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Properties

Label 16.8.28659799503...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{28}\cdot 5^{14}\cdot 41^{2}\cdot 101^{4}$
Root discriminant $69.36$
Ramified primes $2, 5, 41, 101$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1161

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![857261, -2877768, -646350, 2128630, 792520, -1427454, -101178, 277040, -41115, -28590, -2378, 704, 1030, 50, -60, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 60*x^14 + 50*x^13 + 1030*x^12 + 704*x^11 - 2378*x^10 - 28590*x^9 - 41115*x^8 + 277040*x^7 - 101178*x^6 - 1427454*x^5 + 792520*x^4 + 2128630*x^3 - 646350*x^2 - 2877768*x + 857261)
 
gp: K = bnfinit(x^16 - 2*x^15 - 60*x^14 + 50*x^13 + 1030*x^12 + 704*x^11 - 2378*x^10 - 28590*x^9 - 41115*x^8 + 277040*x^7 - 101178*x^6 - 1427454*x^5 + 792520*x^4 + 2128630*x^3 - 646350*x^2 - 2877768*x + 857261, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 60 x^{14} + 50 x^{13} + 1030 x^{12} + 704 x^{11} - 2378 x^{10} - 28590 x^{9} - 41115 x^{8} + 277040 x^{7} - 101178 x^{6} - 1427454 x^{5} + 792520 x^{4} + 2128630 x^{3} - 646350 x^{2} - 2877768 x + 857261 \)

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:  \(286597995038310400000000000000=2^{28}\cdot 5^{14}\cdot 41^{2}\cdot 101^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.36$
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:  $2, 5, 41, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{5} - \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{6} - \frac{1}{5} a$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{7} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{305} a^{14} - \frac{23}{305} a^{13} - \frac{18}{305} a^{12} + \frac{18}{305} a^{11} - \frac{11}{305} a^{10} + \frac{12}{305} a^{9} + \frac{5}{61} a^{8} - \frac{25}{61} a^{7} + \frac{78}{305} a^{6} - \frac{93}{305} a^{5} - \frac{33}{305} a^{4} - \frac{133}{305} a^{3} + \frac{152}{305} a^{2} + \frac{47}{305} a - \frac{24}{61}$, $\frac{1}{6602107990525976295628360977431342174901653245} a^{15} - \frac{1451638162633220077482848109723901488266952}{1320421598105195259125672195486268434980330649} a^{14} - \frac{294254410954109072567157679472250789418309208}{6602107990525976295628360977431342174901653245} a^{13} + \frac{333562053769370715874676033979176388487962119}{6602107990525976295628360977431342174901653245} a^{12} + \frac{102154426600396982280579023846815958709838807}{6602107990525976295628360977431342174901653245} a^{11} + \frac{38405926817295418385012080223719218053215833}{1320421598105195259125672195486268434980330649} a^{10} - \frac{130972860647924913560040801434980833399366478}{6602107990525976295628360977431342174901653245} a^{9} + \frac{109702976645463811182653642259851160687207215}{1320421598105195259125672195486268434980330649} a^{8} + \frac{2360716728595776146776121430992133353118180513}{6602107990525976295628360977431342174901653245} a^{7} - \frac{1002563306932437032727425317104214395460667309}{6602107990525976295628360977431342174901653245} a^{6} - \frac{233086878364485153958895849401913220612044627}{6602107990525976295628360977431342174901653245} a^{5} - \frac{2009364988586131363607222591660314431399538949}{6602107990525976295628360977431342174901653245} a^{4} + \frac{2620779950893335984697547402449706048947093682}{6602107990525976295628360977431342174901653245} a^{3} + \frac{1234606216723600226900821079365251530442572243}{6602107990525976295628360977431342174901653245} a^{2} + \frac{458669212830810154654310851538757691896182589}{1320421598105195259125672195486268434980330649} a + \frac{11502961089913288244208371458370420667220186}{108231278533212726157841983236579379916420545}$

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:  \( 350838597.911 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1161:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1161
Character table for t16n1161 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.8.6464000000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
5Data not computed
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.2.2$x^{4} - 41 x^{2} + 20172$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
101Data not computed