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Properties

Label 16.4.32859742481...0625.1
Degree $16$
Signature $[4, 6]$
Discriminant $5^{12}\cdot 10771^{4}$
Root discriminant $34.06$
Ramified primes $5, 10771$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1496

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1091, 16689, 9270, 17040, 2683, 5111, -174, 1335, -431, -73, -23, -38, -48, 13, -3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 3*x^14 + 13*x^13 - 48*x^12 - 38*x^11 - 23*x^10 - 73*x^9 - 431*x^8 + 1335*x^7 - 174*x^6 + 5111*x^5 + 2683*x^4 + 17040*x^3 + 9270*x^2 + 16689*x + 1091)
 
gp: K = bnfinit(x^16 - 2*x^15 - 3*x^14 + 13*x^13 - 48*x^12 - 38*x^11 - 23*x^10 - 73*x^9 - 431*x^8 + 1335*x^7 - 174*x^6 + 5111*x^5 + 2683*x^4 + 17040*x^3 + 9270*x^2 + 16689*x + 1091, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 3 x^{14} + 13 x^{13} - 48 x^{12} - 38 x^{11} - 23 x^{10} - 73 x^{9} - 431 x^{8} + 1335 x^{7} - 174 x^{6} + 5111 x^{5} + 2683 x^{4} + 17040 x^{3} + 9270 x^{2} + 16689 x + 1091 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3285974248179316650390625=5^{12}\cdot 10771^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.06$
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:  $5, 10771$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{606297934840068907024916897537539} a^{15} + \frac{125705230887712316334036729204961}{606297934840068907024916897537539} a^{14} + \frac{24200108463839241196468701638463}{55117994076369900638628808867049} a^{13} - \frac{217625084829764557101685838536753}{606297934840068907024916897537539} a^{12} + \frac{56068156592850776050482427873487}{606297934840068907024916897537539} a^{11} - \frac{234771513371514429846364370642341}{606297934840068907024916897537539} a^{10} + \frac{77305509190330229176790780015117}{606297934840068907024916897537539} a^{9} + \frac{23328664360069567744749452095483}{606297934840068907024916897537539} a^{8} + \frac{268903587075670983248371750822260}{606297934840068907024916897537539} a^{7} - \frac{80631797144194510625941407688436}{606297934840068907024916897537539} a^{6} + \frac{149467544978347302509621066005319}{606297934840068907024916897537539} a^{5} + \frac{98536485797500568687993571980197}{606297934840068907024916897537539} a^{4} - \frac{104486600092194984316944636565513}{606297934840068907024916897537539} a^{3} - \frac{212283289618180183989841184201143}{606297934840068907024916897537539} a^{2} + \frac{25338436664713964530360464589842}{55117994076369900638628808867049} a - \frac{300330195709747740597751012611936}{606297934840068907024916897537539}$

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:  $9$
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:  \( 877012.00628 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1496:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2304
The 40 conjugacy class representatives for t16n1496
Character table for t16n1496 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.1346375.1, 8.6.168296875.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 siblings: data not computed
Degree 36 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
5Data not computed
10771Data not computed