Properties

Label 16.8.180...841.1
Degree $16$
Signature $[8, 4]$
Discriminant $1.807\times 10^{44}$
Root discriminant $583.52$
Ramified primes $41, 47$
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 + 1990*x^13 - 113388*x^12 + 322364*x^11 - 13147586*x^10 - 14823368*x^9 + 54599923*x^8 + 2235476042*x^7 + 45257850300*x^6 + 24021442783*x^5 - 126249012240*x^4 - 3783570586772*x^3 - 15536602319783*x^2 + 55367792792525*x - 36300633217867)
 
gp: K = bnfinit(x^16 - 4*x^15 - 13*x^14 + 1990*x^13 - 113388*x^12 + 322364*x^11 - 13147586*x^10 - 14823368*x^9 + 54599923*x^8 + 2235476042*x^7 + 45257850300*x^6 + 24021442783*x^5 - 126249012240*x^4 - 3783570586772*x^3 - 15536602319783*x^2 + 55367792792525*x - 36300633217867, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-36300633217867, 55367792792525, -15536602319783, -3783570586772, -126249012240, 24021442783, 45257850300, 2235476042, 54599923, -14823368, -13147586, 322364, -113388, 1990, -13, -4, 1]);
 

\( x^{16} - 4 x^{15} - 13 x^{14} + 1990 x^{13} - 113388 x^{12} + 322364 x^{11} - 13147586 x^{10} - 14823368 x^{9} + 54599923 x^{8} + 2235476042 x^{7} + 45257850300 x^{6} + 24021442783 x^{5} - 126249012240 x^{4} - 3783570586772 x^{3} - 15536602319783 x^{2} + 55367792792525 x - 36300633217867 \)

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:  \(180689179599093672760171229322647431883530841\)\(\medspace = 41^{15}\cdot 47^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $583.52$
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:  $41, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{37} a^{12} + \frac{9}{37} a^{11} + \frac{11}{37} a^{10} - \frac{8}{37} a^{9} - \frac{12}{37} a^{8} - \frac{17}{37} a^{7} + \frac{18}{37} a^{6} + \frac{18}{37} a^{5} + \frac{16}{37} a^{4} - \frac{11}{37} a^{3} + \frac{13}{37} a^{2} + \frac{8}{37} a - \frac{11}{37}$, $\frac{1}{37} a^{13} + \frac{4}{37} a^{11} + \frac{4}{37} a^{10} - \frac{14}{37} a^{9} + \frac{17}{37} a^{8} - \frac{14}{37} a^{7} + \frac{4}{37} a^{6} + \frac{2}{37} a^{5} - \frac{7}{37} a^{4} + \frac{1}{37} a^{3} + \frac{2}{37} a^{2} - \frac{9}{37} a - \frac{12}{37}$, $\frac{1}{596551} a^{14} + \frac{5874}{596551} a^{13} - \frac{6522}{596551} a^{12} - \frac{67128}{596551} a^{11} + \frac{157971}{596551} a^{10} + \frac{9615}{25937} a^{9} - \frac{23938}{596551} a^{8} - \frac{88284}{596551} a^{7} + \frac{182013}{596551} a^{6} - \frac{155381}{596551} a^{5} - \frac{68}{851} a^{4} + \frac{68227}{596551} a^{3} - \frac{43943}{596551} a^{2} + \frac{98895}{596551} a - \frac{46136}{596551}$, $\frac{1}{647217460620117192810904705216971731051492263217493726566678457210766072466175757040303625049249} a^{15} - \frac{200822138449794124520918933508009462269378461063067048656269727982780710989568097265995111}{647217460620117192810904705216971731051492263217493726566678457210766072466175757040303625049249} a^{14} - \frac{2397473227612299274495769912869495394608693439282307426454402493132912026057602606234757620949}{647217460620117192810904705216971731051492263217493726566678457210766072466175757040303625049249} a^{13} - \frac{1760209321102004256160034826658393313578565435903497759449392595767725339186673062172849377818}{647217460620117192810904705216971731051492263217493726566678457210766072466175757040303625049249} a^{12} + \frac{53939001598643974310579741186610148932827401589871239469704410576553062414659921540221634844904}{647217460620117192810904705216971731051492263217493726566678457210766072466175757040303625049249} a^{11} - \frac{116589544771396319374287930643678628828561240337262572611748078552901136661584347577546373462714}{647217460620117192810904705216971731051492263217493726566678457210766072466175757040303625049249} a^{10} + \frac{278463871058290648066407301409778471271550647069848399857668306255213380652031821288153905905304}{647217460620117192810904705216971731051492263217493726566678457210766072466175757040303625049249} a^{9} - \frac{190854771187425903245780999834555500369781148490286071142860472964042495171489783417399311074730}{647217460620117192810904705216971731051492263217493726566678457210766072466175757040303625049249} a^{8} - \frac{320253147734338602045923944175974830436023566821759656233107280872933481499434268248821664116999}{647217460620117192810904705216971731051492263217493726566678457210766072466175757040303625049249} a^{7} - \frac{4428259355543478502183189355367842790391108649949557176484701371507342129683479911826887334827}{20877982600648941703577571136031346162951363329596571824731563135831163627941153452913020162879} a^{6} + \frac{6572722315643500116551293593120942399615621028457758003667637570730977926411135233329749803793}{28139889592179008383082813270303118741369228835543205502899063356989829237659815523491461958663} a^{5} + \frac{42339335473937230744865903807346345084696424145246476947210377723557227974739106895531487531392}{647217460620117192810904705216971731051492263217493726566678457210766072466175757040303625049249} a^{4} + \frac{282182264927266242747044997504371310083796476033125564782174639597768876689111914541600864655796}{647217460620117192810904705216971731051492263217493726566678457210766072466175757040303625049249} a^{3} - \frac{216447603649727662064390380292647488926161126815128954053066839710307479935384499030539531940238}{647217460620117192810904705216971731051492263217493726566678457210766072466175757040303625049249} a^{2} + \frac{4820750447505206474710395840847592546862102390485103669421009784349407908503290839948956506219}{28139889592179008383082813270303118741369228835543205502899063356989829237659815523491461958663} a - \frac{746266910399475591266427159373303998421357639498648334368114415246478075049322461465364013735}{5096200477323757423707911064700564811429072938720423043832113836305244665087998086931524606687}$

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:  \( 32440984770100000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{8}\cdot(2\pi)^{4}\cdot 32440984770100000 \cdot 1}{2\sqrt{180689179599093672760171229322647431883530841}}\approx 0.481456882217510$ (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, 8.8.950338729925911961.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{16}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ R $16$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$

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
47Data not computed