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Properties

Label 21.3.58762312802...9343.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,7^{17}\cdot 43^{18}$
Root discriminant $121.41$
Ramified primes $7, 43$
Class number $7$ (GRH)
Class group $[7]$ (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![35136283, -116740442, 18430223, 367896053, -636836088, 516125610, -249930855, 86251922, -27063618, 9707987, -3880770, 1367751, -337490, 61691, -21697, 9945, -2629, 330, -34, 21, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 21*x^19 - 34*x^18 + 330*x^17 - 2629*x^16 + 9945*x^15 - 21697*x^14 + 61691*x^13 - 337490*x^12 + 1367751*x^11 - 3880770*x^10 + 9707987*x^9 - 27063618*x^8 + 86251922*x^7 - 249930855*x^6 + 516125610*x^5 - 636836088*x^4 + 367896053*x^3 + 18430223*x^2 - 116740442*x + 35136283)
 
gp: K = bnfinit(x^21 - 7*x^20 + 21*x^19 - 34*x^18 + 330*x^17 - 2629*x^16 + 9945*x^15 - 21697*x^14 + 61691*x^13 - 337490*x^12 + 1367751*x^11 - 3880770*x^10 + 9707987*x^9 - 27063618*x^8 + 86251922*x^7 - 249930855*x^6 + 516125610*x^5 - 636836088*x^4 + 367896053*x^3 + 18430223*x^2 - 116740442*x + 35136283, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 330 x^{17} - 2629 x^{16} + 9945 x^{15} - 21697 x^{14} + 61691 x^{13} - 337490 x^{12} + 1367751 x^{11} - 3880770 x^{10} + 9707987 x^{9} - 27063618 x^{8} + 86251922 x^{7} - 249930855 x^{6} + 516125610 x^{5} - 636836088 x^{4} + 367896053 x^{3} + 18430223 x^{2} - 116740442 x + 35136283 \)

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:  \(-58762312802028807390251370834685864006539343=-\,7^{17}\cdot 43^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $121.41$
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:  $7, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{14} - \frac{1}{4}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{14} - \frac{1}{8} a^{13} + \frac{1}{8} a^{11} + \frac{1}{8} a^{9} - \frac{1}{4} a^{8} + \frac{1}{8} a^{7} - \frac{1}{8} a^{5} + \frac{1}{4} a^{4} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{8}$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{13} - \frac{1}{8} a^{12} + \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} + \frac{1}{8} a^{8} + \frac{1}{8} a^{7} + \frac{1}{8} a^{6} - \frac{3}{8} a^{5} - \frac{1}{8} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2} + \frac{3}{8} a - \frac{1}{8}$, $\frac{1}{104} a^{17} + \frac{5}{104} a^{16} - \frac{5}{104} a^{15} - \frac{3}{26} a^{14} - \frac{9}{104} a^{13} + \frac{3}{26} a^{12} - \frac{1}{8} a^{11} + \frac{1}{52} a^{10} - \frac{1}{104} a^{9} - \frac{3}{26} a^{8} + \frac{25}{104} a^{7} - \frac{11}{52} a^{6} - \frac{27}{104} a^{5} - \frac{23}{52} a^{4} + \frac{19}{52} a^{3} - \frac{31}{104} a^{2} + \frac{15}{52} a$, $\frac{1}{3255616} a^{18} - \frac{1821}{465088} a^{17} + \frac{291}{17888} a^{16} - \frac{36713}{813904} a^{15} - \frac{2127}{3255616} a^{14} + \frac{205845}{3255616} a^{13} + \frac{24121}{1627808} a^{12} - \frac{178371}{3255616} a^{11} + \frac{34425}{232544} a^{10} + \frac{22441}{250432} a^{9} + \frac{701}{17888} a^{8} + \frac{541193}{3255616} a^{7} + \frac{396313}{1627808} a^{6} + \frac{1116653}{3255616} a^{5} - \frac{1626295}{3255616} a^{4} + \frac{2865}{813904} a^{3} - \frac{2143}{7267} a^{2} + \frac{33379}{465088} a - \frac{10481}{35776}$, $\frac{1}{3255616} a^{19} + \frac{487}{465088} a^{17} + \frac{69717}{1627808} a^{16} + \frac{148541}{3255616} a^{15} + \frac{901}{203476} a^{14} - \frac{228559}{3255616} a^{13} + \frac{202275}{3255616} a^{12} + \frac{61091}{465088} a^{11} - \frac{327697}{3255616} a^{10} - \frac{3833}{35776} a^{9} - \frac{102541}{3255616} a^{8} + \frac{259933}{3255616} a^{7} + \frac{27119}{250432} a^{6} - \frac{33}{2366} a^{5} + \frac{953527}{3255616} a^{4} - \frac{51599}{116272} a^{3} - \frac{196501}{465088} a^{2} - \frac{14403}{116272} a + \frac{4165}{35776}$, $\frac{1}{7942738199999058761139309320519111762477517872496307797387617344} a^{20} - \frac{185825817346540593466469340511354390546220378591243563831}{3971369099999529380569654660259555881238758936248153898693808672} a^{19} - \frac{971822190626764188679990753933562548335083745551152848787}{7942738199999058761139309320519111762477517872496307797387617344} a^{18} + \frac{97465886504903130560761571759457713701168417777502843598335}{23089355232555403375404968954997417914178831024698569178452376} a^{17} - \frac{235571820713422227686226650034284356218872007493099902322545879}{7942738199999058761139309320519111762477517872496307797387617344} a^{16} + \frac{208051582399077618818313042256000464377713106388628914735362701}{3971369099999529380569654660259555881238758936248153898693808672} a^{15} + \frac{658452454128299248533296687842129086632056439427565091861539101}{7942738199999058761139309320519111762477517872496307797387617344} a^{14} - \frac{912193275876843427394145317346207041111636076206781364849352399}{7942738199999058761139309320519111762477517872496307797387617344} a^{13} - \frac{977593219042737108611770649836840827524119347730980536165986253}{7942738199999058761139309320519111762477517872496307797387617344} a^{12} + \frac{1367926921932406481828383968743445945952179503314842902277229349}{7942738199999058761139309320519111762477517872496307797387617344} a^{11} + \frac{1597084436591304225140578402211608665393443263371224635433580963}{7942738199999058761139309320519111762477517872496307797387617344} a^{10} - \frac{886557359318321176830381314216851581808719444664824343476050983}{7942738199999058761139309320519111762477517872496307797387617344} a^{9} + \frac{392201747051919622717661275656988788203356311504157886754012811}{7942738199999058761139309320519111762477517872496307797387617344} a^{8} - \frac{877399752853145731028456606143904156872545157548996809696096135}{7942738199999058761139309320519111762477517872496307797387617344} a^{7} - \frac{183126868196772668317183575501268674364890396333836195381780705}{3971369099999529380569654660259555881238758936248153898693808672} a^{6} - \frac{1754949435936989293654585431539020101142752225421161012189601965}{7942738199999058761139309320519111762477517872496307797387617344} a^{5} + \frac{1815791980488574654863219436326120409779818573820273667492643891}{3971369099999529380569654660259555881238758936248153898693808672} a^{4} + \frac{3541110847592200337207802792744857302246852541540376234543877573}{7942738199999058761139309320519111762477517872496307797387617344} a^{3} - \frac{1046565099610984294867443757792813087981797838763380091777307}{567338442857075625795664951465650840176965562321164842670544096} a^{2} + \frac{178459432534935443017862709130266126853707358277017829730319805}{1134676885714151251591329902931301680353931124642329685341088192} a + \frac{13848688122125449260438255213528237473719878074536551771304543}{43641418681313509676589611651203910782843504793935757128503392}$

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

Class group and class number

$C_{7}$, which has order $7$ (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:  \( 17109387227548.602 \) (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 ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ R ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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}$
$43$43.7.6.5$x^{7} - 282123$$7$$1$$6$$C_7$$[\ ]_{7}$
43.7.6.5$x^{7} - 282123$$7$$1$$6$$C_7$$[\ ]_{7}$
43.7.6.5$x^{7} - 282123$$7$$1$$6$$C_7$$[\ ]_{7}$