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Properties

Label 16.0.95908900025...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{15}\cdot 61^{7}$
Root discriminant $27.31$
Ramified primes $5, 61$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1192

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16441, 32987, -13360, -45405, 8965, 19316, 10417, -16120, -1260, 3770, 952, -1241, 120, 105, -25, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 25*x^14 + 105*x^13 + 120*x^12 - 1241*x^11 + 952*x^10 + 3770*x^9 - 1260*x^8 - 16120*x^7 + 10417*x^6 + 19316*x^5 + 8965*x^4 - 45405*x^3 - 13360*x^2 + 32987*x + 16441)
 
gp: K = bnfinit(x^16 - 2*x^15 - 25*x^14 + 105*x^13 + 120*x^12 - 1241*x^11 + 952*x^10 + 3770*x^9 - 1260*x^8 - 16120*x^7 + 10417*x^6 + 19316*x^5 + 8965*x^4 - 45405*x^3 - 13360*x^2 + 32987*x + 16441, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 25 x^{14} + 105 x^{13} + 120 x^{12} - 1241 x^{11} + 952 x^{10} + 3770 x^{9} - 1260 x^{8} - 16120 x^{7} + 10417 x^{6} + 19316 x^{5} + 8965 x^{4} - 45405 x^{3} - 13360 x^{2} + 32987 x + 16441 \)

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:  \(95908900025054931640625=5^{15}\cdot 61^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.31$
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:  $5, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{14048769994376469258309589634742069691} a^{15} - \frac{6187477900071213437932604057717899234}{14048769994376469258309589634742069691} a^{14} - \frac{3823615470819622156718911584902646310}{14048769994376469258309589634742069691} a^{13} - \frac{3029383349747514365350666068838881787}{14048769994376469258309589634742069691} a^{12} + \frac{2370620360945212552113657661357321270}{14048769994376469258309589634742069691} a^{11} + \frac{6963231991597235551914103793719497093}{14048769994376469258309589634742069691} a^{10} + \frac{333007522402377036401968057543244663}{14048769994376469258309589634742069691} a^{9} - \frac{3782590391013325234435522123152610203}{14048769994376469258309589634742069691} a^{8} - \frac{2863911341272708911828259541545954948}{14048769994376469258309589634742069691} a^{7} - \frac{2456079375496586960696385030971436101}{14048769994376469258309589634742069691} a^{6} + \frac{5799321717223440988819141149245395239}{14048769994376469258309589634742069691} a^{5} - \frac{4782070284321093756505179253848281191}{14048769994376469258309589634742069691} a^{4} - \frac{3892518946604980048977693455027943241}{14048769994376469258309589634742069691} a^{3} - \frac{4161739274193508624446320880028882609}{14048769994376469258309589634742069691} a^{2} + \frac{3996846372416142813257975223863940233}{14048769994376469258309589634742069691} a - \frac{3716117352644118493292487106849144780}{14048769994376469258309589634742069691}$

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:  \( -\frac{81664122288694266418321963425}{680624484975363076319441385337051} a^{15} + \frac{139773676159878786551534782892}{680624484975363076319441385337051} a^{14} + \frac{2014329535162919843139983624605}{680624484975363076319441385337051} a^{13} - \frac{7908193824028987642770697456586}{680624484975363076319441385337051} a^{12} - \frac{10287094807768202559587071751609}{680624484975363076319441385337051} a^{11} + \frac{92803384693241078317628514446715}{680624484975363076319441385337051} a^{10} - \frac{64040809081118068139906843972639}{680624484975363076319441385337051} a^{9} - \frac{254390618100905823981236638719402}{680624484975363076319441385337051} a^{8} + \frac{35102033199619724724125145247624}{680624484975363076319441385337051} a^{7} + \frac{1086842029998122006457078473206200}{680624484975363076319441385337051} a^{6} - \frac{756040078430000113625760023549357}{680624484975363076319441385337051} a^{5} - \frac{878203798372110501961543920953063}{680624484975363076319441385337051} a^{4} - \frac{750898883087684590609079916138602}{680624484975363076319441385337051} a^{3} + \frac{2786636063282512482288172300619549}{680624484975363076319441385337051} a^{2} + \frac{63890780168146505137927215026877}{680624484975363076319441385337051} a - \frac{1134152209028960247746786067852780}{680624484975363076319441385337051} \) (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:  \( 63387.3336418 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1192:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 88 conjugacy class representatives for t16n1192 are not computed
Character table for t16n1192 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.4765625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
61Data not computed