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Properties

Label 21.3.43266744111...4983.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,7^{17}\cdot 223^{18}$
Root discriminant $497.69$
Ramified primes $7, 223$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $F_7$ (as 21T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-256310093312, -2037042160384, 1100178148832, 402334865128, -535267042184, 8157442846, 78811833717, -16153054924, -809756522, 352134365, -16939676, -17206833, 7864242, -967981, -66028, 39331, -6696, 29, -34, 21, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 21*x^19 - 34*x^18 + 29*x^17 - 6696*x^16 + 39331*x^15 - 66028*x^14 - 967981*x^13 + 7864242*x^12 - 17206833*x^11 - 16939676*x^10 + 352134365*x^9 - 809756522*x^8 - 16153054924*x^7 + 78811833717*x^6 + 8157442846*x^5 - 535267042184*x^4 + 402334865128*x^3 + 1100178148832*x^2 - 2037042160384*x - 256310093312)
 
gp: K = bnfinit(x^21 - 7*x^20 + 21*x^19 - 34*x^18 + 29*x^17 - 6696*x^16 + 39331*x^15 - 66028*x^14 - 967981*x^13 + 7864242*x^12 - 17206833*x^11 - 16939676*x^10 + 352134365*x^9 - 809756522*x^8 - 16153054924*x^7 + 78811833717*x^6 + 8157442846*x^5 - 535267042184*x^4 + 402334865128*x^3 + 1100178148832*x^2 - 2037042160384*x - 256310093312, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 29 x^{17} - 6696 x^{16} + 39331 x^{15} - 66028 x^{14} - 967981 x^{13} + 7864242 x^{12} - 17206833 x^{11} - 16939676 x^{10} + 352134365 x^{9} - 809756522 x^{8} - 16153054924 x^{7} + 78811833717 x^{6} + 8157442846 x^{5} - 535267042184 x^{4} + 402334865128 x^{3} + 1100178148832 x^{2} - 2037042160384 x - 256310093312 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-432667441118225293081744612251055070987538949548395174983=-\,7^{17}\cdot 223^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $497.69$
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:  $7, 223$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{4} a^{10} - \frac{1}{8} a^{9} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{3}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{14} - \frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{8} + \frac{3}{8} a^{6} + \frac{3}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{17} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{3}{8} a^{3}$, $\frac{1}{2435160} a^{18} + \frac{35851}{2435160} a^{17} + \frac{96907}{2435160} a^{16} + \frac{43237}{2435160} a^{15} - \frac{193607}{2435160} a^{14} + \frac{147293}{2435160} a^{13} + \frac{145193}{2435160} a^{12} + \frac{183221}{2435160} a^{11} - \frac{37041}{811720} a^{10} - \frac{1377}{62440} a^{9} + \frac{105821}{2435160} a^{8} - \frac{555823}{2435160} a^{7} - \frac{686867}{2435160} a^{6} - \frac{10795}{23192} a^{5} - \frac{3602}{14495} a^{4} - \frac{11153}{34788} a^{3} + \frac{11589}{57980} a^{2} - \frac{3624}{14495} a + \frac{442}{3345}$, $\frac{1}{9740640} a^{19} + \frac{1}{9740640} a^{18} + \frac{603037}{9740640} a^{17} - \frac{6289}{4870320} a^{16} + \frac{348073}{9740640} a^{15} + \frac{55981}{1217580} a^{14} - \frac{881977}{9740640} a^{13} - \frac{33931}{2435160} a^{12} + \frac{12387}{463840} a^{11} - \frac{391113}{1623440} a^{10} + \frac{1581971}{9740640} a^{9} + \frac{467843}{2435160} a^{8} - \frac{105401}{1391520} a^{7} + \frac{129509}{324688} a^{6} - \frac{24317}{57980} a^{5} - \frac{103609}{278304} a^{4} - \frac{47771}{231920} a^{3} - \frac{20609}{57980} a^{2} + \frac{10891}{173940} a - \frac{63}{223}$, $\frac{1}{24809160885989460455147821822740170394415197652902917139289024864783360098708841027828643376911586051776640} a^{20} - \frac{61553107171721712307198229553653379145982867377488400497748474085097966371686243021295712341985499}{24809160885989460455147821822740170394415197652902917139289024864783360098708841027828643376911586051776640} a^{19} + \frac{853556136462597036640323332884180370943823341066798268220501457914198555703006861740043208230756397}{4961832177197892091029564364548034078883039530580583427857804972956672019741768205565728675382317210355328} a^{18} + \frac{15305471181515264986210316513392507469116794341675904950363941475725841354664943968565198465174259053991}{354416584085563720787826026039145291348788537898613101989843212354048001410126300397552048241594086453952} a^{17} + \frac{187254054479005686932219026421322792981418650610404465834121233692740639981846382743567538205842979787109}{24809160885989460455147821822740170394415197652902917139289024864783360098708841027828643376911586051776640} a^{16} - \frac{66925638503503953780126965945435324003854781688797846757316526164085510947987235429589115625417320683}{22719011800356648768450386284560595599281316531962378332682257202182564192956814128048208220615005541920} a^{15} + \frac{755381580377646407949699109106005065087046432877439194957817442689712379348629634363218150035800468627587}{24809160885989460455147821822740170394415197652902917139289024864783360098708841027828643376911586051776640} a^{14} - \frac{29934225914591954648842829177758574506937530014251127649539564540100907285271929004131620072702477987997}{620229022149736511378695545568504259860379941322572928482225621619584002467721025695716084422789651294416} a^{13} - \frac{80471618238907400601937686367666487123609705078224095436393673699830355963160719228719738714419247018429}{24809160885989460455147821822740170394415197652902917139289024864783360098708841027828643376911586051776640} a^{12} - \frac{12895069971871812950586061058423555936698422411997289718501597032173318778229645763278123335967628242061}{118138861361854573595942008679715097116262845966204367329947737451349333803375433465850682747198028817984} a^{11} - \frac{3640284832260280255341887836164711031658123357252183686167159365007833735593527757533881869756969272980473}{24809160885989460455147821822740170394415197652902917139289024864783360098708841027828643376911586051776640} a^{10} - \frac{399720498920606726100755900777261912657120030357899864869341553217452084057234714906097710379218791244109}{3101145110748682556893477727842521299301899706612864642411128108097920012338605128478580422113948256472080} a^{9} + \frac{369113156395277265763404366255897219416673915593868801555817594650843517200372828605296574436480051560687}{8269720295329820151715940607580056798138399217634305713096341621594453366236280342609547792303862017258880} a^{8} - \frac{987578491343888540512273762113445341391642885287883837472732487105326407994484591236886285241563923641207}{12404580442994730227573910911370085197207598826451458569644512432391680049354420513914321688455793025888320} a^{7} + \frac{514408684393262487647834050146760769872865724283047363571622001360743004277546722945718238898803821653469}{2067430073832455037928985151895014199534599804408576428274085405398613341559070085652386948075965504314720} a^{6} - \frac{246992667356507282751666924795069679475006296757557423549905372558405461512950331057445040310794540603817}{708833168171127441575652052078290582697577075797226203979686424708096002820252600795104096483188172907904} a^{5} + \frac{149688364576183731428292052265348970377098834308271666693072943161059790212652533610117742393752094869711}{354416584085563720787826026039145291348788537898613101989843212354048001410126300397552048241594086453952} a^{4} - \frac{33454838654945352057049355250083939788140561624131531329742118229624274739462980593559737251106659533487}{73836788351159108497463755424821935697664278728877729581217335907093333627109645916156676716998768011240} a^{3} + \frac{30246830978313213974581023165467131688291149660927006846204372081480889561970626131396434029426686429331}{88604146021390930196956506509786322837197134474653275497460803088512000352531575099388012060398521613488} a^{2} - \frac{21487747132388429589921709268709743439263893762738203644511431244910084940779695684847191227961954768577}{55377591263369331373097816568616451773248209046658297185913001930320000220332234437117507537749076008430} a + \frac{163342466803490889743299098177576359413182522130494130746405059118932134264856527115181392577823899942}{709969118761145274014074571392518612477541141623824322896320537568205131029900441501506506894218923185}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 327908201372163200000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_7$ (as 21T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 42
The 7 conjugacy class representatives for $F_7$
Character table for $F_7$

Intermediate fields

\(\Q(\zeta_{7})^+\), Deg 7

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

Sibling fields

Galois closure: data not computed
Degree 7 sibling: data not computed
Degree 14 sibling: 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
223Data not computed