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Properties

Label 16.8.15712755557...5489.2
Degree $16$
Signature $[8, 4]$
Discriminant $23^{10}\cdot 41^{14}$
Root discriminant $182.92$
Ramified primes $23, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1194

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5481397, 30809257, -23723409, 2712470, 7693019, -6774158, 2103327, 50169, -245805, 89059, -20194, 5874, -1513, 174, 3, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 3*x^14 + 174*x^13 - 1513*x^12 + 5874*x^11 - 20194*x^10 + 89059*x^9 - 245805*x^8 + 50169*x^7 + 2103327*x^6 - 6774158*x^5 + 7693019*x^4 + 2712470*x^3 - 23723409*x^2 + 30809257*x - 5481397)
 
gp: K = bnfinit(x^16 - 6*x^15 + 3*x^14 + 174*x^13 - 1513*x^12 + 5874*x^11 - 20194*x^10 + 89059*x^9 - 245805*x^8 + 50169*x^7 + 2103327*x^6 - 6774158*x^5 + 7693019*x^4 + 2712470*x^3 - 23723409*x^2 + 30809257*x - 5481397, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 3 x^{14} + 174 x^{13} - 1513 x^{12} + 5874 x^{11} - 20194 x^{10} + 89059 x^{9} - 245805 x^{8} + 50169 x^{7} + 2103327 x^{6} - 6774158 x^{5} + 7693019 x^{4} + 2712470 x^{3} - 23723409 x^{2} + 30809257 x - 5481397 \)

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:  \(1571275555715210001755383712793895489=23^{10}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $182.92$
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:  $23, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{12} + \frac{1}{8} a^{11} - \frac{1}{4} a^{10} - \frac{1}{8} a^{7} - \frac{3}{8} a^{6} + \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{8} a^{3} - \frac{3}{8} a^{2} - \frac{1}{8} a + \frac{1}{8}$, $\frac{1}{8} a^{13} + \frac{1}{8} a^{11} - \frac{1}{4} a^{10} - \frac{1}{8} a^{8} + \frac{1}{4} a^{7} + \frac{1}{8} a^{6} + \frac{3}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{8}$, $\frac{1}{184} a^{14} - \frac{3}{92} a^{12} + \frac{1}{8} a^{11} - \frac{5}{92} a^{10} - \frac{5}{184} a^{9} + \frac{7}{92} a^{8} + \frac{5}{23} a^{7} + \frac{8}{23} a^{6} + \frac{19}{46} a^{5} - \frac{5}{184} a^{4} - \frac{15}{184} a^{3} - \frac{89}{184} a^{2} - \frac{33}{92} a - \frac{91}{184}$, $\frac{1}{3414973512348524383628564964004325587790556877976} a^{15} - \frac{1196060421413852236668083186224885633449292949}{853743378087131095907141241001081396947639219494} a^{14} - \frac{119600472757961969088826959594848775847283344055}{3414973512348524383628564964004325587790556877976} a^{13} - \frac{21993352745190350125423592042189966074092893427}{853743378087131095907141241001081396947639219494} a^{12} - \frac{55614513194472857274903283507966520473402117479}{1707486756174262191814282482002162793895278438988} a^{11} + \frac{210886480374859697431876469625108299323318189823}{3414973512348524383628564964004325587790556877976} a^{10} + \frac{17710391385495417738031721467767486093341232687}{74238554616272269209316629652267947560664279956} a^{9} + \frac{61763610730392124904557531655690156518904454541}{3414973512348524383628564964004325587790556877976} a^{8} - \frac{1280346725205422400349333608885018212700100370379}{3414973512348524383628564964004325587790556877976} a^{7} - \frac{174081692300489813922388782393997975685585706486}{426871689043565547953570620500540698473819609747} a^{6} + \frac{216300155812661920485614956897563591979954563567}{1707486756174262191814282482002162793895278438988} a^{5} + \frac{171879073805953446616553704793862470193727263341}{426871689043565547953570620500540698473819609747} a^{4} - \frac{85870718667127166340962695587680656430104491641}{426871689043565547953570620500540698473819609747} a^{3} - \frac{902938654757995791465208344977043169088976341491}{3414973512348524383628564964004325587790556877976} a^{2} - \frac{208171931532810037814544421086496251064833384127}{1707486756174262191814282482002162793895278438988} a + \frac{405073458975069966610839780221316955921885264031}{1707486756174262191814282482002162793895278438988}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 789989354667 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1194:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 40 conjugacy class representatives for t16n1194
Character table for t16n1194 is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.54500230757132921.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$23$23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
41Data not computed