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Properties

Label 21.3.24933089799...7847.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,7^{17}\cdot 41^{18}$
Root discriminant $116.55$
Ramified primes $7, 41$
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![-6446237, -50056244, -129757565, -152378359, -86530794, -8975624, 20299735, 13444516, 3096540, 220817, 283880, 265993, 115088, 20797, -1019, 5, -375, 316, -34, 21, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 21*x^19 - 34*x^18 + 316*x^17 - 375*x^16 + 5*x^15 - 1019*x^14 + 20797*x^13 + 115088*x^12 + 265993*x^11 + 283880*x^10 + 220817*x^9 + 3096540*x^8 + 13444516*x^7 + 20299735*x^6 - 8975624*x^5 - 86530794*x^4 - 152378359*x^3 - 129757565*x^2 - 50056244*x - 6446237)
 
gp: K = bnfinit(x^21 - 7*x^20 + 21*x^19 - 34*x^18 + 316*x^17 - 375*x^16 + 5*x^15 - 1019*x^14 + 20797*x^13 + 115088*x^12 + 265993*x^11 + 283880*x^10 + 220817*x^9 + 3096540*x^8 + 13444516*x^7 + 20299735*x^6 - 8975624*x^5 - 86530794*x^4 - 152378359*x^3 - 129757565*x^2 - 50056244*x - 6446237, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 316 x^{17} - 375 x^{16} + 5 x^{15} - 1019 x^{14} + 20797 x^{13} + 115088 x^{12} + 265993 x^{11} + 283880 x^{10} + 220817 x^{9} + 3096540 x^{8} + 13444516 x^{7} + 20299735 x^{6} - 8975624 x^{5} - 86530794 x^{4} - 152378359 x^{3} - 129757565 x^{2} - 50056244 x - 6446237 \)

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:  \(-24933089799576296242287986136979936417017847=-\,7^{17}\cdot 41^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $116.55$
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:  $7, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{14} - \frac{1}{4}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{12} - \frac{1}{4} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{8} + \frac{1}{8} a^{6} + \frac{1}{4} a^{5} + \frac{3}{8} a^{4} + \frac{1}{4} a^{3} + \frac{1}{8} a^{2} + \frac{1}{8} a + \frac{1}{8}$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{4} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{8} + \frac{1}{8} a^{7} - \frac{1}{8} a^{5} + \frac{1}{8} a^{3} - \frac{1}{8} a^{2} + \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{14} - \frac{1}{8} a^{12} - \frac{1}{4} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} + \frac{1}{8} a^{8} - \frac{1}{8} a^{6} + \frac{1}{8} a^{4} - \frac{1}{8} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{28208} a^{18} + \frac{1503}{28208} a^{17} + \frac{311}{14104} a^{16} - \frac{417}{7052} a^{15} - \frac{73}{688} a^{14} + \frac{21}{656} a^{13} - \frac{1451}{14104} a^{12} - \frac{497}{28208} a^{11} + \frac{3149}{14104} a^{10} - \frac{1341}{28208} a^{9} - \frac{45}{14104} a^{8} + \frac{6475}{28208} a^{7} + \frac{2077}{14104} a^{6} - \frac{10473}{28208} a^{5} + \frac{10713}{28208} a^{4} - \frac{500}{1763} a^{3} + \frac{568}{1763} a^{2} - \frac{11273}{28208} a - \frac{1145}{28208}$, $\frac{1}{535952} a^{19} + \frac{5305}{535952} a^{17} - \frac{5825}{133988} a^{16} - \frac{6501}{535952} a^{15} - \frac{6949}{267976} a^{14} - \frac{23757}{535952} a^{13} + \frac{52437}{535952} a^{12} + \frac{58667}{535952} a^{11} - \frac{59867}{535952} a^{10} - \frac{131901}{535952} a^{9} - \frac{41607}{535952} a^{8} + \frac{106243}{535952} a^{7} - \frac{315}{13072} a^{6} - \frac{59447}{267976} a^{5} + \frac{117013}{535952} a^{4} - \frac{64043}{267976} a^{3} + \frac{161961}{535952} a^{2} - \frac{22983}{133988} a - \frac{253625}{535952}$, $\frac{1}{7655542365971564133368098087459555923705167746364889568931482192} a^{20} + \frac{319631639291752084720420465700492284189262845607358331371}{478471397873222758335506130466222245231572984147805598058217637} a^{19} + \frac{77434446952194731109807019820108088819512445808046899398035}{7655542365971564133368098087459555923705167746364889568931482192} a^{18} + \frac{188189324459899946906361714789043535752021583969644272961328717}{3827771182985782066684049043729777961852583873182444784465741096} a^{17} + \frac{162164980371971101445160143486975538457438226992187577567088769}{7655542365971564133368098087459555923705167746364889568931482192} a^{16} + \frac{3654166413749095786919376972661684095665902550181579065589402}{478471397873222758335506130466222245231572984147805598058217637} a^{15} + \frac{859877542192766823499659793050791707814091754111018794240940733}{7655542365971564133368098087459555923705167746364889568931482192} a^{14} + \frac{935840186253421676719229611002872594437657397963870441408184741}{7655542365971564133368098087459555923705167746364889568931482192} a^{13} + \frac{433733809394100863048739917094818852758554036411973191285816777}{7655542365971564133368098087459555923705167746364889568931482192} a^{12} + \frac{811061794386840320922688237255131789302330710995587419139803593}{7655542365971564133368098087459555923705167746364889568931482192} a^{11} - \frac{1519171844330782078482311358731438638384984199525267559155892275}{7655542365971564133368098087459555923705167746364889568931482192} a^{10} + \frac{1091087568494931278243386633673005692748702549653249297473564577}{7655542365971564133368098087459555923705167746364889568931482192} a^{9} - \frac{920510270729460414284309697867304582902658320055431918849312099}{7655542365971564133368098087459555923705167746364889568931482192} a^{8} + \frac{870426491166580508764144905360804343280528616758082715505006489}{7655542365971564133368098087459555923705167746364889568931482192} a^{7} - \frac{19260905550159591016974685656685783645877860158540201558691121}{100730820604889001754843395887625735838225891399538020643835292} a^{6} + \frac{384312551759139064389397114619853752116739794978826764764580681}{7655542365971564133368098087459555923705167746364889568931482192} a^{5} + \frac{27248502867120230063313584428243490718896363153557035776328677}{89017934488041443411256954505343673531455438911219646150366072} a^{4} - \frac{437032161393400722216871442065504573419052521122012670684119581}{7655542365971564133368098087459555923705167746364889568931482192} a^{3} - \frac{306978109101292124701925840462679614075704594515915013367619657}{956942795746445516671012260932444490463145968295611196116435274} a^{2} - \frac{3800372306360115161072304078079501772154083357148605097842217595}{7655542365971564133368098087459555923705167746364889568931482192} a + \frac{618250236789219624346981150984309211883230709244161433670428251}{1913885591492891033342024521864888980926291936591222392232870548}$

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:  \( 115224462320340.23 \) (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

\(\Q(\zeta_{7})^+\), 7.1.79835001978487.1

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 ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ 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}$
$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}$