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Properties

Label 21.3.14693654444...0047.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,7^{17}\cdot 1259^{18}$
Root discriminant $2194.28$
Ramified primes $7, 1259$
Class number Not computed
Class group Not computed
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![-26412654578702301, -62218052737609482, -12369631255580718, 2502989432053803, -1260636204184623, 39614286391869, 248434219855942, -44833968691560, 284300561201, -222335916920, 108000335718, -18581839787, 2442806710, -78804229, -8455612, 1591903, -214036, 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 - 214036*x^16 + 1591903*x^15 - 8455612*x^14 - 78804229*x^13 + 2442806710*x^12 - 18581839787*x^11 + 108000335718*x^10 - 222335916920*x^9 + 284300561201*x^8 - 44833968691560*x^7 + 248434219855942*x^6 + 39614286391869*x^5 - 1260636204184623*x^4 + 2502989432053803*x^3 - 12369631255580718*x^2 - 62218052737609482*x - 26412654578702301)
 
gp: K = bnfinit(x^21 - 7*x^20 + 21*x^19 - 34*x^18 + 29*x^17 - 214036*x^16 + 1591903*x^15 - 8455612*x^14 - 78804229*x^13 + 2442806710*x^12 - 18581839787*x^11 + 108000335718*x^10 - 222335916920*x^9 + 284300561201*x^8 - 44833968691560*x^7 + 248434219855942*x^6 + 39614286391869*x^5 - 1260636204184623*x^4 + 2502989432053803*x^3 - 12369631255580718*x^2 - 62218052737609482*x - 26412654578702301, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 29 x^{17} - 214036 x^{16} + 1591903 x^{15} - 8455612 x^{14} - 78804229 x^{13} + 2442806710 x^{12} - 18581839787 x^{11} + 108000335718 x^{10} - 222335916920 x^{9} + 284300561201 x^{8} - 44833968691560 x^{7} + 248434219855942 x^{6} + 39614286391869 x^{5} - 1260636204184623 x^{4} + 2502989432053803 x^{3} - 12369631255580718 x^{2} - 62218052737609482 x - 26412654578702301 \)

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:  \(-14693654444550188334618410219169833396067319235696680397949255095300047=-\,7^{17}\cdot 1259^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2194.28$
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:  $7, 1259$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{4} a^{14} - \frac{1}{4}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{3}{8}$, $\frac{1}{48} a^{16} - \frac{1}{48} a^{15} - \frac{1}{16} a^{14} - \frac{1}{48} a^{13} - \frac{1}{48} a^{12} - \frac{1}{48} a^{11} + \frac{1}{12} a^{10} + \frac{11}{48} a^{9} - \frac{1}{12} a^{8} + \frac{1}{12} a^{7} + \frac{7}{48} a^{6} + \frac{3}{16} a^{5} - \frac{17}{48} a^{4} + \frac{1}{6} a^{3} + \frac{1}{4} a^{2} - \frac{5}{48} a + \frac{3}{16}$, $\frac{1}{144} a^{17} - \frac{1}{144} a^{16} - \frac{1}{48} a^{15} + \frac{11}{144} a^{14} - \frac{13}{144} a^{13} - \frac{13}{144} a^{12} + \frac{1}{36} a^{11} - \frac{25}{144} a^{10} - \frac{7}{36} a^{9} + \frac{7}{36} a^{8} - \frac{5}{144} a^{7} + \frac{23}{48} a^{6} + \frac{19}{144} a^{5} + \frac{7}{18} a^{4} + \frac{1}{3} a^{3} + \frac{43}{144} a^{2} + \frac{1}{16} a - \frac{1}{2}$, $\frac{1}{11421648} a^{18} + \frac{17681}{11421648} a^{17} - \frac{10859}{1269072} a^{16} - \frac{543679}{11421648} a^{15} + \frac{922679}{11421648} a^{14} - \frac{862255}{11421648} a^{13} + \frac{13654}{713853} a^{12} - \frac{95071}{1631664} a^{11} - \frac{24989}{815832} a^{10} + \frac{256565}{5710824} a^{9} - \frac{2738339}{11421648} a^{8} - \frac{787229}{3807216} a^{7} + \frac{4781125}{11421648} a^{6} + \frac{64705}{203958} a^{5} + \frac{51509}{135972} a^{4} - \frac{56663}{1631664} a^{3} + \frac{98351}{543888} a^{2} + \frac{4019}{15108} a + \frac{1396}{3777}$, $\frac{1}{479709216} a^{19} + \frac{11}{479709216} a^{18} + \frac{32047}{39975768} a^{17} - \frac{2112973}{479709216} a^{16} - \frac{26429479}{479709216} a^{15} + \frac{1705943}{479709216} a^{14} + \frac{5062417}{119927304} a^{13} - \frac{39907219}{479709216} a^{12} + \frac{90497}{68529888} a^{11} + \frac{58622965}{479709216} a^{10} + \frac{8238413}{59963652} a^{9} + \frac{1595663}{53301024} a^{8} + \frac{57771527}{239854608} a^{7} + \frac{59218027}{239854608} a^{6} + \frac{768839}{1903608} a^{5} + \frac{16160603}{34264944} a^{4} - \frac{10709627}{22843296} a^{3} + \frac{2543479}{7614432} a^{2} + \frac{106987}{634536} a + \frac{88295}{846048}$, $\frac{1}{58240510610838340729852391683545816538096149050678699282733343342806282281198862409327365341056852292858697456346493534176244154117739280023665792} a^{20} - \frac{12843211908854240129458143721495042642460759178898699594196304639657629630492839278609007372784433159044746991954988174876618366643862383}{14560127652709585182463097920886454134524037262669674820683335835701570570299715602331841335264213073214674364086623383544061038529434820005916448} a^{19} + \frac{16384182963143670956877517269701405373847456353585616373217361228801408555788444546975450229384058335737885551465041901733365265649786885}{497782141972977271195319587038853132804240590176741019510541396092361387018793695806216797786810703357766644926038406275010633795878113504475776} a^{18} + \frac{167263767518406852652117281836840823446845326094368698315862487315817838264270827611633811560774342751670324301597876881619242732513073426920161}{58240510610838340729852391683545816538096149050678699282733343342806282281198862409327365341056852292858697456346493534176244154117739280023665792} a^{17} + \frac{7535024739140485952827202985847909787292086577663102888500511265479593174973105655601589956581111449179962628697025308437890467080768464753051}{1820015956588698147807887240110806766815504657833709352585416979462696321287464450291480166908026634151834295510827922943007629816179352500739556} a^{16} - \frac{599307503341959716692002526115321425764873415836706905005360517553279555519562428326021627187009313535625701150325231538082736724396929725117409}{14560127652709585182463097920886454134524037262669674820683335835701570570299715602331841335264213073214674364086623383544061038529434820005916448} a^{15} - \frac{6842115283990275579286537958366318087707322308673695094418011210538962713692954681526431346168199963927571914861843903664732557233757174662249325}{58240510610838340729852391683545816538096149050678699282733343342806282281198862409327365341056852292858697456346493534176244154117739280023665792} a^{14} - \frac{4467339648741888638063063118908077691344546397812296465478315189745635976618370585068433125398891758885004197283381093872185299608337823426959915}{58240510610838340729852391683545816538096149050678699282733343342806282281198862409327365341056852292858697456346493534176244154117739280023665792} a^{13} + \frac{143583709625055324078196950893944031769002224725404354964886551830122754834807760697037628240419976669770044610473620808283603843122646198994331}{4160036472202738623560885120253272609864010646477049948766667381629020162942775886380526095789775163775621246881892395298303153865552805715976128} a^{12} - \frac{499865417102390351253835949252226403204423858071878545139042121769034822610752899153927139008455011978855213434128263811391103674848920024344229}{14560127652709585182463097920886454134524037262669674820683335835701570570299715602331841335264213073214674364086623383544061038529434820005916448} a^{11} + \frac{2692003724286979375816625596189484723663283936413277890325169151652958131130872621960008865097366806459892353436850243675783741617922073688834153}{58240510610838340729852391683545816538096149050678699282733343342806282281198862409327365341056852292858697456346493534176244154117739280023665792} a^{10} + \frac{1819286322128592864251642829656134799116086708657536776163641061977413306212254583672368736782941202434030779850923764011516125711142509797277275}{19413503536946113576617463894515272179365383016892899760911114447602094093732954136442455113685617430952899152115497844725414718039246426674555264} a^{9} - \frac{1944835141901672129363190608999345910733244788948251124695910547481704804513654703210343736835016827358358602980583986083437151837793179361597581}{58240510610838340729852391683545816538096149050678699282733343342806282281198862409327365341056852292858697456346493534176244154117739280023665792} a^{8} + \frac{5303143466841724240537424112170096440423068200611955410422016419554418049251316872539995932900763915684237031803740314318805940033630388531603089}{29120255305419170364926195841772908269048074525339349641366671671403141140599431204663682670528426146429348728173246767088122077058869640011832896} a^{7} - \frac{168235604064826449524206091948574626420432601935587834566307153612232850578393719884584830198421187027175200242227784431450595399314831207979681}{1386678824067579541186961706751090869954670215492349982922222460543006720980925295460175365263258387925207082293964131766101051288517601905325376} a^{6} - \frac{115857690293700887498418216165336708123801484111662573159239451450514512066048951641878620878451849685919749507785090330521565820344249692817889}{260002279512671163972555320015829538116500665404815621797916711351813760183923492898782880986860947735976327930118274706143947116597050357248508} a^{5} + \frac{1355481593530894233129066668758261700849191033811264251879039986437602658493302528980629564299973263460272322758286484176256281191594674619767961}{2773357648135159082373923413502181739909340430984699965844444921086013441961850590920350730526516775850414164587928263532202102577035203810650752} a^{4} - \frac{28431121855469014327409369685245098194490898530663547114405062140590499057349275462137189899067685101095707441921074442569649094504262112329053}{57778284336149147549456737781295452914777925645514582621759269189291946707538553977507306885969099496883628428915172156920877137021566746055224} a^{3} + \frac{12142503828301953469905504121129684492999002909406638615405487363531998221759982627322149564440600834067395491967652727478916085648557562002071}{308150849792795453597102601500242415545482270109411107316049435676223715773538954546705636725168530650046018287547584836911344730781689312294528} a^{2} - \frac{5916196192603151286653445867523115728016712370332499845653097667361631689292267233515295613047672686607185497948343374952723530713451310034257}{34238983310310605955233622388915823949498030012156789701783270630691523974837661616300626302796503405560668698616398315212371636753521034699392} a - \frac{100872641214310682404624684133654479227375756981028509324521834189866980513750152760770594262865326516972582438930845703813551342158710652855}{376252563849567098409160685592481581862615714419305381338277699238368395327886391387918970360401136324842513171608772694641446557731000381312}$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
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}$
1259Data not computed