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Properties

Label 16.0.11256645592...249.15
Degree $16$
Signature $[0, 8]$
Discriminant $13^{8}\cdot 53^{14}$
Root discriminant $116.34$
Ramified primes $13, 53$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group 16T1263

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![65951899, 4379232, 1831385, -6469998, 1808447, 872096, 521094, -79428, -14475, 6134, 7067, 86, -267, -22, 34, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 34*x^14 - 22*x^13 - 267*x^12 + 86*x^11 + 7067*x^10 + 6134*x^9 - 14475*x^8 - 79428*x^7 + 521094*x^6 + 872096*x^5 + 1808447*x^4 - 6469998*x^3 + 1831385*x^2 + 4379232*x + 65951899)
 
gp: K = bnfinit(x^16 - 4*x^15 + 34*x^14 - 22*x^13 - 267*x^12 + 86*x^11 + 7067*x^10 + 6134*x^9 - 14475*x^8 - 79428*x^7 + 521094*x^6 + 872096*x^5 + 1808447*x^4 - 6469998*x^3 + 1831385*x^2 + 4379232*x + 65951899, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 34 x^{14} - 22 x^{13} - 267 x^{12} + 86 x^{11} + 7067 x^{10} + 6134 x^{9} - 14475 x^{8} - 79428 x^{7} + 521094 x^{6} + 872096 x^{5} + 1808447 x^{4} - 6469998 x^{3} + 1831385 x^{2} + 4379232 x + 65951899 \)

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:  \(1125664559289128829386632937086249=13^{8}\cdot 53^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $116.34$
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:  $13, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{106} a^{8} - \frac{1}{53} a^{7} + \frac{15}{106} a^{6} + \frac{19}{106} a^{5} - \frac{49}{106} a^{4} - \frac{11}{53} a^{3} - \frac{37}{106} a^{2} - \frac{39}{106} a - \frac{17}{106}$, $\frac{1}{106} a^{9} + \frac{11}{106} a^{7} + \frac{49}{106} a^{6} - \frac{11}{106} a^{5} - \frac{7}{53} a^{4} + \frac{25}{106} a^{3} - \frac{7}{106} a^{2} + \frac{11}{106} a - \frac{17}{53}$, $\frac{1}{106} a^{10} - \frac{35}{106} a^{7} + \frac{18}{53} a^{6} - \frac{11}{106} a^{5} + \frac{17}{53} a^{4} + \frac{23}{106} a^{3} - \frac{3}{53} a^{2} - \frac{29}{106} a - \frac{25}{106}$, $\frac{1}{106} a^{11} - \frac{17}{53} a^{7} - \frac{8}{53} a^{6} - \frac{43}{106} a^{5} + \frac{2}{53} a^{4} - \frac{17}{53} a^{3} - \frac{26}{53} a^{2} - \frac{6}{53} a + \frac{41}{106}$, $\frac{1}{106} a^{12} + \frac{11}{53} a^{7} + \frac{43}{106} a^{6} + \frac{7}{53} a^{5} - \frac{2}{53} a^{4} + \frac{24}{53} a^{3} + \frac{1}{53} a^{2} - \frac{13}{106} a - \frac{24}{53}$, $\frac{1}{106} a^{13} - \frac{19}{106} a^{7} + \frac{1}{53} a^{6} + \frac{1}{53} a^{5} - \frac{20}{53} a^{4} - \frac{22}{53} a^{3} - \frac{47}{106} a^{2} - \frac{19}{53} a - \frac{25}{53}$, $\frac{1}{34874} a^{14} + \frac{2}{2491} a^{13} + \frac{5}{2491} a^{12} - \frac{74}{17437} a^{11} - \frac{61}{34874} a^{10} + \frac{65}{34874} a^{9} + \frac{6}{2491} a^{8} - \frac{2812}{17437} a^{7} + \frac{4674}{17437} a^{6} + \frac{15739}{34874} a^{5} + \frac{13889}{34874} a^{4} + \frac{10263}{34874} a^{3} - \frac{7825}{17437} a^{2} - \frac{17151}{34874} a + \frac{8579}{17437}$, $\frac{1}{1365067399280854486587753481369786101314} a^{15} + \frac{6718134893991288852947679371805569}{1365067399280854486587753481369786101314} a^{14} + \frac{302224453246000504641090326204058149}{195009628468693498083964783052826585902} a^{13} + \frac{2308332246097338471870413691626523817}{1365067399280854486587753481369786101314} a^{12} + \frac{199851121444618490527716981851932623}{105005184560065729737519498566906623178} a^{11} - \frac{2249519510375019166129819184276458229}{1365067399280854486587753481369786101314} a^{10} - \frac{1483631473716041401401417619327870701}{1365067399280854486587753481369786101314} a^{9} - \frac{1998497992390820091466830099480131872}{682533699640427243293876740684893050657} a^{8} + \frac{230629624193586635106077548360683295734}{682533699640427243293876740684893050657} a^{7} + \frac{82582550126175023928751435164343313157}{682533699640427243293876740684893050657} a^{6} - \frac{593345715599616530393606915532540026785}{1365067399280854486587753481369786101314} a^{5} + \frac{47728162310136448698907757756449151294}{682533699640427243293876740684893050657} a^{4} - \frac{216385906860084656802057423838706763895}{682533699640427243293876740684893050657} a^{3} + \frac{1201794137676566326083576062997392260}{97504814234346749041982391526413292951} a^{2} + \frac{21697297562367103472121107827461341187}{105005184560065729737519498566906623178} a - \frac{86930380044048835635464922413296023}{462577905550950351266605720559060014}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  \( 458351978.765 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1263:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1263
Character table for t16n1263 is not computed

Intermediate fields

\(\Q(\sqrt{53}) \), 4.0.148877.1, 8.0.48695101400413.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{10}$ R ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$53$53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$