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Properties

Label 21.21.4539403286...9264.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{18}\cdot 3^{28}\cdot 31^{14}$
Root discriminant $77.34$
Ramified primes $2, 3, 31$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_7:C_3$ (as 21T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-190728, -5843664, -26180496, -15240528, 50881608, 42009624, -42371712, -36320292, 19347408, 15949576, -5110992, -4069848, 784368, 629568, -67734, -58772, 3024, 3180, -54, -90, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 90*x^19 - 54*x^18 + 3180*x^17 + 3024*x^16 - 58772*x^15 - 67734*x^14 + 629568*x^13 + 784368*x^12 - 4069848*x^11 - 5110992*x^10 + 15949576*x^9 + 19347408*x^8 - 36320292*x^7 - 42371712*x^6 + 42009624*x^5 + 50881608*x^4 - 15240528*x^3 - 26180496*x^2 - 5843664*x - 190728)
 
gp: K = bnfinit(x^21 - 90*x^19 - 54*x^18 + 3180*x^17 + 3024*x^16 - 58772*x^15 - 67734*x^14 + 629568*x^13 + 784368*x^12 - 4069848*x^11 - 5110992*x^10 + 15949576*x^9 + 19347408*x^8 - 36320292*x^7 - 42371712*x^6 + 42009624*x^5 + 50881608*x^4 - 15240528*x^3 - 26180496*x^2 - 5843664*x - 190728, 1)
 

Normalized defining polynomial

\( x^{21} - 90 x^{19} - 54 x^{18} + 3180 x^{17} + 3024 x^{16} - 58772 x^{15} - 67734 x^{14} + 629568 x^{13} + 784368 x^{12} - 4069848 x^{11} - 5110992 x^{10} + 15949576 x^{9} + 19347408 x^{8} - 36320292 x^{7} - 42371712 x^{6} + 42009624 x^{5} + 50881608 x^{4} - 15240528 x^{3} - 26180496 x^{2} - 5843664 x - 190728 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[21, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4539403286895028130634165104558075019264=2^{18}\cdot 3^{28}\cdot 31^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $77.34$
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:  $2, 3, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{6} a^{9} + \frac{1}{3} a^{3}$, $\frac{1}{12} a^{10} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{11} - \frac{1}{3} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{12} a^{12} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{36} a^{13} + \frac{1}{36} a^{11} - \frac{1}{18} a^{9} + \frac{1}{18} a^{7} - \frac{1}{6} a^{6} - \frac{4}{9} a^{5} - \frac{1}{2} a^{4} - \frac{1}{9} a^{3} - \frac{1}{3}$, $\frac{1}{72} a^{14} - \frac{1}{36} a^{12} - \frac{1}{36} a^{10} - \frac{2}{9} a^{8} + \frac{1}{6} a^{7} + \frac{4}{9} a^{6} - \frac{1}{2} a^{5} + \frac{4}{9} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{2232} a^{15} + \frac{7}{1116} a^{14} + \frac{7}{1116} a^{13} - \frac{5}{279} a^{12} - \frac{23}{1116} a^{11} - \frac{5}{1116} a^{10} - \frac{13}{186} a^{9} - \frac{107}{558} a^{8} - \frac{41}{558} a^{7} - \frac{161}{558} a^{6} - \frac{61}{279} a^{5} + \frac{265}{558} a^{4} + \frac{137}{558} a^{3} + \frac{28}{93} a^{2} + \frac{1}{186} a - \frac{44}{93}$, $\frac{1}{2232} a^{16} + \frac{1}{558} a^{14} + \frac{1}{186} a^{13} - \frac{11}{558} a^{12} - \frac{2}{93} a^{11} - \frac{2}{279} a^{10} + \frac{2}{31} a^{9} - \frac{2}{9} a^{8} - \frac{7}{186} a^{7} + \frac{43}{279} a^{6} - \frac{7}{93} a^{5} + \frac{3}{31} a^{4} + \frac{13}{31} a^{3} - \frac{13}{62} a^{2} + \frac{14}{31} a + \frac{9}{31}$, $\frac{1}{6696} a^{17} + \frac{1}{372} a^{14} + \frac{1}{279} a^{13} - \frac{1}{558} a^{12} - \frac{10}{837} a^{11} + \frac{5}{558} a^{10} - \frac{41}{558} a^{9} + \frac{53}{558} a^{8} + \frac{73}{558} a^{7} + \frac{59}{279} a^{6} - \frac{349}{837} a^{5} + \frac{131}{279} a^{4} + \frac{233}{558} a^{3} + \frac{23}{279} a^{2} - \frac{2}{93} a + \frac{7}{93}$, $\frac{1}{14322744} a^{18} + \frac{37}{530472} a^{17} + \frac{229}{1591416} a^{16} + \frac{659}{4774248} a^{15} + \frac{7985}{2387124} a^{14} - \frac{7799}{795708} a^{13} + \frac{185561}{7161372} a^{12} - \frac{7685}{795708} a^{11} + \frac{59}{12834} a^{10} - \frac{23471}{596781} a^{9} - \frac{116228}{596781} a^{8} - \frac{32869}{198927} a^{7} - \frac{870887}{3580686} a^{6} - \frac{117749}{397854} a^{5} + \frac{5737}{22103} a^{4} + \frac{196499}{1193562} a^{3} + \frac{21755}{132618} a^{2} + \frac{41893}{132618} a - \frac{2446}{198927}$, $\frac{1}{472650552} a^{19} + \frac{1}{52516728} a^{18} + \frac{1567}{52516728} a^{17} + \frac{5009}{157550184} a^{16} + \frac{24511}{157550184} a^{15} - \frac{35807}{8752788} a^{14} + \frac{631507}{118162638} a^{13} - \frac{806191}{26258364} a^{12} - \frac{131965}{8752788} a^{11} + \frac{2605255}{78775092} a^{10} + \frac{1002802}{19693773} a^{9} + \frac{937151}{4376394} a^{8} + \frac{982516}{5371029} a^{7} + \frac{2709506}{6564591} a^{6} + \frac{1796488}{6564591} a^{5} - \frac{9552671}{19693773} a^{4} + \frac{1235734}{6564591} a^{3} + \frac{985333}{4376394} a^{2} + \frac{6353}{570834} a - \frac{677999}{2188197}$, $\frac{1}{709519998036107407248} a^{20} + \frac{21099232169}{88689999754513425906} a^{19} + \frac{724807707145}{29563333251504475302} a^{18} - \frac{1637141170648399}{59126666503008950604} a^{17} + \frac{166439601384613}{1642407402861359739} a^{16} + \frac{586432966563947}{5375151500273540964} a^{15} + \frac{211577510835876269}{88689999754513425906} a^{14} + \frac{1440754812095882215}{177379999509026851812} a^{13} - \frac{1088238148768049035}{59126666503008950604} a^{12} - \frac{4877735683083682}{122162534097125931} a^{11} + \frac{588552194661527002}{14781666625752237651} a^{10} - \frac{1013063636359803505}{14781666625752237651} a^{9} + \frac{13350198759721869599}{88689999754513425906} a^{8} + \frac{3995543307309689339}{88689999754513425906} a^{7} - \frac{12181085859945257641}{59126666503008950604} a^{6} - \frac{7717258874028772787}{29563333251504475302} a^{5} - \frac{2333852369804210515}{14781666625752237651} a^{4} + \frac{450388403082077054}{1642407402861359739} a^{3} - \frac{2328277153796649779}{4927222208584079217} a^{2} - \frac{2196041670172519238}{4927222208584079217} a - \frac{460778474401616609}{1642407402861359739}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $20$
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:  \( 65890265440600 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_7:C_3$ (as 21T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 21
The 5 conjugacy class representatives for $C_7:C_3$
Character table for $C_7:C_3$

Intermediate fields

3.3.77841.1, 7.7.387790161984.1 x7

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 7 sibling: 7.7.387790161984.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{7}$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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
2Data not computed
$3$3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
$31$31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$