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Properties

Label 21.3.13950884215...9488.2
Degree $21$
Signature $[3, 9]$
Discriminant $-\,2^{18}\cdot 3^{28}\cdot 7^{17}$
Root discriminant $37.87$
Ramified primes $2, 3, 7$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3\times F_7$ (as 21T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11494, 28266, -9702, -75620, -48594, 40320, 52888, -9723, -24948, 1395, 7770, -2457, -1072, 1869, 126, -843, 0, 189, -2, -21, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 21*x^19 - 2*x^18 + 189*x^17 - 843*x^15 + 126*x^14 + 1869*x^13 - 1072*x^12 - 2457*x^11 + 7770*x^10 + 1395*x^9 - 24948*x^8 - 9723*x^7 + 52888*x^6 + 40320*x^5 - 48594*x^4 - 75620*x^3 - 9702*x^2 + 28266*x + 11494)
 
gp: K = bnfinit(x^21 - 21*x^19 - 2*x^18 + 189*x^17 - 843*x^15 + 126*x^14 + 1869*x^13 - 1072*x^12 - 2457*x^11 + 7770*x^10 + 1395*x^9 - 24948*x^8 - 9723*x^7 + 52888*x^6 + 40320*x^5 - 48594*x^4 - 75620*x^3 - 9702*x^2 + 28266*x + 11494, 1)
 

Normalized defining polynomial

\( x^{21} - 21 x^{19} - 2 x^{18} + 189 x^{17} - 843 x^{15} + 126 x^{14} + 1869 x^{13} - 1072 x^{12} - 2457 x^{11} + 7770 x^{10} + 1395 x^{9} - 24948 x^{8} - 9723 x^{7} + 52888 x^{6} + 40320 x^{5} - 48594 x^{4} - 75620 x^{3} - 9702 x^{2} + 28266 x + 11494 \)

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:  \(-1395088421598327334260620375359488=-\,2^{18}\cdot 3^{28}\cdot 7^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.87$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{17} - \frac{1}{4} a^{16} - \frac{1}{4} a^{15} - \frac{1}{2} a^{13} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{19} - \frac{1}{4} a^{15} - \frac{1}{2} a^{12} + \frac{1}{4} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{55459548098837600706376665865780004893791644} a^{20} - \frac{2630954365709930854338652912092984191672989}{27729774049418800353188332932890002446895822} a^{19} + \frac{1742593326470542380807909816507743000569727}{55459548098837600706376665865780004893791644} a^{18} + \frac{7544128542453069576829126877037506228930135}{55459548098837600706376665865780004893791644} a^{17} + \frac{1583234416947435858250772645077661407792189}{13864887024709400176594166466445001223447911} a^{16} + \frac{6820292813843167717868490823054873127259217}{55459548098837600706376665865780004893791644} a^{15} + \frac{3411966228695121030615049205951404631291780}{13864887024709400176594166466445001223447911} a^{14} + \frac{4308264799347358010823478808967276625917712}{13864887024709400176594166466445001223447911} a^{13} + \frac{915057061081357223210266495609869549962045}{55459548098837600706376665865780004893791644} a^{12} + \frac{1546083374525990172136404789884849487828708}{13864887024709400176594166466445001223447911} a^{11} + \frac{24115450651202862222837185284335479614124001}{55459548098837600706376665865780004893791644} a^{10} - \frac{24909882486695545627839240233822101500942197}{55459548098837600706376665865780004893791644} a^{9} - \frac{11296949427393563288448497392383690009530645}{27729774049418800353188332932890002446895822} a^{8} - \frac{9780815418705073211266499372235581118264741}{55459548098837600706376665865780004893791644} a^{7} + \frac{3292412568367742304679423344323528042262273}{27729774049418800353188332932890002446895822} a^{6} + \frac{675115833542647819473496631814155107929051}{13864887024709400176594166466445001223447911} a^{5} - \frac{13400261533778758178764733584053881825236601}{27729774049418800353188332932890002446895822} a^{4} + \frac{13334133193751334517430062778091083817197891}{27729774049418800353188332932890002446895822} a^{3} - \frac{7301284024195734859758339086120157247631523}{27729774049418800353188332932890002446895822} a^{2} - \frac{4255893297126776723867693782791203333118513}{27729774049418800353188332932890002446895822} a - \frac{5469361551171146765903212909650288488231277}{27729774049418800353188332932890002446895822}$

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:  $11$
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:  \( 438540258.63952965 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times F_7$ (as 21T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 126
The 21 conjugacy class representatives for $C_3\times F_7$
Character table for $C_3\times F_7$ is not computed

Intermediate fields

3.3.3969.1, 7.1.7057326528.3

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

Sibling fields

Degree 21 siblings: data not computed
Degree 42 siblings: 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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R $21$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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
$2$2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
3Data not computed
7Data not computed