745 x + 42191 \)

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:  \(6479004825445556640625=5^{14}\cdot 101^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.08$
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:  $5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{416751220567408238071109} a^{15} - \frac{205478217432315962266746}{416751220567408238071109} a^{14} + \frac{56644581687900825860738}{416751220567408238071109} a^{13} + \frac{14735008176046016078229}{37886474597037112551919} a^{12} - \frac{167000657153497387023880}{416751220567408238071109} a^{11} + \frac{2564869419192946371076}{6831987222416528492969} a^{10} - \frac{104241804438155061089990}{416751220567408238071109} a^{9} + \frac{66153673186923983702466}{416751220567408238071109} a^{8} + \frac{95273632929939617597665}{416751220567408238071109} a^{7} + \frac{85727914986199221348276}{416751220567408238071109} a^{6} + \frac{12129410031429527168445}{37886474597037112551919} a^{5} - \frac{84712507911467657317678}{416751220567408238071109} a^{4} - \frac{60242874249444551880405}{416751220567408238071109} a^{3} + \frac{159600432838357324990907}{416751220567408238071109} a^{2} - \frac{146637691005483133050327}{416751220567408238071109} a - \frac{122566454086514778234073}{416751220567408238071109}$

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:  \( \frac{3546458718014944010}{621089747492411681179} a^{15} + \frac{6197735654208938219}{621089747492411681179} a^{14} - \frac{73884842028892247429}{621089747492411681179} a^{13} - \frac{182243603418927529390}{621089747492411681179} a^{12} + \frac{690954663607195458781}{621089747492411681179} a^{11} + \frac{2068426820712271373897}{621089747492411681179} a^{10} - \frac{3416883271460387091142}{621089747492411681179} a^{9} - \frac{12984837128357094314255}{621089747492411681179} a^{8} + \frac{9481283310754014421582}{621089747492411681179} a^{7} + \frac{48272688983554597410670}{621089747492411681179} a^{6} - \frac{11965254469651807723557}{621089747492411681179} a^{5} - \frac{109306297812525473562816}{621089747492411681179} a^{4} + \frac{481151771037143599716}{621089747492411681179} a^{3} + \frac{139758518630872281196709}{621089747492411681179} a^{2} + \frac{13190062489434964648271}{621089747492411681179} a - \frac{84695610317805165624596}{621089747492411681179} \) (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:  \( 22011.2476493 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1263:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1263
Character table for t16n1263 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.1578125.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
5Data not computed
101Data not computed