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Properties

Label 21.3.28038607599...3263.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,7^{17}\cdot 13^{18}\cdot 41^{18}$
Root discriminant $1050.32$
Ramified primes $7, 13, 41$
Class number $2744$ (GRH)
Class group $[14, 14, 14]$ (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![-660800505832768, -862711219086080, 370902066797360, -9543020500384, -27591656221276, 16991239839504, 596900678473, -1136578032834, -7342418196, 8083253229, -1122038804, 250586819, -100744748, -11256581, 331614, -227299, 30908, 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 + 30908*x^16 - 227299*x^15 + 331614*x^14 - 11256581*x^13 - 100744748*x^12 + 250586819*x^11 - 1122038804*x^10 + 8083253229*x^9 - 7342418196*x^8 - 1136578032834*x^7 + 596900678473*x^6 + 16991239839504*x^5 - 27591656221276*x^4 - 9543020500384*x^3 + 370902066797360*x^2 - 862711219086080*x - 660800505832768)
 
gp: K = bnfinit(x^21 - 7*x^20 + 21*x^19 - 34*x^18 + 29*x^17 + 30908*x^16 - 227299*x^15 + 331614*x^14 - 11256581*x^13 - 100744748*x^12 + 250586819*x^11 - 1122038804*x^10 + 8083253229*x^9 - 7342418196*x^8 - 1136578032834*x^7 + 596900678473*x^6 + 16991239839504*x^5 - 27591656221276*x^4 - 9543020500384*x^3 + 370902066797360*x^2 - 862711219086080*x - 660800505832768, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 29 x^{17} + 30908 x^{16} - 227299 x^{15} + 331614 x^{14} - 11256581 x^{13} - 100744748 x^{12} + 250586819 x^{11} - 1122038804 x^{10} + 8083253229 x^{9} - 7342418196 x^{8} - 1136578032834 x^{7} + 596900678473 x^{6} + 16991239839504 x^{5} - 27591656221276 x^{4} - 9543020500384 x^{3} + 370902066797360 x^{2} - 862711219086080 x - 660800505832768 \)

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:  \(-2803860759981050190158681909505960217064638670151364262388173263=-\,7^{17}\cdot 13^{18}\cdot 41^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1050.32$
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:  $7, 13, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{344} a^{14} + \frac{17}{172} a^{13} + \frac{57}{344} a^{12} - \frac{37}{172} a^{11} - \frac{41}{344} a^{10} - \frac{39}{172} a^{9} - \frac{77}{344} a^{8} + \frac{11}{172} a^{7} + \frac{1}{344} a^{6} + \frac{37}{172} a^{5} + \frac{97}{344} a^{4} + \frac{5}{43} a^{3} + \frac{141}{344} a^{2} - \frac{13}{43} a + \frac{9}{86}$, $\frac{1}{344} a^{15} + \frac{19}{344} a^{13} + \frac{13}{86} a^{12} - \frac{19}{344} a^{11} - \frac{15}{86} a^{10} + \frac{81}{344} a^{9} + \frac{15}{86} a^{8} + \frac{27}{344} a^{7} - \frac{33}{86} a^{6} - \frac{97}{344} a^{5} - \frac{81}{172} a^{4} + \frac{71}{344} a^{3} - \frac{41}{172} a^{2} - \frac{63}{172} a - \frac{5}{86}$, $\frac{1}{344} a^{16} + \frac{1}{43} a^{13} - \frac{35}{172} a^{12} + \frac{7}{43} a^{11} + \frac{10}{43} a^{9} - \frac{29}{172} a^{8} + \frac{13}{86} a^{7} + \frac{7}{43} a^{6} - \frac{53}{172} a^{5} - \frac{13}{86} a^{4} + \frac{13}{43} a^{3} + \frac{119}{344} a^{2} + \frac{75}{172} a - \frac{21}{43}$, $\frac{1}{4816} a^{17} - \frac{1}{688} a^{16} - \frac{1}{688} a^{15} - \frac{3}{2408} a^{14} - \frac{391}{4816} a^{13} + \frac{26}{301} a^{12} + \frac{89}{4816} a^{11} + \frac{537}{2408} a^{10} - \frac{87}{688} a^{9} - \frac{97}{602} a^{8} + \frac{57}{688} a^{7} + \frac{15}{301} a^{6} - \frac{1559}{4816} a^{5} - \frac{60}{301} a^{4} - \frac{59}{2408} a^{3} - \frac{363}{4816} a^{2} - \frac{23}{86} a + \frac{29}{172}$, $\frac{1}{215621952} a^{18} + \frac{1777}{30803136} a^{17} + \frac{35101}{30803136} a^{16} - \frac{39233}{53905488} a^{15} - \frac{53087}{215621952} a^{14} + \frac{3001073}{107810976} a^{13} + \frac{2419093}{215621952} a^{12} - \frac{10198399}{53905488} a^{11} + \frac{4453385}{30803136} a^{10} - \frac{1394173}{8293152} a^{9} + \frac{2713597}{30803136} a^{8} - \frac{17164237}{107810976} a^{7} - \frac{95097259}{215621952} a^{6} + \frac{4568777}{107810976} a^{5} + \frac{20792027}{107810976} a^{4} + \frac{37010653}{215621952} a^{3} + \frac{7661677}{15401568} a^{2} + \frac{41229}{366704} a + \frac{48677}{550056}$, $\frac{1}{259608830208} a^{19} - \frac{85}{37086975744} a^{18} + \frac{3020531}{37086975744} a^{17} + \frac{3868315}{9984955008} a^{16} - \frac{329576911}{259608830208} a^{15} - \frac{19001693}{32451103776} a^{14} + \frac{1595073737}{259608830208} a^{13} + \frac{7896078243}{43268138368} a^{12} - \frac{45683119}{904560384} a^{11} - \frac{2306740291}{10817034592} a^{10} + \frac{3351766577}{37086975744} a^{9} + \frac{4736694055}{32451103776} a^{8} - \frac{1246383113}{6656636672} a^{7} - \frac{293035329}{676064662} a^{6} + \frac{53609883205}{129804415104} a^{5} + \frac{2434172147}{6037414656} a^{4} - \frac{1393980037}{3090581312} a^{3} + \frac{29288677}{165566856} a^{2} - \frac{118595621}{331133712} a - \frac{23367215}{47304816}$, $\frac{1}{7243166600823904368821074996607631313175884482693110611137564234856973095744119741809457393774726448405160167554048} a^{20} + \frac{1309820254289483450903228829338638750773959084998671377569617760124571222747688197047844243806199481173}{2414388866941301456273691665535877104391961494231036870379188078285657698581373247269819131258242149468386722518016} a^{19} - \frac{251257550370869045012321071001483505842233910698973894149313122139049307646987006712327754380496367097341}{344912695277328779467670237933696729198851642033005267197026868326522528368767606752831304465463164209769531788288} a^{18} + \frac{677192522487450611088867154153123014823913133807247112330976663739825091898871874418260319184955884814666657}{7018572287620062372888638562604293908116167134392549041799965343853656100527247811830869567611169039152286984064} a^{17} - \frac{2102368515074564122785362144826375758467183753980391792238899809830028523150786505698043423709226285445934306305}{2414388866941301456273691665535877104391961494231036870379188078285657698581373247269819131258242149468386722518016} a^{16} + \frac{12177067904502428077246934892887368697224348801546938093561747760031245649339846427600232899246677261513232279}{1207194433470650728136845832767938552195980747115518435189594039142828849290686623634909565629121074734193361259008} a^{15} + \frac{2666946399241016294745344976811082595544747322908453543422125445490967138495926888202060713807332431489576959747}{2414388866941301456273691665535877104391961494231036870379188078285657698581373247269819131258242149468386722518016} a^{14} + \frac{48745273414759683648854784657343129867871384135319161607720674934005211332759403304334785092201917223314974048885}{1810791650205976092205268749151907828293971120673277652784391058714243273936029935452364348443681612101290041888512} a^{13} + \frac{404643861496447160453474359326895900473928106626969182347955919057381662873398459874850595091697097981876129315}{56148578300960498983109108500834351264929337075140392334399722750829248804217982494646956540889352313218295872512} a^{12} + \frac{629815052966320350039780836034913935947358241776480258363144787324776998117316938063856263102315771821938143134051}{3621583300411952184410537498303815656587942241346555305568782117428486547872059870904728696887363224202580083777024} a^{11} - \frac{344335506876640300753162018120932416200134537191661506859404142754851039094622679050005592658082680220696471341251}{2414388866941301456273691665535877104391961494231036870379188078285657698581373247269819131258242149468386722518016} a^{10} + \frac{817242537206872852167468135601829724670074796689058912095082427849223341840059115195347325802006481935091835277}{92861110266973132933603525597533734784306211316578341168430310703294526868514355664223812740701621133399489327616} a^{9} - \frac{1059067060902214415465825723508522224722320417160652191665214210695982043217639180274621834130983924419009842895359}{7243166600823904368821074996607631313175884482693110611137564234856973095744119741809457393774726448405160167554048} a^{8} + \frac{621328154931442198802039245431279082987410344471544975362516722611764094899391067150526648426661549057205580349913}{3621583300411952184410537498303815656587942241346555305568782117428486547872059870904728696887363224202580083777024} a^{7} + \frac{161619821511680162794781960293880736380994328586285573893244934815901639617468963928491842437285010175181633273}{1777900491120251440555001226462354274220884752747449830912509630549085197777152612127996414770428681493657380352} a^{6} + \frac{35656419899584902908748163496016453483605183397552009686992880993678599043156328520864769068837318161846051845459}{185722220533946265867207051195067469568612422633156682336860621406589053737028711328447625481403242266798978655232} a^{5} - \frac{1724762326889475689171839054665109152296542003702559626336920503669839260561715642331837868992113164240442888661017}{3621583300411952184410537498303815656587942241346555305568782117428486547872059870904728696887363224202580083777024} a^{4} + \frac{40931349286983652392682801790154673391169396892596318883304558005200314580698107845875437431835110213952274807171}{129342260728998292300376339225136273449569365762376975198885075622445948138287852532311739174548686578663574420608} a^{3} - \frac{4200722607321627417549317319648160695997042366856092130973745696246261924241510742727680087774909885315906879659}{9238732909214163735741167087509733817826383268741212514206076830174710581306275180879409941039191898475969601472} a^{2} + \frac{2676188847277337689710212783341960726098598529682006957686725225537656642433692088633759717646447990229037716281}{9238732909214163735741167087509733817826383268741212514206076830174710581306275180879409941039191898475969601472} a + \frac{8200254491102595915108716459792564757500676809291746716664501545895097569247020971407377017786247812404171575}{219969831171765803231932549702612709948247220684314583671573257861302632888244647163795474786647426154189752416}$

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

Class group and class number

$C_{14}\times C_{14}\times C_{14}$, which has order $2744$ (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:  \( 445424233834374960000 \) (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 ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ R ${\href{/LocalNumberField/43.1.0.1}{1} }^{21}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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}$
$13$13.7.6.1$x^{7} - 13$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$
13.7.6.1$x^{7} - 13$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$
13.7.6.1$x^{7} - 13$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$
$41$41.7.6.1$x^{7} - 41$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$
41.7.6.1$x^{7} - 41$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$
41.7.6.1$x^{7} - 41$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$