Properties

Label 16.8.127...161.1
Degree $16$
Signature $[8, 4]$
Discriminant $1.272\times 10^{46}$
Root discriminant $761.27$
Ramified primes $41, 67$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 13*x^14 + 2810*x^13 - 239955*x^12 + 355328*x^11 - 6965852*x^10 - 19367521*x^9 + 2779261606*x^8 - 8362892553*x^7 + 110015690005*x^6 - 48335042911*x^5 - 8411107448026*x^4 - 16949340098240*x^3 + 78581612687548*x^2 + 641225903327167*x + 1381515900331841)
 
gp: K = bnfinit(x^16 - 4*x^15 - 13*x^14 + 2810*x^13 - 239955*x^12 + 355328*x^11 - 6965852*x^10 - 19367521*x^9 + 2779261606*x^8 - 8362892553*x^7 + 110015690005*x^6 - 48335042911*x^5 - 8411107448026*x^4 - 16949340098240*x^3 + 78581612687548*x^2 + 641225903327167*x + 1381515900331841, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1381515900331841, 641225903327167, 78581612687548, -16949340098240, -8411107448026, -48335042911, 110015690005, -8362892553, 2779261606, -19367521, -6965852, 355328, -239955, 2810, -13, -4, 1]);
 

\( x^{16} - 4 x^{15} - 13 x^{14} + 2810 x^{13} - 239955 x^{12} + 355328 x^{11} - 6965852 x^{10} - 19367521 x^{9} + 2779261606 x^{8} - 8362892553 x^{7} + 110015690005 x^{6} - 48335042911 x^{5} - 8411107448026 x^{4} - 16949340098240 x^{3} + 78581612687548 x^{2} + 641225903327167 x + 1381515900331841 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[8, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(12724932380643958942597990061976243299024049161\)\(\medspace = 41^{15}\cdot 67^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $761.27$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $41, 67$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{59} a^{13} - \frac{26}{59} a^{12} - \frac{2}{59} a^{11} + \frac{14}{59} a^{10} + \frac{1}{59} a^{9} - \frac{16}{59} a^{8} + \frac{11}{59} a^{7} - \frac{23}{59} a^{6} - \frac{24}{59} a^{5} + \frac{9}{59} a^{4} - \frac{1}{59} a^{3} - \frac{6}{59} a^{2} - \frac{20}{59} a + \frac{14}{59}$, $\frac{1}{34783391} a^{14} - \frac{145329}{34783391} a^{13} - \frac{8558257}{34783391} a^{12} - \frac{10125122}{34783391} a^{11} - \frac{174767}{419077} a^{10} - \frac{15755008}{34783391} a^{9} - \frac{14682776}{34783391} a^{8} - \frac{58633}{34783391} a^{7} - \frac{10794573}{34783391} a^{6} + \frac{15415606}{34783391} a^{5} - \frac{16135313}{34783391} a^{4} - \frac{13793158}{34783391} a^{3} + \frac{16844986}{34783391} a^{2} + \frac{5561192}{34783391} a - \frac{14593513}{34783391}$, $\frac{1}{404366465876632564416432525217138162902239646599891956796992384305929705372545063428744107506217410415093} a^{15} + \frac{1912757537560973830957892575136100479924884799090080654730558498297163344737290250711677724740565}{404366465876632564416432525217138162902239646599891956796992384305929705372545063428744107506217410415093} a^{14} - \frac{1701742863990147440462585594979771608739620578702871806343273137868069036121137282054774273852170247439}{404366465876632564416432525217138162902239646599891956796992384305929705372545063428744107506217410415093} a^{13} - \frac{194315342187358501173449777535635080361084981935187357345600630824574920637063307444674389000275307514555}{404366465876632564416432525217138162902239646599891956796992384305929705372545063428744107506217410415093} a^{12} + \frac{58330450456145632295427743688530104591249675086110447877167855810266788130499088230908448540439784958494}{404366465876632564416432525217138162902239646599891956796992384305929705372545063428744107506217410415093} a^{11} - \frac{155691424191700290014465876374346889810896793383853777070109939178950237167006459736126606523713420180225}{404366465876632564416432525217138162902239646599891956796992384305929705372545063428744107506217410415093} a^{10} + \frac{157474157651835908098182477566302778431710774176038008659409485666260287110349907407264181440360787503115}{404366465876632564416432525217138162902239646599891956796992384305929705372545063428744107506217410415093} a^{9} + \frac{183295719952795647398506264142472451323392978411562808997661950778271692648941997792903233195473432394350}{404366465876632564416432525217138162902239646599891956796992384305929705372545063428744107506217410415093} a^{8} - \frac{10844842185531229909568406188458371872335008766989438023728812572098584342892806942715070818842629038991}{404366465876632564416432525217138162902239646599891956796992384305929705372545063428744107506217410415093} a^{7} + \frac{90076748542903827904392643660926360586001726284579847892307549243375717248297322721517935989786264870672}{404366465876632564416432525217138162902239646599891956796992384305929705372545063428744107506217410415093} a^{6} + \frac{126547177112857889281415238369233288447495176449207504834900995593997575459777006180147115052009613976554}{404366465876632564416432525217138162902239646599891956796992384305929705372545063428744107506217410415093} a^{5} - \frac{146552121260415323219230243901699919862960086526735973137201143612282702379465068024825992065820764341575}{404366465876632564416432525217138162902239646599891956796992384305929705372545063428744107506217410415093} a^{4} - \frac{119996645057482529054036439964512590191889434199292308974773565003063283438240480138125939137570075361245}{404366465876632564416432525217138162902239646599891956796992384305929705372545063428744107506217410415093} a^{3} - \frac{200825967824994646492256020748760826721030383761242259508661897466950947604260650955293337341124803549956}{404366465876632564416432525217138162902239646599891956796992384305929705372545063428744107506217410415093} a^{2} - \frac{46632141869788643723258835249642922880662861395969593041858380223349271385207364293597757256926659422480}{404366465876632564416432525217138162902239646599891956796992384305929705372545063428744107506217410415093} a - \frac{7321886529067464014320835599325203439168726713734431502917224943835557135757562440190997643264335192270}{404366465876632564416432525217138162902239646599891956796992384305929705372545063428744107506217410415093}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 139906515353000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{8}\cdot(2\pi)^{4}\cdot 139906515353000000 \cdot 1}{2\sqrt{12724932380643958942597990061976243299024049161}}\approx 0.247422756639131$ (assuming GRH)

Galois group

$OD_{32}$ (as 16T22):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_{16} : C_2$
Character table for $C_{16} : C_2$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.3924516938243170601.1

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

Sibling fields

Galois closure: Deg 32

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
41Data not computed
67Data not computed