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Properties

Label 16.8.87678952778...7713.7
Degree $16$
Signature $[8, 4]$
Discriminant $61^{4}\cdot 97^{15}$
Root discriminant $203.67$
Ramified primes $61, 97$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T841

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![167363472977, 57264732095, -15831634490, -4184728814, 977105421, 192753639, -28171423, -10260945, 1105782, 163132, -80068, 5230, 1992, 216, -68, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 68*x^14 + 216*x^13 + 1992*x^12 + 5230*x^11 - 80068*x^10 + 163132*x^9 + 1105782*x^8 - 10260945*x^7 - 28171423*x^6 + 192753639*x^5 + 977105421*x^4 - 4184728814*x^3 - 15831634490*x^2 + 57264732095*x + 167363472977)
 
gp: K = bnfinit(x^16 - 6*x^15 - 68*x^14 + 216*x^13 + 1992*x^12 + 5230*x^11 - 80068*x^10 + 163132*x^9 + 1105782*x^8 - 10260945*x^7 - 28171423*x^6 + 192753639*x^5 + 977105421*x^4 - 4184728814*x^3 - 15831634490*x^2 + 57264732095*x + 167363472977, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 68 x^{14} + 216 x^{13} + 1992 x^{12} + 5230 x^{11} - 80068 x^{10} + 163132 x^{9} + 1105782 x^{8} - 10260945 x^{7} - 28171423 x^{6} + 192753639 x^{5} + 977105421 x^{4} - 4184728814 x^{3} - 15831634490 x^{2} + 57264732095 x + 167363472977 \)

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}{8474380186686225413136712491532172063980897460991965233651349782758897811} a^{15} - \frac{841386551308482227258332193816513405092818845275094528488951045757464586}{8474380186686225413136712491532172063980897460991965233651349782758897811} a^{14} - \frac{1904524414330082085829305117273344063869860018993416067025491156879923517}{8474380186686225413136712491532172063980897460991965233651349782758897811} a^{13} - \frac{3852881890703457996174026281518173275383908420442077255838201684074642620}{8474380186686225413136712491532172063980897460991965233651349782758897811} a^{12} - \frac{242825265901550485635023672873063631039604607360736157209298333701691661}{8474380186686225413136712491532172063980897460991965233651349782758897811} a^{11} - \frac{850810161935494302295865535207345031351036453870419082131107252367751902}{8474380186686225413136712491532172063980897460991965233651349782758897811} a^{10} - \frac{2209523381152897588929040503073463546608951939909324868861708176206457290}{8474380186686225413136712491532172063980897460991965233651349782758897811} a^{9} - \frac{436287410286221387563413994444405300027016766703105798117507076183571443}{8474380186686225413136712491532172063980897460991965233651349782758897811} a^{8} + \frac{3883575520199765952157022680444527250873240069970058655956095725848614935}{8474380186686225413136712491532172063980897460991965233651349782758897811} a^{7} - \frac{3164854879151356016896879944205375084197590160026223705625154549217897552}{8474380186686225413136712491532172063980897460991965233651349782758897811} a^{6} - \frac{2601184998004717528175620127746823037261686829691879663805432972404964263}{8474380186686225413136712491532172063980897460991965233651349782758897811} a^{5} + \frac{1537579042027904298019511494533234653301469170684765111905266089825480892}{8474380186686225413136712491532172063980897460991965233651349782758897811} a^{4} - \frac{997015897628899412468218658489152185712893731190451945496611822659703383}{8474380186686225413136712491532172063980897460991965233651349782758897811} a^{3} - \frac{3977238629161869846739358631826447114918866237228688629470742027174810038}{8474380186686225413136712491532172063980897460991965233651349782758897811} a^{2} + \frac{1841404321777379258156004564183217327965988638649916495219885839027546099}{8474380186686225413136712491532172063980897460991965233651349782758897811} a + \frac{3370812247918963704831916444187317804518168207307786521446446353648835404}{8474380186686225413136712491532172063980897460991965233651349782758897811}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 620760386692 \) (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

\(\Q(\sqrt{97}) \), 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 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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