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Properties

Label 16.0.57704698464...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{8}\cdot 29^{6}\cdot 41^{8}\cdot 83^{4}$
Root discriminant $305.55$
Ramified primes $2, 5, 29, 41, 83$
Class number $32$ (GRH)
Class group $[2, 2, 2, 4]$ (GRH)
Galois group $(C_2^2\times D_4).C_2^3$ (as 16T600)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13646747702389521841, 0, -338349116588169963, 0, 3881042372358443, 0, -25577396821670, 0, 105599243341, 0, -290198410, 0, 595168, 0, -982, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 982*x^14 + 595168*x^12 - 290198410*x^10 + 105599243341*x^8 - 25577396821670*x^6 + 3881042372358443*x^4 - 338349116588169963*x^2 + 13646747702389521841)
 
gp: K = bnfinit(x^16 - 982*x^14 + 595168*x^12 - 290198410*x^10 + 105599243341*x^8 - 25577396821670*x^6 + 3881042372358443*x^4 - 338349116588169963*x^2 + 13646747702389521841, 1)
 

Normalized defining polynomial

\( x^{16} - 982 x^{14} + 595168 x^{12} - 290198410 x^{10} + 105599243341 x^{8} - 25577396821670 x^{6} + 3881042372358443 x^{4} - 338349116588169963 x^{2} + 13646747702389521841 \)

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:  \(5770469846478308612464977355801600000000=2^{16}\cdot 5^{8}\cdot 29^{6}\cdot 41^{8}\cdot 83^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $305.55$
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:  $2, 5, 29, 41, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{41} a^{8} + \frac{2}{41} a^{6} + \frac{12}{41} a^{4}$, $\frac{1}{41} a^{9} + \frac{2}{41} a^{7} + \frac{12}{41} a^{5}$, $\frac{1}{3403} a^{10} + \frac{14}{3403} a^{8} + \frac{1635}{3403} a^{6} + \frac{964}{3403} a^{4} + \frac{41}{83} a^{2}$, $\frac{1}{3403} a^{11} + \frac{14}{3403} a^{9} + \frac{1635}{3403} a^{7} + \frac{964}{3403} a^{5} + \frac{41}{83} a^{3}$, $\frac{1}{57902045} a^{12} - \frac{7788}{57902045} a^{10} - \frac{347463}{57902045} a^{8} - \frac{9416}{282449} a^{6} + \frac{660638}{1412245} a^{4} - \frac{181}{415} a^{2} + \frac{1}{5}$, $\frac{1}{57902045} a^{13} - \frac{7788}{57902045} a^{11} - \frac{347463}{57902045} a^{9} - \frac{9416}{282449} a^{7} + \frac{660638}{1412245} a^{5} - \frac{181}{415} a^{3} + \frac{1}{5} a$, $\frac{1}{402555814120266289501852018314608722470877257406795} a^{14} + \frac{388954719925269591016826294399889488287523}{402555814120266289501852018314608722470877257406795} a^{12} - \frac{21603713865486844762906651008244356415399911581}{402555814120266289501852018314608722470877257406795} a^{10} + \frac{53346974845737307097204651568879795281699855432}{9818434490738202182972000446697773718801884326995} a^{8} + \frac{506998836277978525780975816712914104264751621868}{9818434490738202182972000446697773718801884326995} a^{6} - \frac{255122658289699891100425700612908970919618758}{577045811974034803583426414733927341686857733} a^{4} + \frac{9862369486881329046426042422482015629435369}{34761795902050289372495567152646225402822755} a^{2} - \frac{133108492716143260615449568089163256452}{352242908407898602374127971796145646365}$, $\frac{1}{4428113955322929184520372201460695947179649831474745} a^{15} + \frac{2858734616149077068003010631091675929883325}{885622791064585836904074440292139189435929966294949} a^{13} + \frac{106695122449800363253099988238742332301338312573}{4428113955322929184520372201460695947179649831474745} a^{11} - \frac{941601077789079493825071081749225524249553195514}{108002779398120224012692004913675510906820727596945} a^{9} + \frac{7371271625872240967010185480077353649658549760698}{108002779398120224012692004913675510906820727596945} a^{7} + \frac{2565817467323350097927512255169060631540311887}{31737519658571914197088452810366003792777175315} a^{5} + \frac{2776602188122489370673057748750388099060805}{76475950984510636619490247735821695886210061} a^{3} - \frac{1753425871392476831536438238351433229731}{3874671992486884626115407689757602110015} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}$, which has order $32$ (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:  \( -1 \) (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:  \( 592634255788 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^2\times D_4).C_2^3$ (as 16T600):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 40 conjugacy class representatives for $(C_2^2\times D_4).C_2^3$
Character table for $(C_2^2\times D_4).C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 4.0.29725.2, 4.0.1025.1, 8.0.883575625.2

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.8.6$x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
2.8.8.6$x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
$41$41.4.2.2$x^{4} - 41 x^{2} + 20172$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
41.4.3.4$x^{4} + 8856$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.4$x^{4} + 8856$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
$83$83.4.2.2$x^{4} - 83 x^{2} + 13778$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
83.4.2.2$x^{4} - 83 x^{2} + 13778$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
83.4.0.1$x^{4} - x + 22$$1$$4$$0$$C_4$$[\ ]^{4}$
83.4.0.1$x^{4} - x + 22$$1$$4$$0$$C_4$$[\ ]^{4}$