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Properties

Label 16.16.3480707415...6608.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{40}\cdot 41^{4}\cdot 73^{3}\cdot 1697^{2}$
Root discriminant $81.07$
Ramified primes $2, 41, 73, 1697$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1781

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![79889, 664268, 106928, -2123820, 251748, 2047852, -563604, -778400, 281019, 115204, -45496, -8072, 3166, 280, -96, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 96*x^14 + 280*x^13 + 3166*x^12 - 8072*x^11 - 45496*x^10 + 115204*x^9 + 281019*x^8 - 778400*x^7 - 563604*x^6 + 2047852*x^5 + 251748*x^4 - 2123820*x^3 + 106928*x^2 + 664268*x + 79889)
 
gp: K = bnfinit(x^16 - 4*x^15 - 96*x^14 + 280*x^13 + 3166*x^12 - 8072*x^11 - 45496*x^10 + 115204*x^9 + 281019*x^8 - 778400*x^7 - 563604*x^6 + 2047852*x^5 + 251748*x^4 - 2123820*x^3 + 106928*x^2 + 664268*x + 79889, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 96 x^{14} + 280 x^{13} + 3166 x^{12} - 8072 x^{11} - 45496 x^{10} + 115204 x^{9} + 281019 x^{8} - 778400 x^{7} - 563604 x^{6} + 2047852 x^{5} + 251748 x^{4} - 2123820 x^{3} + 106928 x^{2} + 664268 x + 79889 \)

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:  \(3480707415024772072868946116608=2^{40}\cdot 41^{4}\cdot 73^{3}\cdot 1697^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.07$
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:  $2, 41, 73, 1697$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{61139163334850635368599635383515364426581} a^{15} - \frac{6399649634701558132011654167374709463227}{61139163334850635368599635383515364426581} a^{14} + \frac{7444992178806746002476908001714219828728}{61139163334850635368599635383515364426581} a^{13} - \frac{8319907297322749478286320152891845902910}{61139163334850635368599635383515364426581} a^{12} - \frac{23099692249525280980128324352795275822205}{61139163334850635368599635383515364426581} a^{11} + \frac{19386832457977686446613159273329270193593}{61139163334850635368599635383515364426581} a^{10} + \frac{28823023336988053319868340473700791353742}{61139163334850635368599635383515364426581} a^{9} + \frac{26411721809535106266960707154109638255504}{61139163334850635368599635383515364426581} a^{8} + \frac{26013591433936604423257195018654281664238}{61139163334850635368599635383515364426581} a^{7} - \frac{23553826727218614037612216057979170649601}{61139163334850635368599635383515364426581} a^{6} + \frac{5517189684348261296060962272728307529084}{61139163334850635368599635383515364426581} a^{5} + \frac{9828851319573905728144188909893397225842}{61139163334850635368599635383515364426581} a^{4} + \frac{10417124174802687491728918545477264741951}{61139163334850635368599635383515364426581} a^{3} + \frac{15199087292560734598906394861884557288804}{61139163334850635368599635383515364426581} a^{2} - \frac{2926990680013551640725864404084720765235}{61139163334850635368599635383515364426581} a + \frac{16879337390407939186069548810981154301066}{61139163334850635368599635383515364426581}$

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:  \( 18054188218.2 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1781:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16384
The 148 conjugacy class representatives for t16n1781 are not computed
Character table for t16n1781 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.2624.1, 8.8.8042119168.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$41$41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.8.4.1$x^{8} + 57154 x^{4} - 68921 x^{2} + 816644929$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$73$73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.4.3.3$x^{4} + 365$$4$$1$$3$$C_4$$[\ ]_{4}$
73.4.0.1$x^{4} - x + 13$$1$$4$$0$$C_4$$[\ ]^{4}$
1697Data not computed