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Properties

Label 16.8.41340064862...5625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{14}\cdot 29^{6}\cdot 101^{4}\cdot 149^{6}$
Root discriminant $299.24$
Ramified primes $5, 29, 101, 149$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group 16T1276

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16346881, 61506276, -414545290, -236386720, 97186535, -118981912, 117179468, -32604540, 20197965, -2557780, 1094828, -42032, 12775, 1060, -290, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 290*x^14 + 1060*x^13 + 12775*x^12 - 42032*x^11 + 1094828*x^10 - 2557780*x^9 + 20197965*x^8 - 32604540*x^7 + 117179468*x^6 - 118981912*x^5 + 97186535*x^4 - 236386720*x^3 - 414545290*x^2 + 61506276*x + 16346881)
 
gp: K = bnfinit(x^16 - 4*x^15 - 290*x^14 + 1060*x^13 + 12775*x^12 - 42032*x^11 + 1094828*x^10 - 2557780*x^9 + 20197965*x^8 - 32604540*x^7 + 117179468*x^6 - 118981912*x^5 + 97186535*x^4 - 236386720*x^3 - 414545290*x^2 + 61506276*x + 16346881, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 290 x^{14} + 1060 x^{13} + 12775 x^{12} - 42032 x^{11} + 1094828 x^{10} - 2557780 x^{9} + 20197965 x^{8} - 32604540 x^{7} + 117179468 x^{6} - 118981912 x^{5} + 97186535 x^{4} - 236386720 x^{3} - 414545290 x^{2} + 61506276 x + 16346881 \)

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:  \(4134006486278719588795586854986572265625=5^{14}\cdot 29^{6}\cdot 101^{4}\cdot 149^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $299.24$
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:  $5, 29, 101, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{20} a^{8} - \frac{1}{10} a^{7} + \frac{3}{20} a^{6} - \frac{1}{5} a^{5} - \frac{1}{4} a^{4} + \frac{3}{10} a^{3} - \frac{7}{20} a^{2} + \frac{2}{5} a - \frac{9}{20}$, $\frac{1}{20} a^{9} - \frac{1}{20} a^{7} + \frac{1}{10} a^{6} - \frac{3}{20} a^{5} - \frac{1}{5} a^{4} + \frac{1}{4} a^{3} - \frac{3}{10} a^{2} + \frac{7}{20} a - \frac{2}{5}$, $\frac{1}{20} a^{10} + \frac{1}{10} a^{5} + \frac{1}{20}$, $\frac{1}{20} a^{11} + \frac{1}{10} a^{6} + \frac{1}{20} a$, $\frac{1}{172840} a^{12} - \frac{369}{172840} a^{11} - \frac{91}{43210} a^{10} - \frac{21}{17284} a^{9} - \frac{3381}{172840} a^{8} - \frac{8313}{86420} a^{7} - \frac{3821}{172840} a^{6} + \frac{2633}{86420} a^{5} + \frac{3135}{34568} a^{4} + \frac{6587}{86420} a^{3} - \frac{39071}{86420} a^{2} + \frac{9463}{172840} a + \frac{13527}{34568}$, $\frac{1}{172840} a^{13} + \frac{1747}{172840} a^{11} + \frac{1873}{86420} a^{10} - \frac{3093}{172840} a^{9} - \frac{2483}{172840} a^{8} - \frac{11637}{172840} a^{7} - \frac{39247}{172840} a^{6} - \frac{20189}{172840} a^{5} + \frac{41677}{172840} a^{4} - \frac{23907}{86420} a^{3} - \frac{12507}{172840} a^{2} + \frac{19189}{43210} a + \frac{13671}{34568}$, $\frac{1}{17456840} a^{14} - \frac{13}{8728420} a^{13} - \frac{41}{17456840} a^{12} + \frac{81067}{4364210} a^{11} - \frac{184227}{17456840} a^{10} - \frac{4033}{601960} a^{9} + \frac{6747}{601960} a^{8} - \frac{4128013}{17456840} a^{7} + \frac{581543}{17456840} a^{6} + \frac{3422623}{17456840} a^{5} - \frac{1447639}{8728420} a^{4} + \frac{766969}{17456840} a^{3} + \frac{1967827}{4364210} a^{2} + \frac{1640463}{3491368} a - \frac{249839}{1745684}$, $\frac{1}{1284414372869691541529966475227034992450991095240} a^{15} - \frac{6995742942576385577433232452967641085999}{1284414372869691541529966475227034992450991095240} a^{14} + \frac{257688228356639836138043761597960339172619}{128441437286969154152996647522703499245099109524} a^{13} + \frac{826628985189638940532553241668443533423069}{321103593217422885382491618806758748112747773810} a^{12} - \frac{20990967361216913000284390378813665151753953283}{1284414372869691541529966475227034992450991095240} a^{11} + \frac{2988006331401421993515460966614664280435190299}{128441437286969154152996647522703499245099109524} a^{10} + \frac{2535561107108800068544065024671067227868628773}{256882874573938308305993295045406998490198219048} a^{9} + \frac{4370276811852321991796367032128574490255164337}{321103593217422885382491618806758748112747773810} a^{8} + \frac{189161791922572027042064892735126898543253924573}{1284414372869691541529966475227034992450991095240} a^{7} - \frac{67615136328323364927124458595561791069246521019}{321103593217422885382491618806758748112747773810} a^{6} + \frac{23874711856658577606945759275555992195471761569}{321103593217422885382491618806758748112747773810} a^{5} + \frac{153485134036040083993277143665155391064820350379}{1284414372869691541529966475227034992450991095240} a^{4} + \frac{97818467460255308971884501954663256839919380523}{1284414372869691541529966475227034992450991095240} a^{3} - \frac{25226456313568703210452684627568488329282104237}{321103593217422885382491618806758748112747773810} a^{2} + \frac{128700869364850777359772352303979924679522521183}{321103593217422885382491618806758748112747773810} a - \frac{189573085982275026847920989656784872384580202099}{642207186434845770764983237613517496225495547620}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( 7963521636260 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1276:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1276
Character table for t16n1276 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.540125.2, 8.8.29465236578125.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
5Data not computed
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
101Data not computed
$149$149.4.3.3$x^{4} + 298$$4$$1$$3$$C_4$$[\ ]_{4}$
149.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
149.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
149.4.3.4$x^{4} + 1192$$4$$1$$3$$C_4$$[\ ]_{4}$