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Properties

Label 16.8.32625338328...0073.5
Degree $16$
Signature $[8, 4]$
Discriminant $61^{6}\cdot 97^{15}$
Root discriminant $340.48$
Ramified primes $61, 97$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $(C_2\times C_8).D_4$ (as 16T306)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6042624679631, 1087153097008, -1240783907555, -45734794022, -42242950419, 17850285859, 4834485125, -653861602, 64929697, -11521320, -1977115, 133747, -63708, -2029, 305, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 305*x^14 - 2029*x^13 - 63708*x^12 + 133747*x^11 - 1977115*x^10 - 11521320*x^9 + 64929697*x^8 - 653861602*x^7 + 4834485125*x^6 + 17850285859*x^5 - 42242950419*x^4 - 45734794022*x^3 - 1240783907555*x^2 + 1087153097008*x + 6042624679631)
 
gp: K = bnfinit(x^16 - 2*x^15 + 305*x^14 - 2029*x^13 - 63708*x^12 + 133747*x^11 - 1977115*x^10 - 11521320*x^9 + 64929697*x^8 - 653861602*x^7 + 4834485125*x^6 + 17850285859*x^5 - 42242950419*x^4 - 45734794022*x^3 - 1240783907555*x^2 + 1087153097008*x + 6042624679631, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 305 x^{14} - 2029 x^{13} - 63708 x^{12} + 133747 x^{11} - 1977115 x^{10} - 11521320 x^{9} + 64929697 x^{8} - 653861602 x^{7} + 4834485125 x^{6} + 17850285859 x^{5} - 42242950419 x^{4} - 45734794022 x^{3} - 1240783907555 x^{2} + 1087153097008 x + 6042624679631 \)

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:  \(32625338328875805606560925727460997390073=61^{6}\cdot 97^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $340.48$
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}{7676922731537843160175546848716454983615660306640185925414866670208587900025303292303222115202323827} a^{15} - \frac{2353234642435850491174503550451407566586241182297008688430469830335684753562467401577964509023464369}{7676922731537843160175546848716454983615660306640185925414866670208587900025303292303222115202323827} a^{14} - \frac{1367538324269645834467660650451019317131769021659583457287216694215477171204582290472643497362415874}{7676922731537843160175546848716454983615660306640185925414866670208587900025303292303222115202323827} a^{13} - \frac{2481922637938402736617011379458260309595360695926965496647452009008698330187471440578463883108146314}{7676922731537843160175546848716454983615660306640185925414866670208587900025303292303222115202323827} a^{12} - \frac{1271728486282286546124122822364218396111813813615220276461251942615874949957985449953817690382929009}{7676922731537843160175546848716454983615660306640185925414866670208587900025303292303222115202323827} a^{11} - \frac{571554663877070838519695012977375477375753597706105180799071289747395204485821461387120494488688087}{7676922731537843160175546848716454983615660306640185925414866670208587900025303292303222115202323827} a^{10} + \frac{2324959051498037960317979339922074222291143912222743373916272694779252410518981697542603439925714114}{7676922731537843160175546848716454983615660306640185925414866670208587900025303292303222115202323827} a^{9} + \frac{2216875783605185675731151785541834282227795587305720596803102393290971887354114603919282934777092283}{7676922731537843160175546848716454983615660306640185925414866670208587900025303292303222115202323827} a^{8} - \frac{2110583601177691196934517639934591447691451410690086975364900699181306414987535515306410991357330304}{7676922731537843160175546848716454983615660306640185925414866670208587900025303292303222115202323827} a^{7} - \frac{1392983582802297995065203253428850994573299246931707639261033582350848525382359581905767925980410200}{7676922731537843160175546848716454983615660306640185925414866670208587900025303292303222115202323827} a^{6} - \frac{2446765368070613514921009229320033146965767841217218417026055859995286412830640280783675529611306028}{7676922731537843160175546848716454983615660306640185925414866670208587900025303292303222115202323827} a^{5} - \frac{1505785291072856960881534579654760882078973695640153510115874172822431533860799392292369333602362271}{7676922731537843160175546848716454983615660306640185925414866670208587900025303292303222115202323827} a^{4} + \frac{207812143091151774629606732846533112977513542219441484405257573930951280804914280548776369591457591}{7676922731537843160175546848716454983615660306640185925414866670208587900025303292303222115202323827} a^{3} + \frac{1553262275798745119468409146108826849130067935497951625586037144333197589328226827251440199801803411}{7676922731537843160175546848716454983615660306640185925414866670208587900025303292303222115202323827} a^{2} - \frac{11049966462972290029759902833222942139557433038027986010639249465686480175227697241023105593024064}{7676922731537843160175546848716454983615660306640185925414866670208587900025303292303222115202323827} a + \frac{1365358228862786027430259990982094802734445021463804655631591263680070577783526399724990021080780639}{7676922731537843160175546848716454983615660306640185925414866670208587900025303292303222115202323827}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( 16504108066000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_8).D_4$ (as 16T306):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $(C_2\times C_8).D_4$
Character table for $(C_2\times C_8).D_4$ 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 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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