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Properties

Label 16.0.17987372096...6009.5
Degree $16$
Signature $[0, 8]$
Discriminant $67^{12}\cdot 89^{13}$
Root discriminant $898.33$
Ramified primes $67, 89$
Class number $16$ (GRH)
Class group $[4, 4]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T260)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![636936648311835307, -19004907231570663, 1851579194059094, -646734985272706, 56327674930810, -10483386432081, 955574028589, -65084516877, 6418777262, -240575709, 7526907, 784396, -120134, 6167, -74, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 74*x^14 + 6167*x^13 - 120134*x^12 + 784396*x^11 + 7526907*x^10 - 240575709*x^9 + 6418777262*x^8 - 65084516877*x^7 + 955574028589*x^6 - 10483386432081*x^5 + 56327674930810*x^4 - 646734985272706*x^3 + 1851579194059094*x^2 - 19004907231570663*x + 636936648311835307)
 
gp: K = bnfinit(x^16 - 2*x^15 - 74*x^14 + 6167*x^13 - 120134*x^12 + 784396*x^11 + 7526907*x^10 - 240575709*x^9 + 6418777262*x^8 - 65084516877*x^7 + 955574028589*x^6 - 10483386432081*x^5 + 56327674930810*x^4 - 646734985272706*x^3 + 1851579194059094*x^2 - 19004907231570663*x + 636936648311835307, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 74 x^{14} + 6167 x^{13} - 120134 x^{12} + 784396 x^{11} + 7526907 x^{10} - 240575709 x^{9} + 6418777262 x^{8} - 65084516877 x^{7} + 955574028589 x^{6} - 10483386432081 x^{5} + 56327674930810 x^{4} - 646734985272706 x^{3} + 1851579194059094 x^{2} - 19004907231570663 x + 636936648311835307 \)

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:  \(179873720961956854769314815076056959198372636009=67^{12}\cdot 89^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $898.33$
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:  $67, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{3844574578819928898259623094305898128645590536674421326075686156524257592108149809422003577864141634961871174294306} a^{15} + \frac{212237447464938608305040705915068079185726050073849424350164780375997259200651086369893537658001179670110577492240}{1922287289409964449129811547152949064322795268337210663037843078262128796054074904711001788932070817480935587147153} a^{14} - \frac{308260651412316908506862092716625605924265707007944704233698097138939664081902941751697513645791794700278340059611}{3844574578819928898259623094305898128645590536674421326075686156524257592108149809422003577864141634961871174294306} a^{13} - \frac{343742795981442006046449174144974963467623087969245910859916524159644914937705783703774917367326454038775961119584}{1922287289409964449129811547152949064322795268337210663037843078262128796054074904711001788932070817480935587147153} a^{12} - \frac{287077328178302331685636594857288538402134586522843712054667214404433044433765916749030342349937909466133765777844}{1922287289409964449129811547152949064322795268337210663037843078262128796054074904711001788932070817480935587147153} a^{11} - \frac{696968100385673632339096796101696816975043132443652917652199233782164684644344570712253149630979369061681860876979}{3844574578819928898259623094305898128645590536674421326075686156524257592108149809422003577864141634961871174294306} a^{10} - \frac{1419789644750170285930967637529550904588688060222819950818182741368188934437888591945771889015707785187857906721085}{3844574578819928898259623094305898128645590536674421326075686156524257592108149809422003577864141634961871174294306} a^{9} - \frac{280976518774835191661416309529059641008726416001721342223030276050472721473229170356112816839964070347159296797057}{1922287289409964449129811547152949064322795268337210663037843078262128796054074904711001788932070817480935587147153} a^{8} + \frac{277091404822038057288134304563041553748501692670009196873131019673656459374037504951781201879742546902027280880621}{3844574578819928898259623094305898128645590536674421326075686156524257592108149809422003577864141634961871174294306} a^{7} + \frac{1152505105283374603885899449732208971525173645351444953004893690637836940235423805297674556273996449172348622147625}{3844574578819928898259623094305898128645590536674421326075686156524257592108149809422003577864141634961871174294306} a^{6} - \frac{92357846569931431532802293694633674108003271995635493481673229849971603073902155942109736910716295658970723378877}{349506779892720808932693008573263466240508230606765575097789650593114326555286346311091234351285603178351924935846} a^{5} + \frac{374605047137285154668215343721811439402841045094131439226362897173145209863026171267065415545781543929152052630054}{1922287289409964449129811547152949064322795268337210663037843078262128796054074904711001788932070817480935587147153} a^{4} - \frac{627812254627890899676570292722082954369772806510343535342579664204376526481426931409267231537169945359144606113898}{1922287289409964449129811547152949064322795268337210663037843078262128796054074904711001788932070817480935587147153} a^{3} - \frac{641066047557954790802336445310360406044730040782615105820788881700873825734639732569396424421359819637506833691632}{1922287289409964449129811547152949064322795268337210663037843078262128796054074904711001788932070817480935587147153} a^{2} - \frac{81662148124079348566220974918210068977274997880491026557771026063955615094997461993428650311936567263967959639047}{174753389946360404466346504286631733120254115303382787548894825296557163277643173155545617175642801589175962467923} a - \frac{375922542970584624058581376470689047947374921603027175151016866180637970816859608984027444066942536422346750451877}{1922287289409964449129811547152949064322795268337210663037843078262128796054074904711001788932070817480935587147153}$

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

Class group and class number

$C_{4}\times C_{4}$, 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:  \( -1 \) (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:  \( 5924611583950000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T260):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-67}) \), 4.0.399521.1, 8.0.112525057627992329.3

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$67$67.8.6.2$x^{8} + 1541 x^{4} + 646416$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
67.8.6.2$x^{8} + 1541 x^{4} + 646416$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$89$89.4.3.3$x^{4} + 267$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.3.4$x^{4} + 2403$$4$$1$$3$$C_4$$[\ ]_{4}$
89.8.7.2$x^{8} - 801$$8$$1$$7$$C_8$$[\ ]_{8}$