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Properties

Label 16.16.1006774053...1681.2
Degree $16$
Signature $[16, 0]$
Discriminant $41^{14}\cdot 61^{12}$
Root discriminant $562.58$
Ramified primes $41, 61$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $Q_{16}$ (as 16T14)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6904626882871, -479019525519, 5148673520590, -574159811286, -1022348955259, 242254120296, 49055322017, -20027658582, 840421984, 308362799, -26837605, -1883696, 224913, 5051, -787, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 787*x^14 + 5051*x^13 + 224913*x^12 - 1883696*x^11 - 26837605*x^10 + 308362799*x^9 + 840421984*x^8 - 20027658582*x^7 + 49055322017*x^6 + 242254120296*x^5 - 1022348955259*x^4 - 574159811286*x^3 + 5148673520590*x^2 - 479019525519*x - 6904626882871)
 
gp: K = bnfinit(x^16 - 5*x^15 - 787*x^14 + 5051*x^13 + 224913*x^12 - 1883696*x^11 - 26837605*x^10 + 308362799*x^9 + 840421984*x^8 - 20027658582*x^7 + 49055322017*x^6 + 242254120296*x^5 - 1022348955259*x^4 - 574159811286*x^3 + 5148673520590*x^2 - 479019525519*x - 6904626882871, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 787 x^{14} + 5051 x^{13} + 224913 x^{12} - 1883696 x^{11} - 26837605 x^{10} + 308362799 x^{9} + 840421984 x^{8} - 20027658582 x^{7} + 49055322017 x^{6} + 242254120296 x^{5} - 1022348955259 x^{4} - 574159811286 x^{3} + 5148673520590 x^{2} - 479019525519 x - 6904626882871 \)

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:  \(100677405300724224532172039540740068462931681=41^{14}\cdot 61^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $562.58$
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:  $41, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{12} - \frac{1}{10} a^{10} - \frac{2}{5} a^{8} + \frac{1}{5} a^{6} + \frac{3}{10} a^{5} - \frac{1}{2} a^{4} - \frac{1}{10} a^{3} + \frac{1}{5} a^{2} - \frac{3}{10} a - \frac{1}{10}$, $\frac{1}{10} a^{13} - \frac{1}{10} a^{11} - \frac{2}{5} a^{9} + \frac{1}{5} a^{7} + \frac{3}{10} a^{6} - \frac{1}{2} a^{5} - \frac{1}{10} a^{4} + \frac{1}{5} a^{3} - \frac{3}{10} a^{2} - \frac{1}{10} a$, $\frac{1}{10} a^{14} - \frac{1}{2} a^{9} + \frac{3}{10} a^{8} + \frac{3}{10} a^{7} + \frac{1}{5} a^{6} - \frac{3}{10} a^{5} - \frac{3}{10} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{3}{10} a + \frac{2}{5}$, $\frac{1}{18574329697970454047619741411467946272567463190941482972288900237374314984030776412650} a^{15} - \frac{147511207286384914007845564629144232619640641195844205908875459849947232631857333023}{9287164848985227023809870705733973136283731595470741486144450118687157492015388206325} a^{14} + \frac{91374815255657040125856092055664489433518709948679821115519675025716706981913191969}{18574329697970454047619741411467946272567463190941482972288900237374314984030776412650} a^{13} - \frac{155793572001732707552043815360986498197074058383643971379749233858702922930935429324}{9287164848985227023809870705733973136283731595470741486144450118687157492015388206325} a^{12} + \frac{1251143899714789804024114363836285916650947673060261331764711008305817974463952818601}{18574329697970454047619741411467946272567463190941482972288900237374314984030776412650} a^{11} + \frac{2243682306825091606347982533139525424882593325775962709155698280614519886238558637083}{18574329697970454047619741411467946272567463190941482972288900237374314984030776412650} a^{10} - \frac{7582712530493898680399784953069706392882195638095861412224429754438967759896111699003}{18574329697970454047619741411467946272567463190941482972288900237374314984030776412650} a^{9} + \frac{7892708489058433510346246535107182431496391769357505186689998593717131767612517998677}{18574329697970454047619741411467946272567463190941482972288900237374314984030776412650} a^{8} + \frac{4317540642703529153317475069459118023566471668671136732584122101935676451741583619616}{9287164848985227023809870705733973136283731595470741486144450118687157492015388206325} a^{7} - \frac{847555965563409627535742800652219781457547040532355591608482781517013393747152319112}{9287164848985227023809870705733973136283731595470741486144450118687157492015388206325} a^{6} + \frac{3658745717356023309041344053066387173491947906828792443995731267603255788836122048648}{9287164848985227023809870705733973136283731595470741486144450118687157492015388206325} a^{5} - \frac{1000241905721741237743392478239084176532742646913892425065973332740991972337776852233}{3714865939594090809523948282293589254513492638188296594457780047474862996806155282530} a^{4} - \frac{2550426343046664031352045068467661957192456215229242839661068075950365512131722832842}{9287164848985227023809870705733973136283731595470741486144450118687157492015388206325} a^{3} - \frac{860630313085117917765540781757189675932192333787483196012486874681319759702228570791}{9287164848985227023809870705733973136283731595470741486144450118687157492015388206325} a^{2} - \frac{6441718055570123217519326425336964892779643428739491957975335262949330458454135045003}{18574329697970454047619741411467946272567463190941482972288900237374314984030776412650} a + \frac{1855405010486153735477377575557702717940521599427301170122485748416839766802605234377}{9287164848985227023809870705733973136283731595470741486144450118687157492015388206325}$

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

Class group and class number

$C_{4}$, 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:  $15$
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:  \( 18082637024000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$Q_{16}$ (as 16T14):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $Q_{16}$
Character table for $Q_{16}$

Intermediate fields

\(\Q(\sqrt{61}) \), \(\Q(\sqrt{2501}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{41}, \sqrt{61})\), 4.4.256455041.1 x2, 4.4.4204181.1 x2, 8.8.65769188054311681.3

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$41$41.8.7.4$x^{8} - 1912896$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.4$x^{8} - 1912896$$8$$1$$7$$C_8$$[\ ]_{8}$
$61$61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$