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Properties

Label 16.8.78946419571...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{6}\cdot 5^{4}\cdot 17^{8}\cdot 19^{2}\cdot 97^{8}$
Root discriminant $113.78$
Ramified primes $2, 5, 17, 19, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1123

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![99662, 217481, -2301660, 10100722, -9310725, 1240595, 2232798, -823340, -100815, 50655, 2984, 1666, -1083, 69, 22, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 22*x^14 + 69*x^13 - 1083*x^12 + 1666*x^11 + 2984*x^10 + 50655*x^9 - 100815*x^8 - 823340*x^7 + 2232798*x^6 + 1240595*x^5 - 9310725*x^4 + 10100722*x^3 - 2301660*x^2 + 217481*x + 99662)
 
gp: K = bnfinit(x^16 - 8*x^15 + 22*x^14 + 69*x^13 - 1083*x^12 + 1666*x^11 + 2984*x^10 + 50655*x^9 - 100815*x^8 - 823340*x^7 + 2232798*x^6 + 1240595*x^5 - 9310725*x^4 + 10100722*x^3 - 2301660*x^2 + 217481*x + 99662, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 22 x^{14} + 69 x^{13} - 1083 x^{12} + 1666 x^{11} + 2984 x^{10} + 50655 x^{9} - 100815 x^{8} - 823340 x^{7} + 2232798 x^{6} + 1240595 x^{5} - 9310725 x^{4} + 10100722 x^{3} - 2301660 x^{2} + 217481 x + 99662 \)

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:  \(789464195714758583406110606440000=2^{6}\cdot 5^{4}\cdot 17^{8}\cdot 19^{2}\cdot 97^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $113.78$
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:  $2, 5, 17, 19, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{6} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{8} + \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{8} a^{7} - \frac{1}{16} a^{6} - \frac{1}{8} a^{5} - \frac{1}{16} a^{4} + \frac{1}{8} a^{3} - \frac{7}{16} a^{2} + \frac{1}{16} a - \frac{3}{8}$, $\frac{1}{32} a^{10} + \frac{1}{32} a^{8} - \frac{3}{32} a^{7} - \frac{3}{32} a^{6} + \frac{1}{32} a^{5} + \frac{5}{32} a^{4} - \frac{5}{32} a^{3} + \frac{5}{16} a^{2} + \frac{7}{32} a - \frac{7}{16}$, $\frac{1}{64} a^{11} - \frac{1}{64} a^{10} + \frac{1}{64} a^{9} - \frac{1}{16} a^{8} - \frac{1}{8} a^{7} - \frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{13}{32} a^{4} + \frac{23}{64} a^{3} + \frac{5}{64} a^{2} - \frac{13}{64} a + \frac{15}{32}$, $\frac{1}{128} a^{12} - \frac{3}{128} a^{9} + \frac{1}{32} a^{8} + \frac{1}{32} a^{7} + \frac{1}{16} a^{6} + \frac{1}{64} a^{5} + \frac{45}{128} a^{4} + \frac{3}{32} a^{3} + \frac{5}{16} a^{2} - \frac{15}{128} a + \frac{15}{64}$, $\frac{1}{512} a^{13} - \frac{1}{512} a^{12} - \frac{1}{128} a^{11} - \frac{7}{512} a^{10} + \frac{3}{512} a^{9} - \frac{3}{64} a^{8} - \frac{9}{128} a^{7} + \frac{1}{256} a^{6} - \frac{77}{512} a^{5} - \frac{129}{512} a^{4} + \frac{17}{64} a^{3} + \frac{69}{512} a^{2} + \frac{105}{512} a - \frac{19}{256}$, $\frac{1}{6144} a^{14} - \frac{1}{3072} a^{13} + \frac{7}{2048} a^{12} - \frac{19}{6144} a^{11} - \frac{1}{1024} a^{10} - \frac{115}{6144} a^{9} - \frac{67}{1536} a^{8} - \frac{205}{3072} a^{7} - \frac{239}{6144} a^{6} + \frac{77}{512} a^{5} - \frac{1087}{6144} a^{4} - \frac{2035}{6144} a^{3} - \frac{683}{1536} a^{2} - \frac{1159}{6144} a - \frac{533}{3072}$, $\frac{1}{5895401477859932624122651956109369344} a^{15} - \frac{10261682967127561981365980880173}{1965133825953310874707550652036456448} a^{14} - \frac{531737591367669839592403525534993}{5895401477859932624122651956109369344} a^{13} + \frac{1282827584026230386790084301132397}{1473850369464983156030662989027342336} a^{12} + \frac{40416137172780044411473227289220761}{5895401477859932624122651956109369344} a^{11} - \frac{9552317992832876595461266606501381}{5895401477859932624122651956109369344} a^{10} + \frac{17306834932848677722786570724601825}{1965133825953310874707550652036456448} a^{9} + \frac{68430348421375682057784592223965585}{2947700738929966312061325978054684672} a^{8} + \frac{225724622195688315477930382802838001}{1965133825953310874707550652036456448} a^{7} + \frac{472827886542757520675143717774656359}{5895401477859932624122651956109369344} a^{6} - \frac{1045491461014424330039270563820177371}{5895401477859932624122651956109369344} a^{5} - \frac{26135687937108003829912063924416763}{245641728244163859338443831504557056} a^{4} + \frac{2568368288086625855298477804496013843}{5895401477859932624122651956109369344} a^{3} - \frac{699645812994521796121155718070557979}{5895401477859932624122651956109369344} a^{2} - \frac{184116443877553100243632774158878653}{1965133825953310874707550652036456448} a - \frac{1326217643960522018154450096913548295}{2947700738929966312061325978054684672}$

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:  \( 466874712013 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1123:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 55 conjugacy class representatives for t16n1123 are not computed
Character table for t16n1123 is not computed

Intermediate fields

\(\Q(\sqrt{1649}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{97}) \), \(\Q(\sqrt{17}, \sqrt{97})\), 8.8.184851351960025.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.3$x^{4} + 2 x^{2} + 4 x + 4$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
$5$5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
97Data not computed