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Properties

Label 16.0.20285510539...9329.4
Degree $16$
Signature $[0, 8]$
Discriminant $13^{14}\cdot 61^{6}$
Root discriminant $44.08$
Ramified primes $13, 61$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1263

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15237, 18453, 6901, 11576, 8201, -5313, 4926, 5447, -1380, 117, 636, -43, 180, -45, 11, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 11*x^14 - 45*x^13 + 180*x^12 - 43*x^11 + 636*x^10 + 117*x^9 - 1380*x^8 + 5447*x^7 + 4926*x^6 - 5313*x^5 + 8201*x^4 + 11576*x^3 + 6901*x^2 + 18453*x + 15237)
 
gp: K = bnfinit(x^16 - 6*x^15 + 11*x^14 - 45*x^13 + 180*x^12 - 43*x^11 + 636*x^10 + 117*x^9 - 1380*x^8 + 5447*x^7 + 4926*x^6 - 5313*x^5 + 8201*x^4 + 11576*x^3 + 6901*x^2 + 18453*x + 15237, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 11 x^{14} - 45 x^{13} + 180 x^{12} - 43 x^{11} + 636 x^{10} + 117 x^{9} - 1380 x^{8} + 5447 x^{7} + 4926 x^{6} - 5313 x^{5} + 8201 x^{4} + 11576 x^{3} + 6901 x^{2} + 18453 x + 15237 \)

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:  \(202855105391388496051529329=13^{14}\cdot 61^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.08$
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:  $13, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{27} a^{14} + \frac{1}{27} a^{13} - \frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{4}{27} a^{9} - \frac{4}{27} a^{8} + \frac{11}{27} a^{7} - \frac{4}{27} a^{6} - \frac{5}{27} a^{5} - \frac{11}{27} a^{4} + \frac{7}{27} a^{3} - \frac{2}{9} a^{2} + \frac{2}{27} a - \frac{2}{9}$, $\frac{1}{14852642507599250772280654757286543} a^{15} + \frac{16045655106725377359559237754770}{14852642507599250772280654757286543} a^{14} + \frac{161315677554496319318519720532853}{1650293611955472308031183861920727} a^{13} - \frac{710134768834454774130551692023767}{4950880835866416924093551585762181} a^{12} - \frac{537812655198846632434213224787253}{4950880835866416924093551585762181} a^{11} - \frac{4409859569478951943376682913129585}{14852642507599250772280654757286543} a^{10} + \frac{3302888550325982472796682485965941}{14852642507599250772280654757286543} a^{9} - \frac{4192296137812316051233496921097217}{14852642507599250772280654757286543} a^{8} - \frac{5652980808810450684293769581712400}{14852642507599250772280654757286543} a^{7} + \frac{6599487689788138095762038521052065}{14852642507599250772280654757286543} a^{6} - \frac{243670031200345410769122990344546}{14852642507599250772280654757286543} a^{5} - \frac{5695877819830345143836669452894265}{14852642507599250772280654757286543} a^{4} - \frac{290057034556445875438392552373615}{4950880835866416924093551585762181} a^{3} - \frac{3743331588976999610331570341746699}{14852642507599250772280654757286543} a^{2} - \frac{658903738824882486161281455841793}{1650293611955472308031183861920727} a - \frac{92471126220391290907508976424487}{1650293611955472308031183861920727}$

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

Class group and class number

$C_{2}\times C_{2}$, 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:  $7$
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:  \( 7543590.53384 \) (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

\(\Q(\sqrt{13}) \), 4.0.2197.1, 8.0.294435349.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
13Data not computed
61Data not computed