Properties

Label 16.8.701...177.1
Degree $16$
Signature $[8, 4]$
Discriminant $7.018\times 10^{45}$
Root discriminant $733.48$
Ramified primes $31, 73$
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 - 29*x^14 + 2382*x^13 - 82573*x^12 - 427336*x^11 + 11394403*x^10 + 27090344*x^9 - 189686251*x^8 - 287852338*x^7 - 2171051651*x^6 - 22553825868*x^5 - 971088042492*x^4 - 3371533703552*x^3 - 5254806287296*x^2 + 16552670883328*x - 7727214997504)
 
gp: K = bnfinit(x^16 - 4*x^15 - 29*x^14 + 2382*x^13 - 82573*x^12 - 427336*x^11 + 11394403*x^10 + 27090344*x^9 - 189686251*x^8 - 287852338*x^7 - 2171051651*x^6 - 22553825868*x^5 - 971088042492*x^4 - 3371533703552*x^3 - 5254806287296*x^2 + 16552670883328*x - 7727214997504, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7727214997504, 16552670883328, -5254806287296, -3371533703552, -971088042492, -22553825868, -2171051651, -287852338, -189686251, 27090344, 11394403, -427336, -82573, 2382, -29, -4, 1]);
 

\( x^{16} - 4 x^{15} - 29 x^{14} + 2382 x^{13} - 82573 x^{12} - 427336 x^{11} + 11394403 x^{10} + 27090344 x^{9} - 189686251 x^{8} - 287852338 x^{7} - 2171051651 x^{6} - 22553825868 x^{5} - 971088042492 x^{4} - 3371533703552 x^{3} - 5254806287296 x^{2} + 16552670883328 x - 7727214997504 \)

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:  \(7017513031942338923028700587492388019023549177\)\(\medspace = 31^{12}\cdot 73^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $733.48$
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:  $31, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{10} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{32} a^{11} - \frac{1}{32} a^{10} + \frac{1}{32} a^{9} - \frac{1}{16} a^{8} + \frac{1}{8} a^{6} + \frac{3}{32} a^{5} + \frac{5}{32} a^{4} + \frac{15}{32} a^{3} - \frac{3}{16} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{12} - \frac{1}{32} a^{9} - \frac{1}{16} a^{8} + \frac{1}{8} a^{7} + \frac{7}{32} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} + \frac{9}{32} a^{3} - \frac{3}{16} a^{2}$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{10} - \frac{1}{16} a^{9} + \frac{7}{32} a^{7} - \frac{1}{4} a^{6} - \frac{1}{8} a^{5} + \frac{1}{32} a^{4} + \frac{5}{16} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{128} a^{14} - \frac{1}{128} a^{12} - \frac{1}{64} a^{11} + \frac{7}{128} a^{10} - \frac{1}{32} a^{9} + \frac{3}{128} a^{8} + \frac{1}{32} a^{7} - \frac{7}{128} a^{6} - \frac{9}{64} a^{5} + \frac{25}{128} a^{4} + \frac{1}{8} a^{3} - \frac{11}{32} a^{2} + \frac{3}{8} a$, $\frac{1}{32400479881083662936764190353440909320270899713237000659838439937759219964535081341910016} a^{15} + \frac{72396470910876128966876806882809211157031983788495016149893317331839332816773368967}{1012514996283864466773880948545028416258465616038656270619951248054975623891721291934688} a^{14} - \frac{378108315081127811923895737516130484024488947056649712128576356582848149052237498264573}{32400479881083662936764190353440909320270899713237000659838439937759219964535081341910016} a^{13} + \frac{170280238394072287875521081400674829592219773929350178253014404268364772768621005796253}{16200239940541831468382095176720454660135449856618500329919219968879609982267540670955008} a^{12} - \frac{33632878393909014424947370455258321875193836892445188512283941991432424116419831549957}{32400479881083662936764190353440909320270899713237000659838439937759219964535081341910016} a^{11} + \frac{425744169047398021479074835177007719809101861035936704813675297689554898883953635471841}{8100119970270915734191047588360227330067724928309250164959609984439804991133770335477504} a^{10} + \frac{454951514777158188294885374370208114443661926504950282216479574636237637274658050006163}{32400479881083662936764190353440909320270899713237000659838439937759219964535081341910016} a^{9} - \frac{354148061805373588388632609884832807740769497961709395028551536252460244750895625388347}{8100119970270915734191047588360227330067724928309250164959609984439804991133770335477504} a^{8} + \frac{530566716589726112850539299979012861602629697643729245899075231551674125339797796647621}{32400479881083662936764190353440909320270899713237000659838439937759219964535081341910016} a^{7} - \frac{1651183072492603446844674874012547805177748853391906596006340350766177189294717936307007}{16200239940541831468382095176720454660135449856618500329919219968879609982267540670955008} a^{6} + \frac{5851979121844424514428798467098500232787985612136771886872433220603344150050877341195365}{32400479881083662936764190353440909320270899713237000659838439937759219964535081341910016} a^{5} - \frac{179246734690454228821232973065045753049171561263190717669057346429537379314052045196419}{4050059985135457867095523794180113665033862464154625082479804992219902495566885167738752} a^{4} + \frac{1184200297842174675087916016462133449947786432147540148771169935542638836699972349305921}{8100119970270915734191047588360227330067724928309250164959609984439804991133770335477504} a^{3} + \frac{11565320586572675363390972061515417155683276326402192910915260209803388385573527739297}{2025029992567728933547761897090056832516931232077312541239902496109951247783442583869376} a^{2} + \frac{10532967384329806090243687758691954267073345257147361741899762190092087001567281905327}{31641093633870764586683779642032138008077050501208008456873476501717988246616290372959} a + \frac{11613273711774897212287707354728679659923902527469735202131407436760585066328265742472}{31641093633870764586683779642032138008077050501208008456873476501717988246616290372959}$

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:  \( 416510986026000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{8}\cdot(2\pi)^{4}\cdot 416510986026000000 \cdot 1}{2\sqrt{7017513031942338923028700587492388019023549177}}\approx 0.991891777625763$ (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

\(\Q(\sqrt{73}) \), 4.4.389017.1, 8.8.10202504527754980537.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ $16$ $16$ $16$ $16$

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
31Data not computed
73Data not computed