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Properties

Label 21.21.9885094261...7136.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{18}\cdot 7^{14}\cdot 11^{18}$
Root discriminant $51.77$
Ramified primes $2, 7, 11$
Class number $1$ (GRH)
Class group Trivial (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![343, 7546, 25235, -87913, -303555, 430780, 1116746, -897807, -1809622, 782753, 1367403, -252543, -459906, 49998, 79875, -7656, -7508, 799, 362, -45, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 - 45*x^19 + 362*x^18 + 799*x^17 - 7508*x^16 - 7656*x^15 + 79875*x^14 + 49998*x^13 - 459906*x^12 - 252543*x^11 + 1367403*x^10 + 782753*x^9 - 1809622*x^8 - 897807*x^7 + 1116746*x^6 + 430780*x^5 - 303555*x^4 - 87913*x^3 + 25235*x^2 + 7546*x + 343)
 
gp: K = bnfinit(x^21 - 7*x^20 - 45*x^19 + 362*x^18 + 799*x^17 - 7508*x^16 - 7656*x^15 + 79875*x^14 + 49998*x^13 - 459906*x^12 - 252543*x^11 + 1367403*x^10 + 782753*x^9 - 1809622*x^8 - 897807*x^7 + 1116746*x^6 + 430780*x^5 - 303555*x^4 - 87913*x^3 + 25235*x^2 + 7546*x + 343, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} - 45 x^{19} + 362 x^{18} + 799 x^{17} - 7508 x^{16} - 7656 x^{15} + 79875 x^{14} + 49998 x^{13} - 459906 x^{12} - 252543 x^{11} + 1367403 x^{10} + 782753 x^{9} - 1809622 x^{8} - 897807 x^{7} + 1116746 x^{6} + 430780 x^{5} - 303555 x^{4} - 87913 x^{3} + 25235 x^{2} + 7546 x + 343 \)

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:  \(988509426193021817935320621873627136=2^{18}\cdot 7^{14}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.77$
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:  $2, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{7} a^{9} - \frac{3}{7} a^{8} - \frac{1}{7} a^{7} - \frac{1}{7} a^{5} - \frac{3}{7} a^{4} + \frac{1}{7} a^{3}$, $\frac{1}{7} a^{10} - \frac{3}{7} a^{8} - \frac{3}{7} a^{7} - \frac{1}{7} a^{6} + \frac{1}{7} a^{5} - \frac{1}{7} a^{4} + \frac{3}{7} a^{3}$, $\frac{1}{7} a^{11} + \frac{2}{7} a^{8} + \frac{3}{7} a^{7} + \frac{1}{7} a^{6} + \frac{3}{7} a^{5} + \frac{1}{7} a^{4} + \frac{3}{7} a^{3}$, $\frac{1}{7} a^{12} + \frac{2}{7} a^{8} + \frac{3}{7} a^{7} + \frac{3}{7} a^{6} + \frac{3}{7} a^{5} + \frac{2}{7} a^{4} - \frac{2}{7} a^{3}$, $\frac{1}{7} a^{13} + \frac{2}{7} a^{8} - \frac{2}{7} a^{7} + \frac{3}{7} a^{6} - \frac{3}{7} a^{5} - \frac{3}{7} a^{4} - \frac{2}{7} a^{3}$, $\frac{1}{7} a^{14} - \frac{3}{7} a^{8} - \frac{2}{7} a^{7} - \frac{3}{7} a^{6} - \frac{1}{7} a^{5} - \frac{3}{7} a^{4} - \frac{2}{7} a^{3}$, $\frac{1}{21} a^{15} - \frac{1}{21} a^{13} + \frac{1}{21} a^{12} + \frac{1}{21} a^{11} - \frac{1}{21} a^{10} + \frac{1}{21} a^{9} + \frac{5}{21} a^{8} - \frac{10}{21} a^{7} + \frac{1}{21} a^{6} - \frac{2}{7} a^{5} + \frac{1}{3} a^{4} - \frac{10}{21} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{147} a^{16} - \frac{1}{49} a^{15} + \frac{5}{147} a^{14} + \frac{1}{21} a^{13} - \frac{8}{147} a^{12} - \frac{10}{147} a^{11} - \frac{8}{147} a^{10} - \frac{10}{147} a^{9} - \frac{31}{147} a^{8} + \frac{46}{147} a^{7} + \frac{4}{49} a^{6} + \frac{16}{147} a^{5} + \frac{2}{21} a^{4} + \frac{2}{21} a^{3} - \frac{3}{7} a^{2} - \frac{1}{3} a$, $\frac{1}{147} a^{17} + \frac{1}{49} a^{15} + \frac{1}{147} a^{14} + \frac{2}{49} a^{13} - \frac{2}{49} a^{12} - \frac{10}{147} a^{11} + \frac{1}{147} a^{10} + \frac{3}{49} a^{9} - \frac{11}{49} a^{8} + \frac{59}{147} a^{7} + \frac{17}{147} a^{6} - \frac{1}{147} a^{5} + \frac{2}{21} a^{3} + \frac{1}{21} a^{2} - \frac{1}{3}$, $\frac{1}{441} a^{18} + \frac{1}{441} a^{17} - \frac{1}{441} a^{15} - \frac{8}{441} a^{14} - \frac{1}{63} a^{13} + \frac{5}{147} a^{12} + \frac{4}{63} a^{11} + \frac{2}{147} a^{10} - \frac{29}{441} a^{9} - \frac{17}{63} a^{8} + \frac{19}{147} a^{7} + \frac{92}{441} a^{6} - \frac{1}{63} a^{5} - \frac{5}{21} a^{4} - \frac{1}{63} a^{3} - \frac{1}{3} a^{2} - \frac{4}{9} a - \frac{2}{9}$, $\frac{1}{441} a^{19} - \frac{1}{441} a^{17} - \frac{1}{441} a^{16} - \frac{1}{63} a^{15} + \frac{1}{441} a^{14} + \frac{22}{441} a^{13} + \frac{13}{441} a^{12} - \frac{22}{441} a^{11} + \frac{4}{63} a^{10} - \frac{3}{49} a^{9} - \frac{202}{441} a^{8} - \frac{31}{63} a^{7} - \frac{18}{49} a^{6} - \frac{2}{9} a^{5} - \frac{22}{63} a^{4} + \frac{16}{63} a^{3} - \frac{1}{9} a^{2} + \frac{2}{9} a + \frac{2}{9}$, $\frac{1}{590523477729356804013609217885539} a^{20} - \frac{15580673018953633150907211140}{196841159243118934671203072628513} a^{19} - \frac{13780634575107435841500192721}{590523477729356804013609217885539} a^{18} - \frac{1559500242777709531716498639058}{590523477729356804013609217885539} a^{17} - \frac{625263754986037887374975386330}{590523477729356804013609217885539} a^{16} - \frac{557711655858078678300592169636}{590523477729356804013609217885539} a^{15} + \frac{41989417577716417386469668259984}{590523477729356804013609217885539} a^{14} + \frac{5918048931144486895663204365769}{590523477729356804013609217885539} a^{13} - \frac{27887833030465555099261965641101}{590523477729356804013609217885539} a^{12} - \frac{17077468048444302778323917678699}{590523477729356804013609217885539} a^{11} + \frac{1874165308671192312420139577603}{65613719747706311557067690876171} a^{10} + \frac{38822102298613999316498071730006}{590523477729356804013609217885539} a^{9} - \frac{138673252636319152482221610917947}{590523477729356804013609217885539} a^{8} + \frac{3457160069884002335554001968}{1178689576306101405216784866039} a^{7} + \frac{208813614850772895906729233329243}{590523477729356804013609217885539} a^{6} - \frac{29960392567768469780987114271202}{84360496818479543430515602555077} a^{5} + \frac{37908762146873603516871159874903}{84360496818479543430515602555077} a^{4} - \frac{5738128959484003122783029981929}{84360496818479543430515602555077} a^{3} + \frac{4827173179899624984350987427329}{12051499545497077632930800365011} a^{2} + \frac{274201318550225810027950911275}{1721642792213868233275828623573} a - \frac{132552699098950905978273602581}{573880930737956077758609541191}$

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:  $20$
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:  \( 452833489306 \) (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

\(\Q(\zeta_{7})^+\), 7.7.272225149504.1 x7

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

Sibling fields

Degree 7 sibling: 7.7.272225149504.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{7}$ R R ${\href{/LocalNumberField/13.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
11Data not computed