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Properties

Label 16.0.27350389356...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 439^{2}\cdot 2411^{2}$
Root discriminant $18.94$
Ramified primes $5, 439, 2411$
Class number $3$
Class group $[3]$
Galois group 16T1497

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 6, 19, 24, 52, -40, 145, -110, 98, -32, 49, -16, 19, -9, 7, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 7*x^14 - 9*x^13 + 19*x^12 - 16*x^11 + 49*x^10 - 32*x^9 + 98*x^8 - 110*x^7 + 145*x^6 - 40*x^5 + 52*x^4 + 24*x^3 + 19*x^2 + 6*x + 1)
 
gp: K = bnfinit(x^16 - 3*x^15 + 7*x^14 - 9*x^13 + 19*x^12 - 16*x^11 + 49*x^10 - 32*x^9 + 98*x^8 - 110*x^7 + 145*x^6 - 40*x^5 + 52*x^4 + 24*x^3 + 19*x^2 + 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 7 x^{14} - 9 x^{13} + 19 x^{12} - 16 x^{11} + 49 x^{10} - 32 x^{9} + 98 x^{8} - 110 x^{7} + 145 x^{6} - 40 x^{5} + 52 x^{4} + 24 x^{3} + 19 x^{2} + 6 x + 1 \)

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:  \(273503893564697265625=5^{12}\cdot 439^{2}\cdot 2411^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.94$
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:  $5, 439, 2411$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{205977825235199} a^{15} - \frac{86472378832890}{205977825235199} a^{14} + \frac{80233741707654}{205977825235199} a^{13} - \frac{96481413828759}{205977825235199} a^{12} - \frac{23499708676678}{205977825235199} a^{11} - \frac{36012778488181}{205977825235199} a^{10} - \frac{99675575155052}{205977825235199} a^{9} + \frac{8122615099927}{205977825235199} a^{8} - \frac{64866146372771}{205977825235199} a^{7} + \frac{7476606907292}{205977825235199} a^{6} + \frac{99560768171592}{205977825235199} a^{5} - \frac{35925738293734}{205977825235199} a^{4} + \frac{71652122873333}{205977825235199} a^{3} - \frac{80250160234270}{205977825235199} a^{2} - \frac{38208338690378}{205977825235199} a - \frac{63266210403826}{205977825235199}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( -\frac{5402910252679}{205977825235199} a^{15} + \frac{46677299236925}{205977825235199} a^{14} - \frac{140614961861567}{205977825235199} a^{13} + \frac{291854043757291}{205977825235199} a^{12} - \frac{443957656815709}{205977825235199} a^{11} + \frac{740202624048061}{205977825235199} a^{10} - \frac{934438493449350}{205977825235199} a^{9} + \frac{1776041197076504}{205977825235199} a^{8} - \frac{1997760472380408}{205977825235199} a^{7} + \frac{3751501176001494}{205977825235199} a^{6} - \frac{5108365611216536}{205977825235199} a^{5} + \frac{5552410728464540}{205977825235199} a^{4} - \frac{2660814452156036}{205977825235199} a^{3} + \frac{1468545542449497}{205977825235199} a^{2} + \frac{186428986658261}{205977825235199} a + \frac{215851615763501}{205977825235199} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4846.77455506 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1497:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2304
The 40 conjugacy class representatives for t16n1497
Character table for t16n1497 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.8.661518125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 siblings: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 siblings: data not computed
Degree 36 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 ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
439Data not computed
2411Data not computed