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Properties

Label 16.16.9943583883...7841.1
Degree $16$
Signature $[16, 0]$
Discriminant $29^{14}\cdot 109^{14}$
Root discriminant $1154.37$
Ramified primes $29, 109$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_8.C_4$ (as 16T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![208148520549407, 163997627966411, 14395417397350, -18212997720900, -4693197033536, 389263434099, 256010445822, 16087587618, -4489506330, -662101208, 7509025, 6325863, 265395, -16925, -1288, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 1288*x^14 - 16925*x^13 + 265395*x^12 + 6325863*x^11 + 7509025*x^10 - 662101208*x^9 - 4489506330*x^8 + 16087587618*x^7 + 256010445822*x^6 + 389263434099*x^5 - 4693197033536*x^4 - 18212997720900*x^3 + 14395417397350*x^2 + 163997627966411*x + 208148520549407)
 
gp: K = bnfinit(x^16 - x^15 - 1288*x^14 - 16925*x^13 + 265395*x^12 + 6325863*x^11 + 7509025*x^10 - 662101208*x^9 - 4489506330*x^8 + 16087587618*x^7 + 256010445822*x^6 + 389263434099*x^5 - 4693197033536*x^4 - 18212997720900*x^3 + 14395417397350*x^2 + 163997627966411*x + 208148520549407, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 1288 x^{14} - 16925 x^{13} + 265395 x^{12} + 6325863 x^{11} + 7509025 x^{10} - 662101208 x^{9} - 4489506330 x^{8} + 16087587618 x^{7} + 256010445822 x^{6} + 389263434099 x^{5} - 4693197033536 x^{4} - 18212997720900 x^{3} + 14395417397350 x^{2} + 163997627966411 x + 208148520549407 \)

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:  \(9943583883093737930218051237711077235104795837841=29^{14}\cdot 109^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1154.37$
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:  $29, 109$
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}{487370384861226495922554963217865881980838024397597984263285266395545761843157735867} a^{15} + \frac{174845222287553094470573774650410052025785814826350452794056560509582057248826321926}{487370384861226495922554963217865881980838024397597984263285266395545761843157735867} a^{14} - \frac{118175505220313909333657354960231953508612824127985715205858090695679449631916735712}{487370384861226495922554963217865881980838024397597984263285266395545761843157735867} a^{13} + \frac{62298894243349001419907747638609530371015338122469176176806314932393724119257068022}{487370384861226495922554963217865881980838024397597984263285266395545761843157735867} a^{12} + \frac{33025549526649063562613753978109273103718906160106092097610404510732984130616775472}{487370384861226495922554963217865881980838024397597984263285266395545761843157735867} a^{11} + \frac{160367951848035290792998050849075659912690490950512856241588305079238206625641525218}{487370384861226495922554963217865881980838024397597984263285266395545761843157735867} a^{10} - \frac{145529912830175851332816734440019033555602415999815159971073788374255938755757414742}{487370384861226495922554963217865881980838024397597984263285266395545761843157735867} a^{9} + \frac{201001284111196423405412029987126182815429753412255491968769116326721829597277887489}{487370384861226495922554963217865881980838024397597984263285266395545761843157735867} a^{8} - \frac{115009150427066875661699405339761572692678980614764750845371509717052888816099448305}{487370384861226495922554963217865881980838024397597984263285266395545761843157735867} a^{7} - \frac{133290116982321465208910370423384034359564351658950965653371986861007993423768120781}{487370384861226495922554963217865881980838024397597984263285266395545761843157735867} a^{6} - \frac{132758254223279414164204330698880750309852796129198747562553650517645743413074204260}{487370384861226495922554963217865881980838024397597984263285266395545761843157735867} a^{5} + \frac{52701384909137974449868315702345621885929106592822583653383438607472806479138294219}{487370384861226495922554963217865881980838024397597984263285266395545761843157735867} a^{4} - \frac{92186394425464193105427645888912865847867566182785205971520830950899773889576052592}{487370384861226495922554963217865881980838024397597984263285266395545761843157735867} a^{3} + \frac{185057326198321380716777899428496323681447410091921277625823013430028999365279551836}{487370384861226495922554963217865881980838024397597984263285266395545761843157735867} a^{2} - \frac{111618910447496968167062692478760360396063564366668391959373324034409476255770642111}{487370384861226495922554963217865881980838024397597984263285266395545761843157735867} a + \frac{195577421703403779260582248188098281690318884392127812965759552181976504951427174159}{487370384861226495922554963217865881980838024397597984263285266395545761843157735867}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 6499563080450000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8.C_4$ (as 16T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_8.C_4$
Character table for $C_8.C_4$

Intermediate fields

\(\Q(\sqrt{109}) \), \(\Q(\sqrt{3161}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{29}, \sqrt{109})\), 8.8.997578257579911722961.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$29$29.8.7.2$x^{8} - 116$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
29.8.7.2$x^{8} - 116$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$109$109.8.7.2$x^{8} - 3924$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
109.8.7.2$x^{8} - 3924$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$