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Properties

Label 16.0.16039019200...0592.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{11}\cdot 7^{5}\cdot 294337^{7}$
Root discriminant $7723.67$
Ramified primes $2, 3, 7, 294337$
Class number $16$ (GRH)
Class group $[2, 8]$ (GRH)
Galois group 16T1022

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![86816405953497148304733, 0, 1342530927602222399244, 0, 10584531382898601450, 0, 38845570175747892, 0, 74842201814254, 0, 80560918772, 0, 46469406, 0, 11324, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 11324*x^14 + 46469406*x^12 + 80560918772*x^10 + 74842201814254*x^8 + 38845570175747892*x^6 + 10584531382898601450*x^4 + 1342530927602222399244*x^2 + 86816405953497148304733)
 
gp: K = bnfinit(x^16 + 11324*x^14 + 46469406*x^12 + 80560918772*x^10 + 74842201814254*x^8 + 38845570175747892*x^6 + 10584531382898601450*x^4 + 1342530927602222399244*x^2 + 86816405953497148304733, 1)
 

Normalized defining polynomial

\( x^{16} + 11324 x^{14} + 46469406 x^{12} + 80560918772 x^{10} + 74842201814254 x^{8} + 38845570175747892 x^{6} + 10584531382898601450 x^{4} + 1342530927602222399244 x^{2} + 86816405953497148304733 \)

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:  \(160390192004889035274309827110029156806685344432046320325230592=2^{48}\cdot 3^{11}\cdot 7^{5}\cdot 294337^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $7723.67$
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:  $2, 3, 7, 294337$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{6} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{3} + \frac{1}{6} a$, $\frac{1}{18} a^{10} + \frac{1}{9} a^{8} - \frac{1}{3} a^{6} + \frac{4}{9} a^{4} + \frac{7}{18} a^{2}$, $\frac{1}{54} a^{11} + \frac{1}{27} a^{9} + \frac{2}{9} a^{7} + \frac{4}{27} a^{5} + \frac{7}{54} a^{3} + \frac{1}{3} a$, $\frac{1}{3402} a^{12} - \frac{8}{1701} a^{10} - \frac{130}{567} a^{8} + \frac{328}{1701} a^{6} - \frac{197}{486} a^{4} - \frac{10}{21} a^{2}$, $\frac{1}{37425402} a^{13} + \frac{15149}{1969758} a^{11} - \frac{210361}{6237567} a^{9} - \frac{4678556}{18712701} a^{7} + \frac{1470385}{5346486} a^{5} - \frac{220525}{1386126} a^{3} + \frac{1978}{11001} a$, $\frac{1}{14265746346802424299289327558617575695435789969050182089031707208109626} a^{14} + \frac{40258743545636425415194364437992251525766086202803522135752149887}{750828755094864436804701450453556615549252103634220109949037221479454} a^{12} + \frac{33738759474616101600143656204645644928888784537741311228459950358664}{2377624391133737383214887926436262615905964994841697014838617868018271} a^{10} + \frac{30387668142747245498926126584511614622264090332903422035103371161683}{331761542948893588355565757177152923149669534163957723000737376932782} a^{8} - \frac{665673194865646040717077338391565230241746637335189934635295824285691}{2037963763828917757041332508373939385062255709864311727004529601158518} a^{6} - \frac{251562414480379218583165571308310771084175889520210403389131979598389}{528360975807497196269975094763613914645769998853710447741915081781838} a^{4} + \frac{1292997161800647959128287188192540005809335308709424731592129472118}{4193341077837279335475992815584237417823571419473892442396151442713} a^{2} + \frac{317561558516406712589572777032495016080645605357538799721231}{636889177438229293725783843541258900944992153363728146290058}$, $\frac{1}{2439442625303214555178475012523605443919520084707581137224421932586746046} a^{15} - \frac{908499105615839672652851640827641313516316606321137784144619117}{128391717121221818693603948027558181258922109721451638801285364872986634} a^{13} - \frac{214432328122989037099199769363794101314558385161168954952873182024647}{813147541767738185059491670841201814639840028235860379074807310862248682} a^{11} - \frac{1950852791273495587905802955692779817804438783155574337197670115491273}{28365611922130401804400872238646574929296745171018385316563045727752861} a^{9} - \frac{154107186986601310871532195238870750483091892383535651443921617244560565}{348491803614744936454067858931943634845645726386797305317774561798106578} a^{7} + \frac{15302792575871744153320845057612617302250883109549508450246681690680759}{90349726863082020562165741204577979404426669803984486563867478984694298} a^{5} - \frac{672395692000617569400456003865335445379354961499434135235120511175989}{1434122648620349532732789542929809196895661425460071215299483793407846} a^{3} + \frac{3936208348742414834214831747539520664371710433381724419260306962}{10509626761496940690416022094196083753943788018731560006005392087} a$

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

Class group and class number

$C_{2}\times C_{8}$, which has order $16$ (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:  \( -\frac{107850030165987851047966411}{9362598812129209694381801751542114155718624121} a^{14} - \frac{2296191305874258822041332967005}{18725197624258419388763603503084228311437248242} a^{12} - \frac{1408169924106077786647603038031327}{3120866270709736564793933917180704718572874707} a^{10} - \frac{5734864609881254627147248277936168078}{9362598812129209694381801751542114155718624121} a^{8} - \frac{550445961729007477175410697378955162079}{1337514116018458527768828821648873450816946303} a^{6} - \frac{81958295729338438991158730567451588328697}{693525837935497014398651981595712159682861046} a^{4} - \frac{37635703595521920355191533614572094971823}{5504173316948389003163904615838985394308421} a^{2} + \frac{8733626417625555926540592666894397893373}{29122610142584068799809019131423203144489} \) (order $6$)
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:  \( 4036710027520000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1022:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 29 conjugacy class representatives for t16n1022
Character table for t16n1022 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.42384528.1, 8.0.2842620092391385389662208.1

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
Arithmetically equvalently 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 R $16$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$3$3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
294337Data not computed