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Properties

Label 16.0.19221262311...0544.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{50}\cdot 17^{4}\cdot 103^{4}\cdot 3671^{4}$
Root discriminant $439.27$
Ramified primes $2, 17, 103, 3671$
Class number $235152$ (GRH)
Class group $[2, 6, 19596]$ (GRH)
Galois group 16T1392

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13920305159422, 1851433282320, -1157432693384, -35048106120, 56424774224, -1557349288, -2203368876, 33297712, 59042839, -2576672, -1108984, 46936, 22422, -200, -148, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 148*x^14 - 200*x^13 + 22422*x^12 + 46936*x^11 - 1108984*x^10 - 2576672*x^9 + 59042839*x^8 + 33297712*x^7 - 2203368876*x^6 - 1557349288*x^5 + 56424774224*x^4 - 35048106120*x^3 - 1157432693384*x^2 + 1851433282320*x + 13920305159422)
 
gp: K = bnfinit(x^16 - 148*x^14 - 200*x^13 + 22422*x^12 + 46936*x^11 - 1108984*x^10 - 2576672*x^9 + 59042839*x^8 + 33297712*x^7 - 2203368876*x^6 - 1557349288*x^5 + 56424774224*x^4 - 35048106120*x^3 - 1157432693384*x^2 + 1851433282320*x + 13920305159422, 1)
 

Normalized defining polynomial

\( x^{16} - 148 x^{14} - 200 x^{13} + 22422 x^{12} + 46936 x^{11} - 1108984 x^{10} - 2576672 x^{9} + 59042839 x^{8} + 33297712 x^{7} - 2203368876 x^{6} - 1557349288 x^{5} + 56424774224 x^{4} - 35048106120 x^{3} - 1157432693384 x^{2} + 1851433282320 x + 13920305159422 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1922126231168301760890646394255238227820544=2^{50}\cdot 17^{4}\cdot 103^{4}\cdot 3671^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $439.27$
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:  $2, 17, 103, 3671$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{51} a^{9} + \frac{2}{51} a^{8} + \frac{25}{51} a^{7} - \frac{20}{51} a^{6} + \frac{1}{3} a^{5} - \frac{7}{51} a^{4} + \frac{19}{51} a^{3} - \frac{13}{51} a^{2} - \frac{1}{51} a - \frac{1}{3}$, $\frac{1}{51} a^{10} + \frac{4}{51} a^{8} - \frac{19}{51} a^{7} - \frac{11}{51} a^{6} - \frac{8}{17} a^{5} + \frac{16}{51} a^{4} + \frac{8}{51} a^{2} - \frac{5}{17} a$, $\frac{1}{51} a^{11} + \frac{7}{51} a^{8} - \frac{3}{17} a^{7} - \frac{4}{17} a^{6} + \frac{16}{51} a^{5} + \frac{11}{51} a^{4} - \frac{1}{3} a^{3} + \frac{20}{51} a^{2} + \frac{4}{51} a - \frac{1}{3}$, $\frac{1}{204} a^{12} - \frac{23}{204} a^{8} + \frac{1}{3} a^{7} - \frac{4}{17} a^{6} + \frac{8}{17} a^{5} - \frac{35}{102} a^{4} + \frac{10}{51} a^{3} + \frac{11}{51} a^{2} + \frac{23}{51} a - \frac{1}{6}$, $\frac{1}{6936} a^{13} + \frac{1}{2312} a^{12} - \frac{7}{1734} a^{11} - \frac{1}{578} a^{10} - \frac{47}{6936} a^{9} - \frac{157}{6936} a^{8} + \frac{173}{578} a^{7} + \frac{259}{1734} a^{6} - \frac{489}{1156} a^{5} + \frac{517}{1156} a^{4} + \frac{380}{867} a^{3} + \frac{2}{867} a^{2} - \frac{853}{3468} a - \frac{83}{204}$, $\frac{1}{5007022104} a^{14} - \frac{7069}{1669007368} a^{13} + \frac{2424467}{2503511052} a^{12} - \frac{984583}{1251755526} a^{11} - \frac{1203263}{294530712} a^{10} - \frac{6287719}{5007022104} a^{9} + \frac{323509411}{2503511052} a^{8} - \frac{1089971}{178822218} a^{7} - \frac{1223441983}{2503511052} a^{6} - \frac{728905439}{2503511052} a^{5} + \frac{14892517}{1251755526} a^{4} - \frac{83104853}{625877763} a^{3} - \frac{23510843}{357644436} a^{2} + \frac{1085859155}{2503511052} a - \frac{9123609}{24544226}$, $\frac{1}{3364985398053353809674681247113865874347493468109682276962913306728} a^{15} + \frac{6272407930017913745047044512205171568541149428732339289}{65980105844183408032836887198311095575441048394307495626723790328} a^{14} - \frac{109037784088424307096987000616369719370828671492380637924170869}{1682492699026676904837340623556932937173746734054841138481456653364} a^{13} - \frac{1677670926610034953407621115862584694334516374586316939008419773}{1682492699026676904837340623556932937173746734054841138481456653364} a^{12} + \frac{19917467577717573877320600173013006109137811279898842409110487593}{3364985398053353809674681247113865874347493468109682276962913306728} a^{11} - \frac{1478066351115632040430959507664153714528184049095704046937887241}{160237399907302562365461011767326946397499688957603917950614919368} a^{10} + \frac{1359106263447948093842210637017651784370871175178868314441640637}{560830899675558968279113541185644312391248911351613712827152217788} a^{9} - \frac{63074136074022917797509810227345420865644151555141008864405344291}{560830899675558968279113541185644312391248911351613712827152217788} a^{8} - \frac{427108211873142367853315027400039578042129525006261712528088768199}{1682492699026676904837340623556932937173746734054841138481456653364} a^{7} + \frac{319120549059277341768696778715770777867206466294543666066925702467}{1682492699026676904837340623556932937173746734054841138481456653364} a^{6} + \frac{249985549144843795980627478643983570061562179138829774192121568785}{841246349513338452418670311778466468586873367027420569240728326682} a^{5} - \frac{88120344983442223371028274010466080863439163948525620820353244603}{841246349513338452418670311778466468586873367027420569240728326682} a^{4} - \frac{609070185856874633086033449466692176139998702304435691989643017085}{1682492699026676904837340623556932937173746734054841138481456653364} a^{3} - \frac{744231650855177879394031259251802983849715857858243981109269459439}{1682492699026676904837340623556932937173746734054841138481456653364} a^{2} - \frac{413237791248839934051761488093917344500931028114928966379404598293}{841246349513338452418670311778466468586873367027420569240728326682} a + \frac{5940377094317919791007021862537178650050697363526402725401408571}{16495026461045852008209221799577773893860262098576873906680947582}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{19596}$, which has order $235152$ (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:  $7$
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:  \( 5714559607.36 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1392:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 80 conjugacy class representatives for t16n1392 are not computed
Character table for t16n1392 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 4.0.24199232.1, 4.0.774375424.1, 8.0.599657297295179776.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.24.9$x^{8} + 8 x^{7} + 14 x^{4} + 4 x^{2} + 8 x + 30$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
2.8.26.4$x^{8} + 8 x^{7} + 12 x^{6} + 8 x^{5} + 8 x^{4} + 8 x^{3} + 2$$8$$1$$26$$C_2^2:C_4$$[2, 3, 7/2, 4]$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
$103$103.4.0.1$x^{4} - x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
103.4.0.1$x^{4} - x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
103.8.4.1$x^{8} + 106090 x^{4} - 1092727 x^{2} + 2813772025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3671Data not computed