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Properties

Label 16.0.54646496237...1129.2
Degree $16$
Signature $[0, 8]$
Discriminant $11^{12}\cdot 89^{15}$
Root discriminant $406.07$
Ramified primes $11, 89$
Class number $128$ (GRH)
Class group $[4, 32]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T260)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![182391223777, -140857502107, 34121567244, -2204993690, -374685728, -116994683, 168888625, -48094621, 8058060, -1125939, 213155, -52866, 6942, -445, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 445*x^13 + 6942*x^12 - 52866*x^11 + 213155*x^10 - 1125939*x^9 + 8058060*x^8 - 48094621*x^7 + 168888625*x^6 - 116994683*x^5 - 374685728*x^4 - 2204993690*x^3 + 34121567244*x^2 - 140857502107*x + 182391223777)
 
gp: K = bnfinit(x^16 - 445*x^13 + 6942*x^12 - 52866*x^11 + 213155*x^10 - 1125939*x^9 + 8058060*x^8 - 48094621*x^7 + 168888625*x^6 - 116994683*x^5 - 374685728*x^4 - 2204993690*x^3 + 34121567244*x^2 - 140857502107*x + 182391223777, 1)
 

Normalized defining polynomial

\( x^{16} - 445 x^{13} + 6942 x^{12} - 52866 x^{11} + 213155 x^{10} - 1125939 x^{9} + 8058060 x^{8} - 48094621 x^{7} + 168888625 x^{6} - 116994683 x^{5} - 374685728 x^{4} - 2204993690 x^{3} + 34121567244 x^{2} - 140857502107 x + 182391223777 \)

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:  \(546464962373097060033371075256437529131129=11^{12}\cdot 89^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $406.07$
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:  $11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{22} a^{12} + \frac{2}{11} a^{11} - \frac{1}{22} a^{10} + \frac{2}{11} a^{9} - \frac{3}{11} a^{8} - \frac{1}{22} a^{7} - \frac{9}{22} a^{6} - \frac{2}{11} a^{5} - \frac{5}{22} a^{4} - \frac{1}{22} a^{3} + \frac{1}{22} a^{2} + \frac{1}{11} a + \frac{2}{11}$, $\frac{1}{22} a^{13} + \frac{5}{22} a^{11} - \frac{3}{22} a^{10} - \frac{5}{11} a^{8} - \frac{5}{22} a^{7} + \frac{5}{11} a^{6} + \frac{4}{11} a^{4} + \frac{5}{22} a^{3} + \frac{9}{22} a^{2} + \frac{7}{22} a - \frac{5}{22}$, $\frac{1}{242} a^{14} + \frac{1}{121} a^{13} + \frac{3}{242} a^{12} + \frac{43}{242} a^{11} + \frac{7}{242} a^{10} - \frac{9}{121} a^{9} - \frac{45}{121} a^{8} + \frac{23}{121} a^{7} - \frac{58}{121} a^{6} - \frac{61}{242} a^{5} + \frac{43}{121} a^{4} + \frac{87}{242} a^{3} + \frac{28}{121} a^{2} + \frac{52}{121} a - \frac{29}{242}$, $\frac{1}{2477488102962349181410792022398895477953400835521868014966618764293430} a^{15} + \frac{242639845725760696767875149063145532857561911598255938011789587124}{176963435925882084386485144457063962710957202537276286783329911735245} a^{14} + \frac{197921832992180716544774981099176930180321446450856155764384851589}{176963435925882084386485144457063962710957202537276286783329911735245} a^{13} - \frac{22191672366474140219823487643449241620860681784396330178462111374242}{1238744051481174590705396011199447738976700417760934007483309382146715} a^{12} + \frac{50364183922532057959988951979131813711882056889895598242623727858694}{1238744051481174590705396011199447738976700417760934007483309382146715} a^{11} - \frac{272044094821791751072520430348039779660542564865114000481586955842403}{2477488102962349181410792022398895477953400835521868014966618764293430} a^{10} + \frac{3380982659966577503731686217194440731073325414495819134126304111466}{10237554144472517278556991828094609413030581964966396756060408116915} a^{9} - \frac{813826949827416572684346372703987265042390982479016614782218950295477}{2477488102962349181410792022398895477953400835521868014966618764293430} a^{8} - \frac{499325062114727938130522788417612297374438574931595774420049510066611}{1238744051481174590705396011199447738976700417760934007483309382146715} a^{7} - \frac{246435924815461589496639229533123209192547004864636193862034522690269}{1238744051481174590705396011199447738976700417760934007483309382146715} a^{6} - \frac{1118020501235688886384283993083440582600939258195081804409435238278553}{2477488102962349181410792022398895477953400835521868014966618764293430} a^{5} - \frac{718580371999374102436345768715535604375419742576115906222424354546801}{2477488102962349181410792022398895477953400835521868014966618764293430} a^{4} + \frac{22376626324543681463290441596159583966663801531896692804859243724552}{65197055341114452142389263747339354682984232513733368814911020112985} a^{3} + \frac{52754304557230643166837993268528585700867273054130124464313924318541}{225226191178395380128253820218081407086672803229260728633328978572130} a^{2} + \frac{47929809901556641442546304776648997665806846075285353764467720240665}{495497620592469836282158404479779095590680167104373602993323752858686} a + \frac{9847126868103228997275989212485899261950837327090551744343084488897}{176963435925882084386485144457063962710957202537276286783329911735245}$

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

Class group and class number

$C_{4}\times C_{32}$, which has order $128$ (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:  \( 3694538597510 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T260):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-979}) \), 4.0.85301249.1, 8.0.647590974205440089.4

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ $16$

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
$11$11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
89Data not computed