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Properties

Label 16.0.13375625263...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 23^{12}$
Root discriminant $42.94$
Ramified primes $5, 23$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_8: C_2$ (as 16T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6241, 67150, 278706, 532430, 430002, 16280, -145282, -55475, 6125, 7450, 1132, -515, 182, 110, -21, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 21*x^14 + 110*x^13 + 182*x^12 - 515*x^11 + 1132*x^10 + 7450*x^9 + 6125*x^8 - 55475*x^7 - 145282*x^6 + 16280*x^5 + 430002*x^4 + 532430*x^3 + 278706*x^2 + 67150*x + 6241)
 
gp: K = bnfinit(x^16 - 5*x^15 - 21*x^14 + 110*x^13 + 182*x^12 - 515*x^11 + 1132*x^10 + 7450*x^9 + 6125*x^8 - 55475*x^7 - 145282*x^6 + 16280*x^5 + 430002*x^4 + 532430*x^3 + 278706*x^2 + 67150*x + 6241, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 21 x^{14} + 110 x^{13} + 182 x^{12} - 515 x^{11} + 1132 x^{10} + 7450 x^{9} + 6125 x^{8} - 55475 x^{7} - 145282 x^{6} + 16280 x^{5} + 430002 x^{4} + 532430 x^{3} + 278706 x^{2} + 67150 x + 6241 \)

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:  \(133756252636842779541015625=5^{14}\cdot 23^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.94$
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:  $5, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{19} a^{14} - \frac{3}{19} a^{13} - \frac{6}{19} a^{12} - \frac{3}{19} a^{11} - \frac{7}{19} a^{10} - \frac{3}{19} a^{9} - \frac{9}{19} a^{8} - \frac{3}{19} a^{7} + \frac{2}{19} a^{6} + \frac{3}{19} a^{5} + \frac{2}{19} a^{4} + \frac{7}{19} a^{3} - \frac{7}{19} a^{2} - \frac{7}{19} a + \frac{5}{19}$, $\frac{1}{1665983283550497346468155139122711346287311} a^{15} - \frac{35064844480264641229224546553685277234343}{1665983283550497346468155139122711346287311} a^{14} - \frac{545922848506673161788278634126392895974515}{1665983283550497346468155139122711346287311} a^{13} - \frac{31215810646587713973094807010178817422219}{1665983283550497346468155139122711346287311} a^{12} + \frac{98540421796428396340376249794514945659299}{1665983283550497346468155139122711346287311} a^{11} + \frac{709080700534995069167081121748602896407875}{1665983283550497346468155139122711346287311} a^{10} + \frac{558497305761601716894686241151490503980188}{1665983283550497346468155139122711346287311} a^{9} + \frac{742533240236105646299520670391081160417933}{1665983283550497346468155139122711346287311} a^{8} + \frac{563371297264160675908865359834530688818754}{1665983283550497346468155139122711346287311} a^{7} - \frac{289646790518763573183827059649912354800967}{1665983283550497346468155139122711346287311} a^{6} + \frac{65831782147515166414704968702665515822589}{1665983283550497346468155139122711346287311} a^{5} + \frac{754476885226341506237051527532233619501086}{1665983283550497346468155139122711346287311} a^{4} + \frac{35908552195636773048070207708971486626403}{87683330713184070866745007322247965594069} a^{3} - \frac{711109347016559563248739797959843132406808}{1665983283550497346468155139122711346287311} a^{2} - \frac{380516424447339336425754523848146169618605}{1665983283550497346468155139122711346287311} a - \frac{8238340628783506284653468904792588690463}{21088395994310092993267786571173561345409}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( \frac{34004432469419675678390648448196388309}{87683330713184070866745007322247965594069} a^{15} - \frac{183398344734400699382289541256132243343}{87683330713184070866745007322247965594069} a^{14} - \frac{640592876333536080597402206555719373562}{87683330713184070866745007322247965594069} a^{13} + \frac{3984521784620430631373736893572871265652}{87683330713184070866745007322247965594069} a^{12} + \frac{4600571046432374938063448896672044128770}{87683330713184070866745007322247965594069} a^{11} - \frac{19159107446462251156630504747247256647893}{87683330713184070866745007322247965594069} a^{10} + \frac{46122620851432188954667029663805229489203}{87683330713184070866745007322247965594069} a^{9} + \frac{234503206830880207099733159749956994631530}{87683330713184070866745007322247965594069} a^{8} + \frac{118205252162635714165400131878550341630352}{87683330713184070866745007322247965594069} a^{7} - \frac{1924940906148433575085785509679511458058107}{87683330713184070866745007322247965594069} a^{6} - \frac{4179698863905664882822429731103070940831611}{87683330713184070866745007322247965594069} a^{5} + \frac{2124055519087590736830968716508610481228918}{87683330713184070866745007322247965594069} a^{4} + \frac{13658012416846087993308489896178968421633846}{87683330713184070866745007322247965594069} a^{3} + \frac{12846708953776047403792289472879709202958617}{87683330713184070866745007322247965594069} a^{2} + \frac{4847135878059862922126394330256539578028799}{87683330713184070866745007322247965594069} a + \frac{9092834319523432404194409852251492074732}{1109915578647899631224620345851240070811} \) (order $10$)
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:  \( 1890649.08655 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{-115}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{5}, \sqrt{-23})\), 4.4.66125.1, \(\Q(\zeta_{5})\), 8.0.4372515625.1, 8.4.11565303828125.1 x2

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

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
5Data not computed
23Data not computed