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Properties

Label 21.7.71657031731...1967.1
Degree $21$
Signature $[7, 7]$
Discriminant $-\,23^{7}\cdot 29^{18}$
Root discriminant $50.98$
Ramified primes $23, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_7\times S_3$ (as 21T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -53, -267, 686, 6317, 12758, -11002, -20848, 20201, 12105, -15769, -2978, 5070, 840, -1451, -147, 209, 23, -12, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 12*x^19 + 23*x^18 + 209*x^17 - 147*x^16 - 1451*x^15 + 840*x^14 + 5070*x^13 - 2978*x^12 - 15769*x^11 + 12105*x^10 + 20201*x^9 - 20848*x^8 - 11002*x^7 + 12758*x^6 + 6317*x^5 + 686*x^4 - 267*x^3 - 53*x^2 + 1)
 
gp: K = bnfinit(x^21 - x^20 - 12*x^19 + 23*x^18 + 209*x^17 - 147*x^16 - 1451*x^15 + 840*x^14 + 5070*x^13 - 2978*x^12 - 15769*x^11 + 12105*x^10 + 20201*x^9 - 20848*x^8 - 11002*x^7 + 12758*x^6 + 6317*x^5 + 686*x^4 - 267*x^3 - 53*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{21} - x^{20} - 12 x^{19} + 23 x^{18} + 209 x^{17} - 147 x^{16} - 1451 x^{15} + 840 x^{14} + 5070 x^{13} - 2978 x^{12} - 15769 x^{11} + 12105 x^{10} + 20201 x^{9} - 20848 x^{8} - 11002 x^{7} + 12758 x^{6} + 6317 x^{5} + 686 x^{4} - 267 x^{3} - 53 x^{2} + 1 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-716570317313053275835569057448231967=-\,23^{7}\cdot 29^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.98$
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:  $23, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{17} a^{18} + \frac{2}{17} a^{17} - \frac{6}{17} a^{16} - \frac{5}{17} a^{15} + \frac{4}{17} a^{14} - \frac{7}{17} a^{13} + \frac{6}{17} a^{12} + \frac{2}{17} a^{11} - \frac{5}{17} a^{10} + \frac{7}{17} a^{9} + \frac{8}{17} a^{8} + \frac{7}{17} a^{7} + \frac{7}{17} a^{6} + \frac{3}{17} a^{5} + \frac{4}{17} a^{4} + \frac{1}{17} a^{3} - \frac{5}{17}$, $\frac{1}{17} a^{19} + \frac{7}{17} a^{17} + \frac{7}{17} a^{16} - \frac{3}{17} a^{15} + \frac{2}{17} a^{14} + \frac{3}{17} a^{13} + \frac{7}{17} a^{12} + \frac{8}{17} a^{11} - \frac{6}{17} a^{9} + \frac{8}{17} a^{8} - \frac{7}{17} a^{7} + \frac{6}{17} a^{6} - \frac{2}{17} a^{5} - \frac{7}{17} a^{4} - \frac{2}{17} a^{3} - \frac{5}{17} a - \frac{7}{17}$, $\frac{1}{4300901676177655907028730081031261437111} a^{20} - \frac{99618363096501955978464708385159017484}{4300901676177655907028730081031261437111} a^{19} - \frac{66391514802739723763562876618563019732}{4300901676177655907028730081031261437111} a^{18} + \frac{414905401620950753039928462561548298059}{4300901676177655907028730081031261437111} a^{17} - \frac{1725456454724524973397806139660487741844}{4300901676177655907028730081031261437111} a^{16} - \frac{839631401301789252977662954238750806951}{4300901676177655907028730081031261437111} a^{15} - \frac{1584022410536081112801047456685107163446}{4300901676177655907028730081031261437111} a^{14} + \frac{2097148848393437907831753245210804094316}{4300901676177655907028730081031261437111} a^{13} - \frac{1028230881105076513530905676012769448911}{4300901676177655907028730081031261437111} a^{12} - \frac{1942496618578313858844824877206924497712}{4300901676177655907028730081031261437111} a^{11} - \frac{164272405404998040300063060300592007495}{4300901676177655907028730081031261437111} a^{10} + \frac{1646974922317314991166334788648480734637}{4300901676177655907028730081031261437111} a^{9} - \frac{703222909371607025084317285636928556034}{4300901676177655907028730081031261437111} a^{8} - \frac{585536379396221777073057657128506524463}{4300901676177655907028730081031261437111} a^{7} - \frac{1421598045328713009861855684556740823598}{4300901676177655907028730081031261437111} a^{6} + \frac{713449270962229438501324855086253998137}{4300901676177655907028730081031261437111} a^{5} + \frac{159387138690778689448214552798462148329}{4300901676177655907028730081031261437111} a^{4} + \frac{1554249948832599502128931002780188290292}{4300901676177655907028730081031261437111} a^{3} + \frac{1647740871919452047248091932425029462106}{4300901676177655907028730081031261437111} a^{2} + \frac{1593391454076409549809554679358778942017}{4300901676177655907028730081031261437111} a - \frac{1145409105613170481386917022612795686797}{4300901676177655907028730081031261437111}$

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:  $13$
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:  \( 9787612489.51 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_7\times S_3$ (as 21T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 42
The 21 conjugacy class representatives for $C_7\times S_3$
Character table for $C_7\times S_3$ is not computed

Intermediate fields

3.1.23.1, 7.7.594823321.1

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

Sibling fields

Galois closure: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $21$ $21$ ${\href{/LocalNumberField/5.14.0.1}{14} }{,}\,{\href{/LocalNumberField/5.7.0.1}{7} }$ ${\href{/LocalNumberField/7.14.0.1}{14} }{,}\,{\href{/LocalNumberField/7.7.0.1}{7} }$ ${\href{/LocalNumberField/11.14.0.1}{14} }{,}\,{\href{/LocalNumberField/11.7.0.1}{7} }$ $21$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/19.14.0.1}{14} }{,}\,{\href{/LocalNumberField/19.7.0.1}{7} }$ R R $21$ ${\href{/LocalNumberField/37.14.0.1}{14} }{,}\,{\href{/LocalNumberField/37.7.0.1}{7} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.7.0.1}{7} }$ $21$ ${\href{/LocalNumberField/53.14.0.1}{14} }{,}\,{\href{/LocalNumberField/53.7.0.1}{7} }$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{21}$

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
$23$23.7.0.1$x^{7} - x + 8$$1$$7$$0$$C_7$$[\ ]^{7}$
23.14.7.2$x^{14} - 148035889 x^{2} + 27238603576$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
29Data not computed