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Properties

Label 16.12.1814980849...0641.1
Degree $16$
Signature $[12, 2]$
Discriminant $17^{14}\cdot 47^{6}$
Root discriminant $50.55$
Ramified primes $17, 47$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T258)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![28561, 202124, 169642, -378293, -351468, 245190, 187327, -39383, -21082, -18873, 1, 3687, 44, -98, -4, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 4*x^14 - 98*x^13 + 44*x^12 + 3687*x^11 + x^10 - 18873*x^9 - 21082*x^8 - 39383*x^7 + 187327*x^6 + 245190*x^5 - 351468*x^4 - 378293*x^3 + 169642*x^2 + 202124*x + 28561)
 
gp: K = bnfinit(x^16 - 5*x^15 - 4*x^14 - 98*x^13 + 44*x^12 + 3687*x^11 + x^10 - 18873*x^9 - 21082*x^8 - 39383*x^7 + 187327*x^6 + 245190*x^5 - 351468*x^4 - 378293*x^3 + 169642*x^2 + 202124*x + 28561, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 4 x^{14} - 98 x^{13} + 44 x^{12} + 3687 x^{11} + x^{10} - 18873 x^{9} - 21082 x^{8} - 39383 x^{7} + 187327 x^{6} + 245190 x^{5} - 351468 x^{4} - 378293 x^{3} + 169642 x^{2} + 202124 x + 28561 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1814980849112797822933640641=17^{14}\cdot 47^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.55$
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:  $17, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{764} a^{14} + \frac{8}{191} a^{13} + \frac{117}{764} a^{12} + \frac{11}{764} a^{11} - \frac{38}{191} a^{10} - \frac{65}{191} a^{9} + \frac{303}{764} a^{8} + \frac{43}{191} a^{7} + \frac{117}{764} a^{6} + \frac{58}{191} a^{5} + \frac{53}{382} a^{4} + \frac{51}{191} a^{3} + \frac{137}{382} a^{2} - \frac{165}{764} a - \frac{137}{764}$, $\frac{1}{144341633365224836354692579780971730852} a^{15} + \frac{2779412055115361151862552079886211}{144341633365224836354692579780971730852} a^{14} + \frac{18883084130502065013897935028656571661}{144341633365224836354692579780971730852} a^{13} + \frac{247584191340298763169375374963097889}{72170816682612418177346289890485865426} a^{12} - \frac{35844006857777889897851574304578708437}{144341633365224836354692579780971730852} a^{11} + \frac{8705200031876694640915604383198222218}{36085408341306209088673144945242932713} a^{10} + \frac{18250912203132471623954491158858504495}{144341633365224836354692579780971730852} a^{9} - \frac{21627600495136497971562022150179188197}{144341633365224836354692579780971730852} a^{8} + \frac{37399818308204699465103142881797336163}{144341633365224836354692579780971730852} a^{7} + \frac{38240246431618447559023147299274880887}{144341633365224836354692579780971730852} a^{6} + \frac{8893577260056854614459506442057825273}{36085408341306209088673144945242932713} a^{5} + \frac{14530851549844560066423601001266922699}{72170816682612418177346289890485865426} a^{4} + \frac{175189997278562285321667147096906372}{2775800641638939160667164995787917901} a^{3} - \frac{32934289156453331043776120379747309245}{144341633365224836354692579780971730852} a^{2} - \frac{6800871491809442252059727451325574923}{72170816682612418177346289890485865426} a - \frac{7515854840559902966533229399780437}{854092505119673587897589229473205508}$

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:  $13$
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:  \( 99766009.0279 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T258):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\), 8.6.42602592046879.1, 8.6.2506034826287.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
17Data not computed
$47$47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$