Properties

Label 16.8.241...649.1
Degree $16$
Signature $[8, 4]$
Discriminant $2.410\times 10^{39}$
Root discriminant $289.32$
Ramified primes $7, 89$
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 - 3*x^15 - 260*x^14 - 440*x^13 + 9958*x^12 + 134350*x^11 + 903564*x^10 - 1823804*x^9 - 25340911*x^8 - 85828035*x^7 + 138216368*x^6 + 575302060*x^5 + 324609488*x^4 - 971480768*x^3 - 1401515264*x^2 - 937304064*x - 102055936)
 
gp: K = bnfinit(x^16 - 3*x^15 - 260*x^14 - 440*x^13 + 9958*x^12 + 134350*x^11 + 903564*x^10 - 1823804*x^9 - 25340911*x^8 - 85828035*x^7 + 138216368*x^6 + 575302060*x^5 + 324609488*x^4 - 971480768*x^3 - 1401515264*x^2 - 937304064*x - 102055936, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-102055936, -937304064, -1401515264, -971480768, 324609488, 575302060, 138216368, -85828035, -25340911, -1823804, 903564, 134350, 9958, -440, -260, -3, 1]);
 

\( x^{16} - 3 x^{15} - 260 x^{14} - 440 x^{13} + 9958 x^{12} + 134350 x^{11} + 903564 x^{10} - 1823804 x^{9} - 25340911 x^{8} - 85828035 x^{7} + 138216368 x^{6} + 575302060 x^{5} + 324609488 x^{4} - 971480768 x^{3} - 1401515264 x^{2} - 937304064 x - 102055936 \)

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:  \(2410052925086109011392504405401329728649\)\(\medspace = 7^{12}\cdot 89^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $289.32$
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:  $7, 89$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{32} a^{7} - \frac{1}{32} a^{6} + \frac{1}{32} a^{5} + \frac{1}{32} a^{4} - \frac{1}{16} a^{3}$, $\frac{1}{128} a^{8} - \frac{1}{64} a^{6} - \frac{1}{32} a^{5} + \frac{5}{128} a^{4} - \frac{3}{32} a^{3} - \frac{1}{32} a^{2} + \frac{1}{8} a$, $\frac{1}{256} a^{9} - \frac{1}{256} a^{8} - \frac{1}{128} a^{7} - \frac{1}{128} a^{6} + \frac{9}{256} a^{5} + \frac{15}{256} a^{4} - \frac{7}{32} a^{3} - \frac{3}{64} a^{2} - \frac{5}{16} a - \frac{1}{2}$, $\frac{1}{1024} a^{10} + \frac{1}{1024} a^{9} + \frac{1}{512} a^{8} - \frac{3}{512} a^{7} - \frac{7}{1024} a^{6} + \frac{41}{1024} a^{5} + \frac{1}{256} a^{4} + \frac{7}{256} a^{3} - \frac{1}{4} a^{2} - \frac{5}{16} a - \frac{1}{2}$, $\frac{1}{4096} a^{11} + \frac{1}{4096} a^{9} + \frac{1}{512} a^{8} - \frac{1}{4096} a^{7} + \frac{1}{256} a^{6} + \frac{155}{4096} a^{5} - \frac{19}{512} a^{4} + \frac{201}{1024} a^{3} - \frac{7}{32} a^{2} + \frac{9}{64} a - \frac{1}{8}$, $\frac{1}{8192} a^{12} - \frac{1}{8192} a^{11} + \frac{1}{8192} a^{10} - \frac{9}{8192} a^{9} - \frac{25}{8192} a^{8} + \frac{49}{8192} a^{7} - \frac{21}{8192} a^{6} + \frac{445}{8192} a^{5} + \frac{75}{2048} a^{4} - \frac{297}{2048} a^{3} + \frac{1}{32} a^{2} + \frac{27}{128} a - \frac{3}{16}$, $\frac{1}{32768} a^{13} + \frac{1}{32768} a^{12} - \frac{3}{32768} a^{11} + \frac{9}{32768} a^{10} + \frac{3}{32768} a^{9} - \frac{17}{32768} a^{8} - \frac{337}{32768} a^{7} - \frac{573}{32768} a^{6} - \frac{47}{1024} a^{5} + \frac{257}{8192} a^{4} - \frac{93}{2048} a^{3} + \frac{57}{512} a^{2} - \frac{11}{128} a + \frac{1}{16}$, $\frac{1}{524288} a^{14} + \frac{1}{524288} a^{13} + \frac{29}{524288} a^{12} - \frac{55}{524288} a^{11} - \frac{253}{524288} a^{10} + \frac{911}{524288} a^{9} - \frac{945}{524288} a^{8} - \frac{1341}{524288} a^{7} - \frac{437}{16384} a^{6} - \frac{5367}{131072} a^{5} - \frac{1053}{32768} a^{4} + \frac{311}{8192} a^{3} - \frac{379}{2048} a^{2} + \frac{3}{32} a + \frac{1}{32}$, $\frac{1}{8582891791545894230824426304476045267226853376} a^{15} + \frac{77197610833608778163759611204516244839}{4291445895772947115412213152238022633613426688} a^{14} + \frac{27865252150463555443511812860679606436157}{4291445895772947115412213152238022633613426688} a^{13} + \frac{240336666247995288631529934262640854273449}{4291445895772947115412213152238022633613426688} a^{12} + \frac{103938667067192796143281834913074295099681}{1072861473943236778853053288059505658403356672} a^{11} - \frac{1665191204736422566133190928887037431918109}{4291445895772947115412213152238022633613426688} a^{10} - \frac{3316593189691529993133962009639130258071903}{4291445895772947115412213152238022633613426688} a^{9} - \frac{9599584527276389112052902956695596093197477}{4291445895772947115412213152238022633613426688} a^{8} - \frac{39879249707186052736135323843829176968487369}{8582891791545894230824426304476045267226853376} a^{7} - \frac{36869656766939445472545246886074581639047875}{2145722947886473557706106576119011316806713344} a^{6} - \frac{53827749371752544217075972847763814788872495}{2145722947886473557706106576119011316806713344} a^{5} - \frac{16831698260369883364386659929280945635568553}{536430736971618389426526644029752829201678336} a^{4} + \frac{32275542005577534291397929763811203302812791}{134107684242904597356631661007438207300419584} a^{3} - \frac{6173842386881257720939262783146188249559035}{33526921060726149339157915251859551825104896} a^{2} - \frac{21322922713950396646664781331509983316119}{523858141573846083424342425810305497267264} a + \frac{252580127745252977477658925675240541343905}{523858141573846083424342425810305497267264}$

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:  \( 2494293233420000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{8}\cdot(2\pi)^{4}\cdot 2494293233420000000 \cdot 1}{2\sqrt{2410052925086109011392504405401329728649}}\approx 10135.9313797314$ (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

\(\Q(\sqrt{89}) \), 4.4.704969.1, 8.8.106199435084165129.1

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{16}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
7Data not computed
89Data not computed