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Properties

Label 16.0.26436931289...625.19
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 101^{8}$
Root discriminant $33.60$
Ramified primes $5, 101$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T172)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![101761, -166837, 242291, -202915, 134228, -86449, 46977, -18634, 8437, -3622, 963, -260, 125, -23, 0, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 23*x^13 + 125*x^12 - 260*x^11 + 963*x^10 - 3622*x^9 + 8437*x^8 - 18634*x^7 + 46977*x^6 - 86449*x^5 + 134228*x^4 - 202915*x^3 + 242291*x^2 - 166837*x + 101761)
 
gp: K = bnfinit(x^16 - 2*x^15 - 23*x^13 + 125*x^12 - 260*x^11 + 963*x^10 - 3622*x^9 + 8437*x^8 - 18634*x^7 + 46977*x^6 - 86449*x^5 + 134228*x^4 - 202915*x^3 + 242291*x^2 - 166837*x + 101761, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 23 x^{13} + 125 x^{12} - 260 x^{11} + 963 x^{10} - 3622 x^{9} + 8437 x^{8} - 18634 x^{7} + 46977 x^{6} - 86449 x^{5} + 134228 x^{4} - 202915 x^{3} + 242291 x^{2} - 166837 x + 101761 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2643693128974804931640625=5^{12}\cdot 101^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.60$
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:  $5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{2}{5} a^{10} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{11} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{2} + \frac{1}{5}$, $\frac{1}{355} a^{14} + \frac{28}{355} a^{13} - \frac{31}{355} a^{12} - \frac{136}{355} a^{11} - \frac{20}{71} a^{10} - \frac{108}{355} a^{9} + \frac{24}{71} a^{8} - \frac{171}{355} a^{7} + \frac{35}{71} a^{6} + \frac{18}{355} a^{5} - \frac{112}{355} a^{4} + \frac{18}{71} a^{3} - \frac{33}{355} a^{2} - \frac{141}{355} a - \frac{17}{71}$, $\frac{1}{19629152629340496961820125756788155} a^{15} + \frac{27004569283930014685718290193733}{19629152629340496961820125756788155} a^{14} + \frac{1145747080686274869372040998162}{61533393822384003015110112090245} a^{13} - \frac{256047049631324880175426329693016}{19629152629340496961820125756788155} a^{12} + \frac{8543393961175461601353172501638723}{19629152629340496961820125756788155} a^{11} + \frac{8791978494431540049203739712768787}{19629152629340496961820125756788155} a^{10} - \frac{3666753175196931742891892113549089}{19629152629340496961820125756788155} a^{9} + \frac{6049893640702169498383272291745558}{19629152629340496961820125756788155} a^{8} - \frac{388367896795001519523886947013103}{1784468420849136087438193250617105} a^{7} - \frac{640507691787765486975452613737849}{1784468420849136087438193250617105} a^{6} + \frac{8359665816310658309424820581632562}{19629152629340496961820125756788155} a^{5} + \frac{24819010538910715517957582014854}{61533393822384003015110112090245} a^{4} + \frac{2647647829047505378121270340565947}{19629152629340496961820125756788155} a^{3} + \frac{74782955701257176021500993972981}{276466938441415450166480644461805} a^{2} - \frac{1310077672250090783036961474892556}{3925830525868099392364025151357631} a - \frac{21444604387100704619201943591179}{61533393822384003015110112090245}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{847103953440099303420734080}{55293387688283090033296128892361} a^{15} + \frac{2062405814058521806287496858}{276466938441415450166480644461805} a^{14} - \frac{44600087558698585260378842}{866667518625126803029719888595} a^{13} - \frac{106790104746250753487493650424}{276466938441415450166480644461805} a^{12} + \frac{285678760312933872993306576489}{276466938441415450166480644461805} a^{11} - \frac{37061123937471433418695913363}{276466938441415450166480644461805} a^{10} + \frac{1974835453509549645682948987461}{276466938441415450166480644461805} a^{9} - \frac{6407874060135893950706020002169}{276466938441415450166480644461805} a^{8} + \frac{523506156998150076932794859812}{25133358040128677287861876769255} a^{7} - \frac{1474058883572718267926658003483}{25133358040128677287861876769255} a^{6} + \frac{49389875978946373892024463160249}{276466938441415450166480644461805} a^{5} - \frac{51355883729403195616142323588}{866667518625126803029719888595} a^{4} + \frac{21170874246333923980560771853044}{276466938441415450166480644461805} a^{3} + \frac{4949335027148251004306975153831}{276466938441415450166480644461805} a^{2} - \frac{176743332429582742058971029112837}{276466938441415450166480644461805} a + \frac{823858200972408222714458105348}{866667518625126803029719888595} \) (order $10$)
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:  \( 430477.21164 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T172):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.4.2525.1, 4.0.12625.1, 8.4.325188753125.4, 8.4.325188753125.5, 8.0.159390625.1

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

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$101$$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.4.3.1$x^{4} - 101$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.1$x^{4} - 101$$4$$1$$3$$C_4$$[\ ]_{4}$