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Properties

Label 21.1.156...976.1
Degree $21$
Signature $[1, 10]$
Discriminant $1.568\times 10^{36}$
Root discriminant \(52.92\)
Ramified primes see page
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_{21}$ (as 21T164)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^21 + 4*x - 8)
 
Copy content gp:K = bnfinit(y^21 + 4*y - 8, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 + 4*x - 8);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^21 + 4*x - 8)
 

\( x^{21} + 4x - 8 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $21$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[1, 10]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(1568357617874304002134971883743870976\) \(\medspace = 2^{28}\cdot 23\cdot 9257\cdot 126173\cdot 217490699493582407\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(52.92\)
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  not computed
Ramified primes:   \(2\), \(23\), \(9257\), \(126173\), \(217490699493582407\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant((OK)))
 
Discriminant root field:  $\Q(\sqrt{58425\!\cdots\!24421}$)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}$, $\frac{1}{2}a^{11}$, $\frac{1}{2}a^{12}$, $\frac{1}{2}a^{13}$, $\frac{1}{2}a^{14}$, $\frac{1}{2}a^{15}$, $\frac{1}{4}a^{16}-\frac{1}{2}a^{6}$, $\frac{1}{4}a^{17}-\frac{1}{2}a^{7}$, $\frac{1}{4}a^{18}-\frac{1}{2}a^{8}$, $\frac{1}{4}a^{19}-\frac{1}{2}a^{9}$, $\frac{1}{4}a^{20}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $10$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{1}{4}a^{16}-\frac{1}{2}a^{6}-a+1$, $\frac{1}{4}a^{16}-\frac{1}{2}a^{6}+a-1$, $\frac{1}{4}a^{20}-\frac{1}{2}a^{18}-\frac{1}{4}a^{16}-\frac{1}{2}a^{15}+\frac{1}{2}a^{12}-\frac{3}{2}a^{10}+\frac{1}{2}a^{6}-a^{5}+a^{4}-3a^{2}+1$, $\frac{1}{4}a^{20}-\frac{1}{2}a^{19}+\frac{3}{4}a^{17}-\frac{1}{4}a^{16}+\frac{3}{2}a^{15}+a^{14}+\frac{1}{2}a^{13}+\frac{5}{2}a^{12}+\frac{3}{2}a^{10}+2a^{9}-a^{8}+\frac{5}{2}a^{7}-\frac{1}{2}a^{6}-2a^{5}+2a^{4}-5a^{3}-5$, $\frac{1}{4}a^{20}-\frac{1}{4}a^{19}+\frac{3}{4}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{14}-\frac{3}{2}a^{12}+\frac{3}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}+\frac{3}{2}a^{8}+a^{6}-3a^{5}+2a^{4}-2a^{3}-3a^{2}+a+1$, $\frac{1}{4}a^{20}-\frac{1}{2}a^{19}+\frac{3}{4}a^{18}-a^{17}+\frac{3}{2}a^{15}-2a^{14}+\frac{1}{2}a^{13}+\frac{3}{2}a^{12}-2a^{11}+\frac{3}{2}a^{10}-a^{9}+\frac{1}{2}a^{8}+2a^{7}-5a^{6}+5a^{5}-6a^{3}+7a^{2}-3a-1$, $\frac{1}{2}a^{19}-\frac{3}{4}a^{17}+a^{15}+\frac{1}{2}a^{14}+\frac{1}{2}a^{12}-a^{10}+a^{8}-\frac{1}{2}a^{7}-a^{6}+a^{5}-3a^{3}-a^{2}+2a-1$, $\frac{1}{4}a^{20}+\frac{1}{4}a^{19}-\frac{1}{4}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}+\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{3}{2}a^{9}-\frac{1}{2}a^{8}+a^{5}+a^{4}-2a^{3}-2a^{2}-1$, $\frac{1}{4}a^{20}-\frac{1}{4}a^{18}+\frac{1}{4}a^{16}-\frac{1}{2}a^{14}+a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}+a^{9}-\frac{1}{2}a^{8}-a^{7}+\frac{3}{2}a^{6}-a^{5}-a^{3}+3a^{2}-2a-1$, $\frac{1}{4}a^{20}+\frac{1}{2}a^{19}-\frac{3}{4}a^{17}+\frac{1}{2}a^{16}+\frac{1}{2}a^{15}-\frac{3}{2}a^{14}+a^{13}+\frac{3}{2}a^{12}-2a^{11}-\frac{1}{2}a^{10}+a^{9}+a^{8}-\frac{1}{2}a^{7}-2a^{6}+3a^{5}-a^{4}-4a^{3}+6a^{2}+a-5$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 15652649356.7 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{10}\cdot 15652649356.7 \cdot 1}{2\cdot\sqrt{1568357617874304002134971883743870976}}\cr\approx \mathstrut & 1.19857162283 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^21 + 4*x - 8) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^21 + 4*x - 8, 1); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 + 4*x - 8); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 + 4*x - 8); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$S_{21}$ (as 21T164):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A non-solvable group of order 51090942171709440000
The 792 conjugacy class representatives for $S_{21}$
Character table for $S_{21}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 42 sibling: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $17{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ $21$ ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ ${\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ $18{,}\,{\href{/padicField/17.3.0.1}{3} }$ $15{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ R ${\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.9.0.1}{9} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }$ ${\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ ${\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.10.0.1}{10} }$ $21$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display $\Q_{2}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
2.1.20.28a1.5$x^{20} + 2 x^{10} + 2 x^{9} + 2$$20$$1$$28$20T80not computed
\(23\) Copy content Toggle raw display $\Q_{23}$$x + 18$$1$$1$$0$Trivial$$[\ ]$$
23.1.2.1a1.2$x^{2} + 115$$2$$1$$1$$C_2$$$[\ ]_{2}$$
23.2.1.0a1.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
23.16.1.0a1.1$x^{16} + 19 x^{7} + 19 x^{6} + 16 x^{5} + 13 x^{4} + x^{3} + 14 x^{2} + 17 x + 5$$1$$16$$0$$C_{16}$$$[\ ]^{16}$$
\(9257\) Copy content Toggle raw display $\Q_{9257}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{9257}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $5$$1$$5$$0$$C_5$$$[\ ]^{5}$$
Deg $8$$1$$8$$0$$C_8$$$[\ ]^{8}$$
\(126173\) Copy content Toggle raw display $\Q_{126173}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{126173}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{126173}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $4$$1$$4$$0$$C_4$$$[\ ]^{4}$$
Deg $4$$1$$4$$0$$C_4$$$[\ ]^{4}$$
Deg $5$$1$$5$$0$$C_5$$$[\ ]^{5}$$
\(217490699493582407\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $6$$1$$6$$0$$C_6$$$[\ ]^{6}$$
Deg $11$$1$$11$$0$$C_{11}$$$[\ ]^{11}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)