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Properties

Label 16.8.51950061426...6352.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{8}\cdot 17^{15}\cdot 26626013^{2}$
Root discriminant $170.70$
Ramified primes $2, 17, 26626013$
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![43106377080817, -1215733652348, -5109366359220, -672880822233, -91601657871, 28506195667, 13256524043, 448957127, -22008114, -9682635, -4287707, -29499, -2277, -591, 305, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 305*x^14 - 591*x^13 - 2277*x^12 - 29499*x^11 - 4287707*x^10 - 9682635*x^9 - 22008114*x^8 + 448957127*x^7 + 13256524043*x^6 + 28506195667*x^5 - 91601657871*x^4 - 672880822233*x^3 - 5109366359220*x^2 - 1215733652348*x + 43106377080817)
 
gp: K = bnfinit(x^16 - 4*x^15 + 305*x^14 - 591*x^13 - 2277*x^12 - 29499*x^11 - 4287707*x^10 - 9682635*x^9 - 22008114*x^8 + 448957127*x^7 + 13256524043*x^6 + 28506195667*x^5 - 91601657871*x^4 - 672880822233*x^3 - 5109366359220*x^2 - 1215733652348*x + 43106377080817, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 305 x^{14} - 591 x^{13} - 2277 x^{12} - 29499 x^{11} - 4287707 x^{10} - 9682635 x^{9} - 22008114 x^{8} + 448957127 x^{7} + 13256524043 x^{6} + 28506195667 x^{5} - 91601657871 x^{4} - 672880822233 x^{3} - 5109366359220 x^{2} - 1215733652348 x + 43106377080817 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(519500614265953437911136445884676352=2^{8}\cdot 17^{15}\cdot 26626013^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $170.70$
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:  $2, 17, 26626013$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{4} a^{12} + \frac{1}{4} a^{9} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{404} a^{13} - \frac{11}{101} a^{11} + \frac{57}{404} a^{10} + \frac{19}{101} a^{9} - \frac{25}{101} a^{8} + \frac{193}{404} a^{7} - \frac{73}{202} a^{6} + \frac{137}{404} a^{5} + \frac{57}{404} a^{4} + \frac{3}{404} a^{3} - \frac{2}{101} a^{2} - \frac{29}{404} a + \frac{31}{101}$, $\frac{1}{613787690783562770852} a^{14} - \frac{134678145613433350}{153446922695890692713} a^{13} + \frac{23262512169383682653}{306893845391781385426} a^{12} - \frac{165006840200437756931}{613787690783562770852} a^{11} + \frac{20914219509753587027}{153446922695890692713} a^{10} - \frac{6456403331888680477}{306893845391781385426} a^{9} + \frac{152890338172218489649}{613787690783562770852} a^{8} - \frac{78798205146252150857}{306893845391781385426} a^{7} - \frac{284320986458226251233}{613787690783562770852} a^{6} + \frac{38595030888852376025}{613787690783562770852} a^{5} - \frac{17074449892779533383}{47214437752581751604} a^{4} + \frac{555280296520268287}{3038552924671102826} a^{3} + \frac{167483176256006860885}{613787690783562770852} a^{2} - \frac{18470873745635238341}{153446922695890692713} a + \frac{123871011327127595415}{306893845391781385426}$, $\frac{1}{1440745713583732716827565353060796273046793228407174744269680561653331701002332} a^{15} - \frac{156248293358513445097344610950428871651904844158141782597}{720372856791866358413782676530398136523396614203587372134840280826665850501166} a^{14} - \frac{18191772946744777391011968889857108531757525270809524298178073943923592437}{1440745713583732716827565353060796273046793228407174744269680561653331701002332} a^{13} + \frac{16973673314576337080857294763629021736178401260412153205203417270221554008121}{360186428395933179206891338265199068261698307101793686067420140413332925250583} a^{12} - \frac{27142586272487974669387816283335722227423936888590865701770329037745022191551}{55413296676297412185675590502338318194107431861814413241141560063589680807782} a^{11} - \frac{199711744436949323625839221013529122019015507383288532808698837170217886501641}{1440745713583732716827565353060796273046793228407174744269680561653331701002332} a^{10} - \frac{39161947179837318086487109625692570836284433150049696698704677332373242532492}{360186428395933179206891338265199068261698307101793686067420140413332925250583} a^{9} + \frac{65811713276590885736106010226368687931330707262165133978341380694529539092715}{360186428395933179206891338265199068261698307101793686067420140413332925250583} a^{8} - \frac{126909659826195586150722505560308091720645992712281713235956717980102312853995}{360186428395933179206891338265199068261698307101793686067420140413332925250583} a^{7} + \frac{478447528518803052256492862433262805037041262978488056988016265382343529541}{27706648338148706092837795251169159097053715930907206620570780031794840403891} a^{6} - \frac{307996185450829798397451347642091900214357536440268747899963086861827349603863}{720372856791866358413782676530398136523396614203587372134840280826665850501166} a^{5} + \frac{25497312149185266346074457123636774217662903389501150699406411210529380920198}{360186428395933179206891338265199068261698307101793686067420140413332925250583} a^{4} + \frac{289955145809846694071084786798444909598550943660935371243756205087315973214071}{1440745713583732716827565353060796273046793228407174744269680561653331701002332} a^{3} - \frac{608186255761120072090944336801869905759649435899747884082909506837367302607629}{1440745713583732716827565353060796273046793228407174744269680561653331701002332} a^{2} - \frac{548425166146912732662524025344427808424555701349630081415015087977227494369463}{1440745713583732716827565353060796273046793228407174744269680561653331701002332} a + \frac{411444824847642464085799616192362391693153802859869071142152738150807345944485}{1440745713583732716827565353060796273046793228407174744269680561653331701002332}$

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:  $11$
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:  \( 670771901779 \) (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 $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ 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$2.8.8.5$x^{8} + 2 x^{7} + 8 x^{2} + 16$$2$$4$$8$$C_8:C_2$$[2, 2]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
17Data not computed
26626013Data not computed