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Properties

Label 16.0.35687701699...5625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{15}\cdot 61^{9}$
Root discriminant $45.66$
Ramified primes $5, 61$
Class number $16$ (GRH)
Class group $[4, 4]$ (GRH)
Galois group 16T1192

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![75581, 196993, 313920, 483375, 667520, 693794, 523387, 254330, 44600, -19960, -8458, 1351, 1000, 25, -45, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 45*x^14 + 25*x^13 + 1000*x^12 + 1351*x^11 - 8458*x^10 - 19960*x^9 + 44600*x^8 + 254330*x^7 + 523387*x^6 + 693794*x^5 + 667520*x^4 + 483375*x^3 + 313920*x^2 + 196993*x + 75581)
 
gp: K = bnfinit(x^16 - 3*x^15 - 45*x^14 + 25*x^13 + 1000*x^12 + 1351*x^11 - 8458*x^10 - 19960*x^9 + 44600*x^8 + 254330*x^7 + 523387*x^6 + 693794*x^5 + 667520*x^4 + 483375*x^3 + 313920*x^2 + 196993*x + 75581, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 45 x^{14} + 25 x^{13} + 1000 x^{12} + 1351 x^{11} - 8458 x^{10} - 19960 x^{9} + 44600 x^{8} + 254330 x^{7} + 523387 x^{6} + 693794 x^{5} + 667520 x^{4} + 483375 x^{3} + 313920 x^{2} + 196993 x + 75581 \)

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:  \(356877016993229400634765625=5^{15}\cdot 61^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.66$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{22} a^{13} + \frac{3}{22} a^{12} - \frac{5}{22} a^{11} + \frac{7}{22} a^{10} - \frac{1}{2} a^{9} - \frac{7}{22} a^{8} - \frac{1}{22} a^{7} + \frac{5}{11} a^{6} - \frac{7}{22} a^{5} + \frac{3}{11} a^{3} + \frac{9}{22} a^{2} + \frac{3}{11} a - \frac{1}{2}$, $\frac{1}{22} a^{14} - \frac{3}{22} a^{12} + \frac{1}{22} a^{10} + \frac{2}{11} a^{9} + \frac{9}{22} a^{8} - \frac{9}{22} a^{7} - \frac{2}{11} a^{6} + \frac{5}{11} a^{5} + \frac{3}{11} a^{4} - \frac{9}{22} a^{3} + \frac{1}{22} a^{2} + \frac{2}{11} a$, $\frac{1}{7682415162365946413162294621490947916722} a^{15} + \frac{33210237121712348601242804174293052503}{7682415162365946413162294621490947916722} a^{14} + \frac{2021255760628970913445958586686288955}{3841207581182973206581147310745473958361} a^{13} + \frac{935983220262667919951900843380000117587}{7682415162365946413162294621490947916722} a^{12} - \frac{450325525910681628488374506314068061794}{3841207581182973206581147310745473958361} a^{11} - \frac{3257407628499225099465843038992763467821}{7682415162365946413162294621490947916722} a^{10} + \frac{150031195487338084510938223377819489887}{3841207581182973206581147310745473958361} a^{9} - \frac{323106079346230415465523286676457362829}{3841207581182973206581147310745473958361} a^{8} - \frac{650722750826363192982549149630126859168}{3841207581182973206581147310745473958361} a^{7} - \frac{234469120003315351602051464087620207569}{698401378396904219378390420135540719702} a^{6} + \frac{775653349846536641429429382780916045555}{3841207581182973206581147310745473958361} a^{5} + \frac{1071378292674810310352842476482828778281}{7682415162365946413162294621490947916722} a^{4} + \frac{1602994419674487188809038494753770371948}{3841207581182973206581147310745473958361} a^{3} - \frac{52548675037380872134325711372073085927}{3841207581182973206581147310745473958361} a^{2} + \frac{1939319940972449632395088169266040324997}{7682415162365946413162294621490947916722} a + \frac{11744589868078893281472413453240427590}{349200689198452109689195210067770359851}$

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

Class group and class number

$C_{4}\times C_{4}$, which has order $16$ (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:  \( \frac{164537732264905444470041859684380217}{349200689198452109689195210067770359851} a^{15} - \frac{775679361216820881967235704055199700}{349200689198452109689195210067770359851} a^{14} - \frac{6074770761294144073384346126225084865}{349200689198452109689195210067770359851} a^{13} + \frac{14532106107629909801259089564640437904}{349200689198452109689195210067770359851} a^{12} + \frac{139608213212203639819441136810734258791}{349200689198452109689195210067770359851} a^{11} - \frac{17112254114136112399760092872018222150}{349200689198452109689195210067770359851} a^{10} - \frac{123803401255367419932547540847099800562}{31745517199859282699017746369797305441} a^{9} - \frac{948456952763154115125898401138500672747}{349200689198452109689195210067770359851} a^{8} + \frac{8955695839691240583770584286754349220394}{349200689198452109689195210067770359851} a^{7} + \frac{26480487690578119427489524038144462015991}{349200689198452109689195210067770359851} a^{6} + \frac{40802110385185570864262154094515229876522}{349200689198452109689195210067770359851} a^{5} + \frac{44313506312406516288765923902194590767199}{349200689198452109689195210067770359851} a^{4} + \frac{33461548204248077957078371296007891936255}{349200689198452109689195210067770359851} a^{3} + \frac{21343514476327152174388463374311579882004}{349200689198452109689195210067770359851} a^{2} + \frac{14460821392294389235347412354149801984879}{349200689198452109689195210067770359851} a + \frac{669583100484634309354433613403552947772}{31745517199859282699017746369797305441} \) (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:  \( 3211466.17743 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.17732890625.2

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{12}$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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