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Properties

Label 21.3.63993157373...8047.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,7^{17}\cdot 491^{18}$
Root discriminant $978.97$
Ramified primes $7, 491$
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![629094860366336, 177581658222976, -178475909056704, -76273061749528, 21486789774576, 8605795373014, -84897810087, -639064890904, -17534305986, 3360629141, 849128436, -118397153, 30949290, -9461221, 507776, -238849, 42220, 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 + 42220*x^16 - 238849*x^15 + 507776*x^14 - 9461221*x^13 + 30949290*x^12 - 118397153*x^11 + 849128436*x^10 + 3360629141*x^9 - 17534305986*x^8 - 639064890904*x^7 - 84897810087*x^6 + 8605795373014*x^5 + 21486789774576*x^4 - 76273061749528*x^3 - 178475909056704*x^2 + 177581658222976*x + 629094860366336)
 
gp: K = bnfinit(x^21 - 7*x^20 + 21*x^19 - 34*x^18 + 29*x^17 + 42220*x^16 - 238849*x^15 + 507776*x^14 - 9461221*x^13 + 30949290*x^12 - 118397153*x^11 + 849128436*x^10 + 3360629141*x^9 - 17534305986*x^8 - 639064890904*x^7 - 84897810087*x^6 + 8605795373014*x^5 + 21486789774576*x^4 - 76273061749528*x^3 - 178475909056704*x^2 + 177581658222976*x + 629094860366336, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 29 x^{17} + 42220 x^{16} - 238849 x^{15} + 507776 x^{14} - 9461221 x^{13} + 30949290 x^{12} - 118397153 x^{11} + 849128436 x^{10} + 3360629141 x^{9} - 17534305986 x^{8} - 639064890904 x^{7} - 84897810087 x^{6} + 8605795373014 x^{5} + 21486789774576 x^{4} - 76273061749528 x^{3} - 178475909056704 x^{2} + 177581658222976 x + 629094860366336 \)

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:  \(-639931573735145678269077305601948179707569066438710730266948047=-\,7^{17}\cdot 491^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $978.97$
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:  $7, 491$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{10} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{3}$, $\frac{1}{82488} a^{18} - \frac{5477}{82488} a^{17} - \frac{1723}{82488} a^{16} - \frac{2227}{41244} a^{15} - \frac{5251}{82488} a^{14} + \frac{2957}{41244} a^{13} + \frac{9785}{82488} a^{12} - \frac{989}{41244} a^{11} + \frac{7681}{82488} a^{10} + \frac{1061}{41244} a^{9} - \frac{6571}{82488} a^{8} + \frac{9589}{41244} a^{7} + \frac{19573}{82488} a^{6} - \frac{2215}{5892} a^{5} + \frac{167}{491} a^{4} + \frac{2029}{11784} a^{3} - \frac{343}{1473} a^{2} + \frac{451}{1473} a + \frac{143}{1473}$, $\frac{1}{30025632} a^{19} + \frac{173}{30025632} a^{18} - \frac{712613}{10008544} a^{17} - \frac{148761}{5004272} a^{16} + \frac{394855}{10008544} a^{15} - \frac{23281}{312767} a^{14} + \frac{878413}{10008544} a^{13} + \frac{406207}{7506408} a^{12} - \frac{1103679}{10008544} a^{11} - \frac{169745}{15012816} a^{10} + \frac{2158885}{10008544} a^{9} - \frac{4915}{134043} a^{8} - \frac{815785}{10008544} a^{7} - \frac{5278283}{15012816} a^{6} - \frac{38734}{134043} a^{5} + \frac{1780711}{4289376} a^{4} + \frac{466051}{2144688} a^{3} - \frac{15389}{178724} a^{2} + \frac{2385}{25532} a - \frac{1402}{10311}$, $\frac{1}{7195789253908276915630141190027580535717334786676432572492458632238993684004531644185458697778671361618316787721966702834778900352} a^{20} - \frac{50207405976047365303842008756815261669907027851843508916362776587666744556584354151336103544889017656942290128986093283983}{7195789253908276915630141190027580535717334786676432572492458632238993684004531644185458697778671361618316787721966702834778900352} a^{19} + \frac{237761848993863009952345139984814916330513064059026766095956988487418455220642052211920706939828000680773648107744633885439}{48950947305498482419252661156650207725968263854941718180220807022033970639486609824390875495093002459988549576339909543093734016} a^{18} - \frac{171626477852181769694977419371308718368214950071337231746007060112711057476920635086160985433654419755659956650188983964850501573}{3597894626954138457815070595013790267858667393338216286246229316119496842002265822092729348889335680809158393860983351417389450176} a^{17} - \frac{528102528972675503299953433976464303698814493611995494871893309224364963483075808907231665234815309483514777318298845495279670835}{7195789253908276915630141190027580535717334786676432572492458632238993684004531644185458697778671361618316787721966702834778900352} a^{16} + \frac{113095813493972706082413093798194460320418279212434961730757166396390546943999413671586346670103332208539585313734245917248297001}{1798947313477069228907535297506895133929333696669108143123114658059748421001132911046364674444667840404579196930491675708694725088} a^{15} - \frac{217657685388314056812208352447959683806631831928651738834135509625517935055413052840853355405630338410055825234474579636580294913}{7195789253908276915630141190027580535717334786676432572492458632238993684004531644185458697778671361618316787721966702834778900352} a^{14} + \frac{27462780086245056804290578575088905021754100284706655854240192832617792230352333758932210861110619181076815830753509442372524647}{299824552246178204817922549584482522321555616111518023853852443009958070166855485174394112407444640067429866155081945951449120848} a^{13} + \frac{188522111562366554998374229518591565225414106338722104690187292826597728650014389126825205418901752462865112479612821783262007849}{2398596417969425638543380396675860178572444928892144190830819544079664561334843881395152899259557120539438929240655567611592966784} a^{12} - \frac{20663689697467459220122821215818333052170158717687181424015035963957408316133349218620697713963468785585831563523866034336630891}{171328315569244688467384314048275727040888923492296013630772824577118897238203134385368064232825508609959923517189683400828069056} a^{11} + \frac{446838448100033067924124923049591154024947191861257705520554578436556317997392080786928275759922231775352315709051790932195774373}{2398596417969425638543380396675860178572444928892144190830819544079664561334843881395152899259557120539438929240655567611592966784} a^{10} + \frac{125697414397920046516993515564712191218642150140408060910361048666116095537983418938290795176165637384658946495416725671567988333}{599649104492356409635845099168965044643111232223036047707704886019916140333710970348788224814889280134859732310163891902898241696} a^{9} - \frac{100814032101372304919813852315223679232778816622619813577857306712366468057203999758625008583335281670032925529191453151163699769}{2398596417969425638543380396675860178572444928892144190830819544079664561334843881395152899259557120539438929240655567611592966784} a^{8} + \frac{12349423491413970611113115498376194558615392741887948922873954727464528683873049241753681435522162259511370837301542159290384717}{92253708383439447636283861410610006868170958803544007339646905541525560051340149284428957663829120020747651124640598754292037184} a^{7} - \frac{28095420619214321427550841579607194168314191398155265954283529520326493069704639687853646160082091466244809524827691638444127421}{899473656738534614453767648753447566964666848334554071561557329029874210500566455523182337222333920202289598465245837854347362544} a^{6} + \frac{260820826887079114878320101927357720838384476341474784575267997535906278342000828209280164161939728093191111012232489987106997119}{1027969893415468130804305884289654362245333540953776081784636947462713383429218806312208385396953051659759541103138100404968414336} a^{5} + \frac{61921470201868362224630598071539198236786017017323908609980634416350262722678718047489389658319997991428444268656340529311765377}{513984946707734065402152942144827181122666770476888040892318473731356691714609403156104192698476525829879770551569050202484207168} a^{4} + \frac{34102536827797312629762674401899117921392626227147366889265243935696969393028190235885377103916252078114736364343731262889161}{10708019723077793029211519628017232940055557718268500851923301536069931077387695899085504014551594288122495219824355212551754316} a^{3} + \frac{37048690573586759573990433774989095545750181049890150095967953256598208546206090306073248423358957355856439469519230660185101243}{128496236676933516350538235536206795280666692619222010223079618432839172928652350789026048174619131457469942637892262550621051792} a^{2} + \frac{6195167963897591471130466489524827940071004303650050929490995340786959198360992461919416839910482249511196326359109261793844863}{16062029584616689543817279442025849410083336577402751277884952304104896616081543848628256021827391432183742829736532818827631474} a + \frac{103749748229466831048150107402343204725575034502808744324548940529218973928967868635082470265507930944649341528030770404795072}{617770368639103443992972286231763438849359099130875049149421242465572946772367071101086770070284285853220878066789723801062749}$

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}$
491Data not computed