↑

Properties

Label 16.8.25960186080...0625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{10}\cdot 149^{14}$
Root discriminant $217.97$
Ramified primes $5, 149$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2\times OD_{16}).C_2$ (as 16T123)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-620598976, 3661444944, -5561915004, 638329410, -63289201, -180375228, 62552122, 9918036, -3584998, -150192, 731, 0, 4947, 0, -131, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 131*x^14 + 4947*x^12 + 731*x^10 - 150192*x^9 - 3584998*x^8 + 9918036*x^7 + 62552122*x^6 - 180375228*x^5 - 63289201*x^4 + 638329410*x^3 - 5561915004*x^2 + 3661444944*x - 620598976)
 
gp: K = bnfinit(x^16 - 131*x^14 + 4947*x^12 + 731*x^10 - 150192*x^9 - 3584998*x^8 + 9918036*x^7 + 62552122*x^6 - 180375228*x^5 - 63289201*x^4 + 638329410*x^3 - 5561915004*x^2 + 3661444944*x - 620598976, 1)
 

Normalized defining polynomial

\( x^{16} - 131 x^{14} + 4947 x^{12} + 731 x^{10} - 150192 x^{9} - 3584998 x^{8} + 9918036 x^{7} + 62552122 x^{6} - 180375228 x^{5} - 63289201 x^{4} + 638329410 x^{3} - 5561915004 x^{2} + 3661444944 x - 620598976 \)

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:  \(25960186080802016874638962845712890625=5^{10}\cdot 149^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $217.97$
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:  $5, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{6} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{7} - \frac{1}{6} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{228} a^{12} + \frac{13}{228} a^{11} - \frac{1}{228} a^{10} - \frac{3}{19} a^{9} - \frac{10}{57} a^{8} + \frac{13}{114} a^{7} - \frac{23}{76} a^{6} - \frac{85}{228} a^{5} - \frac{1}{57} a^{4} - \frac{65}{228} a^{3} - \frac{113}{228} a^{2} + \frac{25}{114} a + \frac{1}{3}$, $\frac{1}{228} a^{13} - \frac{3}{38} a^{11} + \frac{5}{76} a^{10} - \frac{7}{57} a^{9} - \frac{2}{19} a^{8} + \frac{11}{228} a^{7} - \frac{2}{19} a^{6} - \frac{77}{228} a^{5} - \frac{17}{76} a^{4} + \frac{43}{114} a^{3} - \frac{13}{76} a^{2} - \frac{7}{38} a$, $\frac{1}{75468} a^{14} + \frac{13}{6289} a^{13} - \frac{1}{75468} a^{12} - \frac{1115}{37734} a^{11} + \frac{119}{25156} a^{10} - \frac{1215}{6289} a^{9} - \frac{1623}{25156} a^{8} - \frac{863}{18867} a^{7} - \frac{4982}{18867} a^{6} + \frac{17585}{37734} a^{5} - \frac{3109}{12578} a^{4} - \frac{12047}{37734} a^{3} - \frac{20587}{75468} a^{2} - \frac{11743}{37734} a + \frac{200}{993}$, $\frac{1}{502953849755502068441607405130245931539068665487570352} a^{15} + \frac{87099233413858318921376312499428763128260302297}{41912820812958505703467283760853827628255722123964196} a^{14} + \frac{58614188297498859526042176842686289964051613993151}{167651283251834022813869135043415310513022888495856784} a^{13} - \frac{25789596975047116501048899466164230813471545755837}{41912820812958505703467283760853827628255722123964196} a^{12} - \frac{22676918351160966660527606135307952992577700541352261}{502953849755502068441607405130245931539068665487570352} a^{11} - \frac{8652083698093712009118104147434303329568352134945193}{125738462438875517110401851282561482884767166371892588} a^{10} + \frac{77760788242742478399715792383308690868813818950738667}{502953849755502068441607405130245931539068665487570352} a^{9} + \frac{297920520998893154289221277478605742466123707176481}{41912820812958505703467283760853827628255722123964196} a^{8} - \frac{30105077376072297083170204762514257458939067426153183}{251476924877751034220803702565122965769534332743785176} a^{7} + \frac{24925202787150387125434663516307551689418013128775399}{125738462438875517110401851282561482884767166371892588} a^{6} + \frac{105897438809671371611805652284572881239515928718289269}{251476924877751034220803702565122965769534332743785176} a^{5} - \frac{54979901223054524063673609117428983735292239299578473}{125738462438875517110401851282561482884767166371892588} a^{4} + \frac{28890764636845615824163323229823062292225331564125053}{167651283251834022813869135043415310513022888495856784} a^{3} + \frac{79141881735768990126830756369042976299441598553540667}{251476924877751034220803702565122965769534332743785176} a^{2} + \frac{7611261494630451601588846420806533564825781660030909}{41912820812958505703467283760853827628255722123964196} a + \frac{107119726632870637205690767072243211284976415273407}{1654453453143098909347392780033703722167989031209113}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 18629811541500 \) (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

\(\Q(\sqrt{745}) \), \(\Q(\sqrt{149}) \), \(\Q(\sqrt{5}) \), 4.4.82698725.1 x2, 4.4.16539745.1 x2, \(\Q(\sqrt{5}, \sqrt{149})\), 8.8.6839079116625625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$149$149.8.7.2$x^{8} - 596$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
149.8.7.2$x^{8} - 596$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$