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Properties

Label 16.16.3123684216...5728.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{8}\cdot 17^{15}\cdot 6529^{2}$
Root discriminant $60.38$
Ramified primes $2, 17, 6529$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T841

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![589493, 997927, -1359220, -2470290, 1216201, 2153889, -601077, -798655, 189583, 130754, -29408, -10163, 2206, 368, -77, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 77*x^14 + 368*x^13 + 2206*x^12 - 10163*x^11 - 29408*x^10 + 130754*x^9 + 189583*x^8 - 798655*x^7 - 601077*x^6 + 2153889*x^5 + 1216201*x^4 - 2470290*x^3 - 1359220*x^2 + 997927*x + 589493)
 
gp: K = bnfinit(x^16 - 5*x^15 - 77*x^14 + 368*x^13 + 2206*x^12 - 10163*x^11 - 29408*x^10 + 130754*x^9 + 189583*x^8 - 798655*x^7 - 601077*x^6 + 2153889*x^5 + 1216201*x^4 - 2470290*x^3 - 1359220*x^2 + 997927*x + 589493, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 77 x^{14} + 368 x^{13} + 2206 x^{12} - 10163 x^{11} - 29408 x^{10} + 130754 x^{9} + 189583 x^{8} - 798655 x^{7} - 601077 x^{6} + 2153889 x^{5} + 1216201 x^{4} - 2470290 x^{3} - 1359220 x^{2} + 997927 x + 589493 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(31236842166910780816202985728=2^{8}\cdot 17^{15}\cdot 6529^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.38$
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:  $2, 17, 6529$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{20431099816652561319026254906297050162079} a^{15} + \frac{5729459531456505287614483011945425642335}{20431099816652561319026254906297050162079} a^{14} + \frac{1062241871757602782377010492967803962278}{20431099816652561319026254906297050162079} a^{13} + \frac{7004567349698257969498022449541937823687}{20431099816652561319026254906297050162079} a^{12} + \frac{3348128148990902191157329769527626251395}{20431099816652561319026254906297050162079} a^{11} - \frac{5048443828424985717203808339988230894553}{20431099816652561319026254906297050162079} a^{10} + \frac{2982158459629926842760348538629546411023}{20431099816652561319026254906297050162079} a^{9} + \frac{7303800965839466286049132530830293913027}{20431099816652561319026254906297050162079} a^{8} + \frac{6266621936566396127201744744533889001321}{20431099816652561319026254906297050162079} a^{7} + \frac{4582693163758060266013144287707292946987}{20431099816652561319026254906297050162079} a^{6} + \frac{7499512752322156180611784993916847543348}{20431099816652561319026254906297050162079} a^{5} + \frac{257312344680508393937459134485730525145}{20431099816652561319026254906297050162079} a^{4} + \frac{4998223480676190163049266629722569285771}{20431099816652561319026254906297050162079} a^{3} - \frac{10181102915174688805715253388503384980965}{20431099816652561319026254906297050162079} a^{2} - \frac{2946396712697547770536135634699567961803}{20431099816652561319026254906297050162079} a + \frac{635850005987822480905404283274105662550}{20431099816652561319026254906297050162079}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 892154070.873 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T841:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n841
Character table for t16n841 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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 R $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
17Data not computed
6529Data not computed