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Properties

Label 16.8.87678952778...7713.1
Degree $16$
Signature $[8, 4]$
Discriminant $61^{4}\cdot 97^{15}$
Root discriminant $203.67$
Ramified primes $61, 97$
Class number $16$ (GRH)
Class group $[4, 4]$ (GRH)
Galois group $C_2^3.C_8$ (as 16T104)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-791777377, 6779883369, -7687245606, 161939835, 341740386, 392424841, 3798904, -3788717, 384199, -688528, 97345, 23660, -2082, -75, -68, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 68*x^14 - 75*x^13 - 2082*x^12 + 23660*x^11 + 97345*x^10 - 688528*x^9 + 384199*x^8 - 3788717*x^7 + 3798904*x^6 + 392424841*x^5 + 341740386*x^4 + 161939835*x^3 - 7687245606*x^2 + 6779883369*x - 791777377)
 
gp: K = bnfinit(x^16 - 6*x^15 - 68*x^14 - 75*x^13 - 2082*x^12 + 23660*x^11 + 97345*x^10 - 688528*x^9 + 384199*x^8 - 3788717*x^7 + 3798904*x^6 + 392424841*x^5 + 341740386*x^4 + 161939835*x^3 - 7687245606*x^2 + 6779883369*x - 791777377, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 68 x^{14} - 75 x^{13} - 2082 x^{12} + 23660 x^{11} + 97345 x^{10} - 688528 x^{9} + 384199 x^{8} - 3788717 x^{7} + 3798904 x^{6} + 392424841 x^{5} + 341740386 x^{4} + 161939835 x^{3} - 7687245606 x^{2} + 6779883369 x - 791777377 \)

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:  \(8767895277848913089642817986417897713=61^{4}\cdot 97^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $203.67$
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:  $61, 97$
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}{84916775603173268528347001933642347257837238547144380018341161604638331282913} a^{15} + \frac{25789995603037686238969796547367968896772055423052819834032747153408694455978}{84916775603173268528347001933642347257837238547144380018341161604638331282913} a^{14} - \frac{18030654907814383068840385455185818106909376105831255560035655593648727310732}{84916775603173268528347001933642347257837238547144380018341161604638331282913} a^{13} - \frac{3567142967340315584629750585785002007303713665431135235936263401723118513786}{84916775603173268528347001933642347257837238547144380018341161604638331282913} a^{12} + \frac{35751167031266724366098015029778418488469405314903681975778791417701346227414}{84916775603173268528347001933642347257837238547144380018341161604638331282913} a^{11} - \frac{21303946486269459952560536528494969154486499592370895261802734335494456160059}{84916775603173268528347001933642347257837238547144380018341161604638331282913} a^{10} - \frac{12215905427581742954698956602057992784740604561501843956212817634459077448445}{84916775603173268528347001933642347257837238547144380018341161604638331282913} a^{9} - \frac{17577381700368891158594252783976934488806932803281772199979768803616685562217}{84916775603173268528347001933642347257837238547144380018341161604638331282913} a^{8} - \frac{30939966229032599951549006832551061496410122998580124111862823376785274573195}{84916775603173268528347001933642347257837238547144380018341161604638331282913} a^{7} + \frac{30430354349537054781353424704388154346177203259667049591289508738698297913893}{84916775603173268528347001933642347257837238547144380018341161604638331282913} a^{6} + \frac{42412919267568278279323403916653322744404919174765487570468731523854766545920}{84916775603173268528347001933642347257837238547144380018341161604638331282913} a^{5} - \frac{36748919989689321722280007045360633145824737348901574049908549699776682769833}{84916775603173268528347001933642347257837238547144380018341161604638331282913} a^{4} + \frac{4033390242871255869762094824797696622368329485309308528186282022540118428304}{84916775603173268528347001933642347257837238547144380018341161604638331282913} a^{3} - \frac{30963755500629224140608050495338058606728450977993375731920317142290350294766}{84916775603173268528347001933642347257837238547144380018341161604638331282913} a^{2} - \frac{4803305645279290129312562970678603815873280747071692412707437129692510155284}{84916775603173268528347001933642347257837238547144380018341161604638331282913} a - \frac{106722460490417542361634098147811904862990543514280474565446401726582956442}{824434714593915228430553416831479099590652801428586213770302539850857585271}$

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:  $11$
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:  \( 137803342579 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_8$ (as 16T104):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2^3.C_8$
Character table for $C_2^3.C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
61Data not computed
97Data not computed