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Properties

Label 16.16.3300567822...4401.1
Degree $16$
Signature $[16, 0]$
Discriminant $61^{14}\cdot 109^{14}$
Root discriminant $2212.65$
Ramified primes $61, 109$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_8.C_4$ (as 16T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-20956414760060291, -19832907564347600, 27327366661749576, 18952504528520578, 2304642596194550, -306922160402116, -49913000623296, 1889662834916, 386060968586, -5433619055, -1451756694, 6908002, 2824784, -2359, -2706, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 2706*x^14 - 2359*x^13 + 2824784*x^12 + 6908002*x^11 - 1451756694*x^10 - 5433619055*x^9 + 386060968586*x^8 + 1889662834916*x^7 - 49913000623296*x^6 - 306922160402116*x^5 + 2304642596194550*x^4 + 18952504528520578*x^3 + 27327366661749576*x^2 - 19832907564347600*x - 20956414760060291)
 
gp: K = bnfinit(x^16 - x^15 - 2706*x^14 - 2359*x^13 + 2824784*x^12 + 6908002*x^11 - 1451756694*x^10 - 5433619055*x^9 + 386060968586*x^8 + 1889662834916*x^7 - 49913000623296*x^6 - 306922160402116*x^5 + 2304642596194550*x^4 + 18952504528520578*x^3 + 27327366661749576*x^2 - 19832907564347600*x - 20956414760060291, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 2706 x^{14} - 2359 x^{13} + 2824784 x^{12} + 6908002 x^{11} - 1451756694 x^{10} - 5433619055 x^{9} + 386060968586 x^{8} + 1889662834916 x^{7} - 49913000623296 x^{6} - 306922160402116 x^{5} + 2304642596194550 x^{4} + 18952504528520578 x^{3} + 27327366661749576 x^{2} - 19832907564347600 x - 20956414760060291 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(330056782202198660511172839698929216685213271046904401=61^{14}\cdot 109^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2212.65$
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:  $61, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{25} a^{14} - \frac{2}{25} a^{13} - \frac{2}{25} a^{12} - \frac{1}{25} a^{11} + \frac{1}{25} a^{10} + \frac{4}{25} a^{9} + \frac{4}{25} a^{8} + \frac{9}{25} a^{7} - \frac{2}{5} a^{6} - \frac{1}{25} a^{5} + \frac{1}{5} a^{4} + \frac{7}{25} a^{3} - \frac{7}{25} a^{2} - \frac{11}{25} a + \frac{3}{25}$, $\frac{1}{1245900689636813278929519382412396661250454851264128447047958597258509861693107481612582760016139114605198215606575} a^{15} - \frac{19731490752685173226821871341213240623994246982086510652386003899473483178985085389887361929686293864253119277981}{1245900689636813278929519382412396661250454851264128447047958597258509861693107481612582760016139114605198215606575} a^{14} + \frac{21149100906964028717269278593317791597912611216478617240782820522887299772306173102932635723612555893332650891526}{1245900689636813278929519382412396661250454851264128447047958597258509861693107481612582760016139114605198215606575} a^{13} + \frac{38594519097097267695915324296056098055056377840087013644895218875132449360456012247877280424535103398516912384137}{1245900689636813278929519382412396661250454851264128447047958597258509861693107481612582760016139114605198215606575} a^{12} - \frac{24319499636386400908995840874962839689114395460790828814933736750312178576882451932869891468451394233633785959167}{249180137927362655785903876482479332250090970252825689409591719451701972338621496322516552003227822921039643121315} a^{11} + \frac{4846257872125673247891153101562503792610103048153071350500434262593921159777881737243033980243067644343236963259}{249180137927362655785903876482479332250090970252825689409591719451701972338621496322516552003227822921039643121315} a^{10} - \frac{443792591384536337234664547075762037084008781342737263689298772441616305090399322291922517586590416673780245882672}{1245900689636813278929519382412396661250454851264128447047958597258509861693107481612582760016139114605198215606575} a^{9} + \frac{276869477495052412253692096666432468329144898714856531058302396860278659267936143749296546755098128331708920083963}{1245900689636813278929519382412396661250454851264128447047958597258509861693107481612582760016139114605198215606575} a^{8} + \frac{541943062623995073163556005819226868492308356448769066024172436015448952262281397665910548158603061719453609448999}{1245900689636813278929519382412396661250454851264128447047958597258509861693107481612582760016139114605198215606575} a^{7} - \frac{312793318136272813760542820440826979844253335732434558123706488610003944776918315916607509229971606837863444939361}{1245900689636813278929519382412396661250454851264128447047958597258509861693107481612582760016139114605198215606575} a^{6} + \frac{496371723656694859729503012263172864223344209377056976257956386572534559756438279867206255746804716571342811005299}{1245900689636813278929519382412396661250454851264128447047958597258509861693107481612582760016139114605198215606575} a^{5} + \frac{13781285049484439835701789941481024050721143819347253883810844019060651604703435069179464984770571957502158707807}{1245900689636813278929519382412396661250454851264128447047958597258509861693107481612582760016139114605198215606575} a^{4} - \frac{116771532762587533793161577834881060687215056332490755020110843739690280032105992969491440909588045559093345517208}{249180137927362655785903876482479332250090970252825689409591719451701972338621496322516552003227822921039643121315} a^{3} + \frac{396915358760768136274199648816722024630835025445518032446668799611239710703659706534254496451447414027415566142222}{1245900689636813278929519382412396661250454851264128447047958597258509861693107481612582760016139114605198215606575} a^{2} + \frac{43235378309041745857568902575184548152120984194690586514171376719414305929264807112640516891739287421987126338497}{1245900689636813278929519382412396661250454851264128447047958597258509861693107481612582760016139114605198215606575} a - \frac{227379716802227281675077029664816811506669310613441700922250092509663373487316268356044292383040653604008407567507}{1245900689636813278929519382412396661250454851264128447047958597258509861693107481612582760016139114605198215606575}$

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:  $15$
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:  \( 7901580094650000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8.C_4$ (as 16T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_8.C_4$
Character table for $C_8.C_4$

Intermediate fields

\(\Q(\sqrt{61}) \), \(\Q(\sqrt{109}) \), \(\Q(\sqrt{6649}) \), \(\Q(\sqrt{61}, \sqrt{109})\), 8.8.86404825551402914547601.1

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

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$61$61.8.7.2$x^{8} - 244$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
61.8.7.2$x^{8} - 244$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$109$109.8.7.2$x^{8} - 3924$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
109.8.7.2$x^{8} - 3924$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$