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Properties

Label 16.16.2020123727...6256.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{50}\cdot 17^{4}\cdot 16673^{6}$
Root discriminant $678.56$
Ramified primes $2, 17, 16673$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1392

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1048645407290447, -106821264489440, -298573308211004, 13466549733144, 24403328720004, -576556296440, -844541825988, 8204441920, 14186331356, -46989984, -124399308, 111336, 568104, -136, -1236, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 1236*x^14 - 136*x^13 + 568104*x^12 + 111336*x^11 - 124399308*x^10 - 46989984*x^9 + 14186331356*x^8 + 8204441920*x^7 - 844541825988*x^6 - 576556296440*x^5 + 24403328720004*x^4 + 13466549733144*x^3 - 298573308211004*x^2 - 106821264489440*x + 1048645407290447)
 
gp: K = bnfinit(x^16 - 1236*x^14 - 136*x^13 + 568104*x^12 + 111336*x^11 - 124399308*x^10 - 46989984*x^9 + 14186331356*x^8 + 8204441920*x^7 - 844541825988*x^6 - 576556296440*x^5 + 24403328720004*x^4 + 13466549733144*x^3 - 298573308211004*x^2 - 106821264489440*x + 1048645407290447, 1)
 

Normalized defining polynomial

\( x^{16} - 1236 x^{14} - 136 x^{13} + 568104 x^{12} + 111336 x^{11} - 124399308 x^{10} - 46989984 x^{9} + 14186331356 x^{8} + 8204441920 x^{7} - 844541825988 x^{6} - 576556296440 x^{5} + 24403328720004 x^{4} + 13466549733144 x^{3} - 298573308211004 x^{2} - 106821264489440 x + 1048645407290447 \)

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:  \(2020123727657308174489736833620393728206176256=2^{50}\cdot 17^{4}\cdot 16673^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $678.56$
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:  $2, 17, 16673$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{16} a^{12} - \frac{1}{4} a^{11} + \frac{1}{8} a^{10} - \frac{1}{4} a^{9} + \frac{1}{16} a^{8} + \frac{1}{4} a^{6} + \frac{1}{16} a^{4} - \frac{1}{4} a^{3} + \frac{3}{8} a^{2} - \frac{1}{4} a + \frac{5}{16}$, $\frac{1}{16} a^{13} + \frac{1}{8} a^{11} - \frac{1}{4} a^{10} + \frac{1}{16} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{1}{16} a^{5} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2} + \frac{5}{16} a - \frac{1}{4}$, $\frac{1}{16} a^{14} - \frac{1}{4} a^{11} - \frac{3}{16} a^{10} - \frac{1}{4} a^{9} + \frac{1}{8} a^{8} - \frac{7}{16} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{7}{16} a^{2} - \frac{1}{4} a + \frac{3}{8}$, $\frac{1}{52405869185017450163889688266598625809657978241762243548669816632437864539269489883852151134496} a^{15} - \frac{882009095761150882408651650836153026959753248617948129847094696666115964217514917138103201023}{52405869185017450163889688266598625809657978241762243548669816632437864539269489883852151134496} a^{14} + \frac{84451525134853942407254512207786376212364817867300869409792996282799026688929075590253683441}{7486552740716778594841384038085517972808282605966034792667116661776837791324212840550307304928} a^{13} - \frac{532035562345933924338905802270683123937044140453539020853082825876118881084216221922466050601}{52405869185017450163889688266598625809657978241762243548669816632437864539269489883852151134496} a^{12} - \frac{12741950145559858555919541144891590381491736800490701960470605476286081072525317395853998245381}{52405869185017450163889688266598625809657978241762243548669816632437864539269489883852151134496} a^{11} + \frac{1000846161239917148023723944973073437500695564244973021494802909849408758895336686460491578717}{7486552740716778594841384038085517972808282605966034792667116661776837791324212840550307304928} a^{10} + \frac{6985248535932589358145961860052304713656558671875985625353961632096963814905183554411459057769}{52405869185017450163889688266598625809657978241762243548669816632437864539269489883852151134496} a^{9} - \frac{5602798007776393654012998950403115914909660005553066762161083859863788833519741675592885256071}{52405869185017450163889688266598625809657978241762243548669816632437864539269489883852151134496} a^{8} - \frac{1875286478626515434845487510892929722438595504954680225157685988118219349430087594070458715405}{7486552740716778594841384038085517972808282605966034792667116661776837791324212840550307304928} a^{7} - \frac{6188233242422826452660379125018925314220642395487058386799429151642898456737302667533925871867}{52405869185017450163889688266598625809657978241762243548669816632437864539269489883852151134496} a^{6} + \frac{362155645818349313438542395485297249135944755677785940343251850808833709883395288051630027445}{1690511909194111295609344782793504058376063814250394953182897310723802081911919028511359714016} a^{5} - \frac{808184658513165750384514112041184230661277754436192410380685099825290798939550923478651941907}{7486552740716778594841384038085517972808282605966034792667116661776837791324212840550307304928} a^{4} + \frac{10589291676607174805299952240735560708776612329725542069377850544238487506412074036191107723299}{52405869185017450163889688266598625809657978241762243548669816632437864539269489883852151134496} a^{3} + \frac{10296132852985172320546822584574169202200529028951482948022804022823684138620312990443716177315}{52405869185017450163889688266598625809657978241762243548669816632437864539269489883852151134496} a^{2} - \frac{5439595402697356503956074216832916728089276877147255767623645548376465730354379347075497721991}{52405869185017450163889688266598625809657978241762243548669816632437864539269489883852151134496} a + \frac{12410513394065529491443984313406900660992712584780544013628042442378631757491672443818396977865}{52405869185017450163889688266598625809657978241762243548669816632437864539269489883852151134496}$

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:  \( 707642643960000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1392:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 80 conjugacy class representatives for t16n1392 are not computed
Character table for t16n1392 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 4.4.1067072.2, 4.4.34146304.1, 8.8.1165970076860416.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.26.4$x^{8} + 8 x^{7} + 12 x^{6} + 8 x^{5} + 8 x^{4} + 8 x^{3} + 2$$8$$1$$26$$C_2^2:C_4$$[2, 3, 7/2, 4]$
2.8.24.9$x^{8} + 8 x^{7} + 14 x^{4} + 4 x^{2} + 8 x + 30$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
16673Data not computed