↑

Properties

Label 16.8.65139256847...3216.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{20}\cdot 13^{6}\cdot 17^{12}\cdot 47^{2}$
Root discriminant $84.31$
Ramified primes $2, 13, 17, 47$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T781

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1801, 76790, -401358, 738828, -744437, 457456, -91951, -97134, 93214, -33624, 2973, 1842, -765, 118, -9, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 9*x^14 + 118*x^13 - 765*x^12 + 1842*x^11 + 2973*x^10 - 33624*x^9 + 93214*x^8 - 97134*x^7 - 91951*x^6 + 457456*x^5 - 744437*x^4 + 738828*x^3 - 401358*x^2 + 76790*x + 1801)
 
gp: K = bnfinit(x^16 - 6*x^15 - 9*x^14 + 118*x^13 - 765*x^12 + 1842*x^11 + 2973*x^10 - 33624*x^9 + 93214*x^8 - 97134*x^7 - 91951*x^6 + 457456*x^5 - 744437*x^4 + 738828*x^3 - 401358*x^2 + 76790*x + 1801, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 9 x^{14} + 118 x^{13} - 765 x^{12} + 1842 x^{11} + 2973 x^{10} - 33624 x^{9} + 93214 x^{8} - 97134 x^{7} - 91951 x^{6} + 457456 x^{5} - 744437 x^{4} + 738828 x^{3} - 401358 x^{2} + 76790 x + 1801 \)

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:  \(6513925684721600930241986953216=2^{20}\cdot 13^{6}\cdot 17^{12}\cdot 47^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $84.31$
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:  $2, 13, 17, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{78} a^{12} - \frac{35}{78} a^{11} + \frac{1}{3} a^{10} + \frac{4}{13} a^{9} + \frac{1}{39} a^{8} - \frac{23}{78} a^{7} + \frac{4}{39} a^{6} - \frac{16}{39} a^{5} + \frac{1}{39} a^{4} + \frac{5}{78} a^{3} - \frac{16}{39} a^{2} - \frac{4}{13} a - \frac{31}{78}$, $\frac{1}{78} a^{13} - \frac{29}{78} a^{11} - \frac{1}{39} a^{10} - \frac{8}{39} a^{9} - \frac{31}{78} a^{8} - \frac{17}{78} a^{7} + \frac{7}{39} a^{6} - \frac{1}{3} a^{5} - \frac{1}{26} a^{4} - \frac{1}{6} a^{3} + \frac{1}{3} a^{2} - \frac{1}{6} a + \frac{7}{78}$, $\frac{1}{1014} a^{14} - \frac{1}{1014} a^{13} - \frac{2}{507} a^{12} + \frac{44}{507} a^{11} + \frac{80}{169} a^{10} - \frac{3}{26} a^{9} - \frac{124}{507} a^{8} + \frac{235}{507} a^{7} + \frac{236}{507} a^{6} + \frac{1}{338} a^{5} - \frac{175}{507} a^{4} - \frac{113}{507} a^{3} - \frac{59}{1014} a^{2} + \frac{139}{507} a + \frac{155}{507}$, $\frac{1}{4027481041858500629959005830699919678} a^{15} - \frac{637043939412681464570268589895572}{2013740520929250314979502915349959839} a^{14} - \frac{10932020321613426259489964440344449}{4027481041858500629959005830699919678} a^{13} + \frac{9183522221125738614683599471799537}{2013740520929250314979502915349959839} a^{12} - \frac{294266574360521958189692678261977973}{2013740520929250314979502915349959839} a^{11} - \frac{148369175349686821561460839555964089}{4027481041858500629959005830699919678} a^{10} - \frac{616856833336824760738556909543985225}{1342493680619500209986335276899973226} a^{9} + \frac{654748400730898950867680320589402444}{2013740520929250314979502915349959839} a^{8} + \frac{2382098169172957699644766028513971}{154903116994557716536884839642304603} a^{7} + \frac{27762768763974136425177134166133415}{211972686413605296313631885826311562} a^{6} + \frac{204245492518010084358896729767586683}{4027481041858500629959005830699919678} a^{5} + \frac{909201487053650583413959095570029428}{2013740520929250314979502915349959839} a^{4} - \frac{663975909790509757299096021567677215}{4027481041858500629959005830699919678} a^{3} - \frac{2688100663935039153899794510984679}{7230666143372532549298035602692854} a^{2} + \frac{398865651388386422129418687676065776}{2013740520929250314979502915349959839} a + \frac{789069903249211472561773152584845484}{2013740520929250314979502915349959839}$

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:  \( 2178288210.02 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T781:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n781 are not computed
Character table for t16n781 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 4.4.510952.2, 4.4.30056.2, 8.8.261071946304.3

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.20.7$x^{8} + 72 x^{4} + 144$$4$$2$$20$$Q_8:C_2$$[2, 3, 7/2]^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$17$17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
47Data not computed