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Properties

Label 16.8.25934622137...849.13
Degree $16$
Signature $[8, 4]$
Discriminant $37^{10}\cdot 149^{10}$
Root discriminant $217.96$
Ramified primes $37, 149$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $(C_2\times OD_{16}).C_2$ (as 16T123)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-497848761, 1153732464, -875529162, 361128255, -56895286, 7629119, -4535462, 382749, -5272, -208356, 86127, -3267, -2046, 491, -51, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 51*x^14 + 491*x^13 - 2046*x^12 - 3267*x^11 + 86127*x^10 - 208356*x^9 - 5272*x^8 + 382749*x^7 - 4535462*x^6 + 7629119*x^5 - 56895286*x^4 + 361128255*x^3 - 875529162*x^2 + 1153732464*x - 497848761)
 
gp: K = bnfinit(x^16 - 6*x^15 - 51*x^14 + 491*x^13 - 2046*x^12 - 3267*x^11 + 86127*x^10 - 208356*x^9 - 5272*x^8 + 382749*x^7 - 4535462*x^6 + 7629119*x^5 - 56895286*x^4 + 361128255*x^3 - 875529162*x^2 + 1153732464*x - 497848761, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 51 x^{14} + 491 x^{13} - 2046 x^{12} - 3267 x^{11} + 86127 x^{10} - 208356 x^{9} - 5272 x^{8} + 382749 x^{7} - 4535462 x^{6} + 7629119 x^{5} - 56895286 x^{4} + 361128255 x^{3} - 875529162 x^{2} + 1153732464 x - 497848761 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(25934622137786186512293731859285256849=37^{10}\cdot 149^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $217.96$
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:  $37, 149$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{9} a^{11} + \frac{2}{9} a^{10} - \frac{4}{9} a^{9} - \frac{1}{9} a^{8} - \frac{4}{9} a^{6} + \frac{1}{9} a^{5} + \frac{4}{9} a^{4} + \frac{2}{9} a^{3} + \frac{2}{9} a^{2} + \frac{4}{9} a + \frac{1}{3}$, $\frac{1}{1341} a^{12} + \frac{59}{1341} a^{11} + \frac{542}{1341} a^{10} - \frac{391}{1341} a^{9} - \frac{1}{447} a^{8} + \frac{653}{1341} a^{7} + \frac{394}{1341} a^{6} + \frac{538}{1341} a^{5} - \frac{292}{1341} a^{4} + \frac{647}{1341} a^{3} + \frac{145}{1341} a^{2} + \frac{200}{447} a - \frac{45}{149}$, $\frac{1}{6705} a^{13} - \frac{12}{745} a^{11} - \frac{1228}{6705} a^{10} - \frac{556}{2235} a^{9} - \frac{667}{2235} a^{8} - \frac{13}{149} a^{7} + \frac{278}{2235} a^{6} - \frac{1042}{6705} a^{5} + \frac{793}{2235} a^{4} - \frac{1523}{6705} a^{3} + \frac{346}{1341} a^{2} - \frac{1684}{6705} a + \frac{68}{2235}$, $\frac{1}{60345} a^{14} - \frac{1}{20115} a^{13} + \frac{4}{20115} a^{12} + \frac{1706}{60345} a^{11} + \frac{3727}{20115} a^{10} - \frac{716}{2235} a^{9} - \frac{7999}{20115} a^{8} - \frac{8777}{20115} a^{7} - \frac{12139}{60345} a^{6} + \frac{797}{4023} a^{5} + \frac{3776}{12069} a^{4} - \frac{18871}{60345} a^{3} - \frac{5119}{60345} a^{2} + \frac{4147}{20115} a + \frac{1982}{6705}$, $\frac{1}{2800665859480130702942634606487242747636745964419717185} a^{15} + \frac{477345630740973513454134806943766720637864590879}{311185095497792300326959400720804749737416218268857465} a^{14} - \frac{19215901473249597866695369138867489982076716863313}{933555286493376900980878202162414249212248654806572395} a^{13} + \frac{715133664102036802346532398418074653659185869007639}{2800665859480130702942634606487242747636745964419717185} a^{12} - \frac{4021368081276028423777885839562015272525354965847184}{103728365165930766775653133573601583245805406089619155} a^{11} + \frac{6640825696640661153530243371668573198834490872385877}{103728365165930766775653133573601583245805406089619155} a^{10} - \frac{215722238436912560019344335833402975176936479572101296}{933555286493376900980878202162414249212248654806572395} a^{9} - \frac{6988903622371629521139927216424767800348803068792433}{186711057298675380196175640432482849842449730961314479} a^{8} - \frac{211011060608140577035126864076656065483002888613726058}{2800665859480130702942634606487242747636745964419717185} a^{7} - \frac{1261030242494080856919158069244377124396040429686738}{3841791302441880250950116058281540120215015040356265} a^{6} + \frac{1304284259867314497261911436231146796896696167267246033}{2800665859480130702942634606487242747636745964419717185} a^{5} - \frac{659990860494921298178903099725427724943937477035067122}{2800665859480130702942634606487242747636745964419717185} a^{4} + \frac{664419288183691270805345927142921842976447747151333784}{2800665859480130702942634606487242747636745964419717185} a^{3} + \frac{948289171422934652490217378810291309436651087411609}{2305074781465128150570069634968924072129009024213759} a^{2} - \frac{221207016658202292417911268327575361639307012381777}{1115358765225062008340356274984963260707585011716335} a + \frac{11788518687779878053463390478481188651056946402399348}{103728365165930766775653133573601583245805406089619155}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  $11$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
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:  \( 2129463444270 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).C_2$ (as 16T123):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $(C_2\times OD_{16}).C_2$
Character table for $(C_2\times OD_{16}).C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{149}) \), \(\Q(\sqrt{37}) \), \(\Q(\sqrt{5513}) \), 4.4.821437.1 x2, 4.4.203981.1 x2, \(\Q(\sqrt{37}, \sqrt{149})\), 8.8.923744721862561.1

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
37Data not computed
149Data not computed