↑

Properties

Label 22.2.168...256.1
Degree $22$
Signature $[2, 10]$
Discriminant $1.688\times 10^{29}$
Root discriminant \(21.31\)
Ramified primes $2,200601609583$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^{10}.S_{11}$ (as 22T51)

Related objects

Downloads

Learn more

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^22 + x^20 - 12*x^14 - 14*x^12 + 16*x^10 + 28*x^8 + x^6 - 14*x^4 - 7*x^2 - 1)
 
Copy content gp:K = bnfinit(y^22 + y^20 - 12*y^14 - 14*y^12 + 16*y^10 + 28*y^8 + y^6 - 14*y^4 - 7*y^2 - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + x^20 - 12*x^14 - 14*x^12 + 16*x^10 + 28*x^8 + x^6 - 14*x^4 - 7*x^2 - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^22 + x^20 - 12*x^14 - 14*x^12 + 16*x^10 + 28*x^8 + x^6 - 14*x^4 - 7*x^2 - 1)
 

\( x^{22} + x^{20} - 12x^{14} - 14x^{12} + 16x^{10} + 28x^{8} + x^{6} - 14x^{4} - 7x^{2} - 1 \) 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:  $22$
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:  $[2, 10]$
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:   \(168783011453769015346390368256\) \(\medspace = 2^{22}\cdot 200601609583^{2}\) 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:  \(21.31\)
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:   \(2\), \(200601609583\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$ 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:  Yes
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:  $11$
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:   $86a^{20}+43a^{18}-21a^{16}+11a^{14}-1037a^{12}-685a^{10}+1713a^{8}+1545a^{6}-684a^{4}-855a^{2}-172$, $a$, $306a^{20}+148a^{18}-76a^{16}+40a^{14}-3692a^{12}-2377a^{10}+6119a^{8}+5399a^{6}-2483a^{4}-2993a^{2}-590$, $390a^{21}+187a^{19}-98a^{17}+50a^{15}-4707a^{13}-3011a^{11}+7814a^{9}+6865a^{7}-3183a^{5}-3815a^{3}-751a$, $86a^{20}+43a^{18}-21a^{16}+11a^{14}-1037a^{12}-685a^{10}+1713a^{8}+1545a^{6}-684a^{4}-854a^{2}-172$, $361a^{20}+174a^{18}-90a^{16}+47a^{14}-4356a^{12}-2797a^{10}+7224a^{8}+6362a^{6}-2936a^{4}-3531a^{2}-696$, $149a^{20}+72a^{18}-37a^{16}+19a^{14}-1798a^{12}-1157a^{10}+2979a^{8}+2633a^{6}-1206a^{4}-1461a^{2}-291$, $55a^{21}+203a^{20}+26a^{19}+98a^{18}-13a^{17}-50a^{16}+8a^{15}+27a^{14}-663a^{13}-2449a^{12}-419a^{11}-1574a^{10}+1094a^{9}+4055a^{8}+950a^{7}+3573a^{6}-448a^{5}-1645a^{4}-523a^{3}-1979a^{2}-101a-390$, $106a^{21}-106a^{20}+51a^{19}-51a^{18}-26a^{17}+26a^{16}+14a^{15}-14a^{14}-1279a^{13}+1279a^{12}-820a^{11}+820a^{10}+2116a^{9}-2116a^{8}+1863a^{7}-1863a^{6}-857a^{5}+857a^{4}-1031a^{3}+1031a^{2}-204a+204$, $201a^{21}+187a^{20}+98a^{19}+89a^{18}-50a^{17}-47a^{16}+26a^{15}+24a^{14}-2425a^{13}-2257a^{12}-1571a^{11}-1436a^{10}+4019a^{9}+3748a^{8}+3564a^{7}+3279a^{6}-1625a^{5}-1533a^{4}-1976a^{3}-1821a^{2}-393a-356$, $212a^{21}-113a^{20}+101a^{19}-53a^{18}-52a^{17}+29a^{16}+28a^{15}-14a^{14}-2558a^{13}+1365a^{12}-1628a^{11}+859a^{10}+4234a^{9}-2272a^{8}+3708a^{7}-1973a^{6}-1724a^{5}+932a^{4}-2053a^{3}+1101a^{2}-404a+216$ 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:  \( 355604.295989 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 355604.295989 \cdot 1}{2\cdot\sqrt{168783011453769015346390368256}}\cr\approx \mathstrut & 0.166008877310 \end{aligned}\] (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^22 + x^20 - 12*x^14 - 14*x^12 + 16*x^10 + 28*x^8 + x^6 - 14*x^4 - 7*x^2 - 1) 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^22 + x^20 - 12*x^14 - 14*x^12 + 16*x^10 + 28*x^8 + x^6 - 14*x^4 - 7*x^2 - 1, 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^22 + x^20 - 12*x^14 - 14*x^12 + 16*x^10 + 28*x^8 + x^6 - 14*x^4 - 7*x^2 - 1); 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^22 + x^20 - 12*x^14 - 14*x^12 + 16*x^10 + 28*x^8 + x^6 - 14*x^4 - 7*x^2 - 1); 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

$C_2^{10}.S_{11}$ (as 22T51):

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 40874803200
The 376 conjugacy class representatives for $C_2^{10}.S_{11}$
Character table for $C_2^{10}.S_{11}$

Intermediate fields

11.5.200601609583.1

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

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 22 sibling: data not computed
Degree 44 sibling: data not computed
Minimal sibling: 22.8.88784765467352548428076714954223549740883971996276485665050840867788808040301622657694450352949666726338607645486396011579512455168.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.7.0.1}{7} }^{2}{,}\,{\href{/padicField/3.4.0.1}{4} }^{2}$ ${\href{/padicField/5.7.0.1}{7} }^{2}{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}$ ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ ${\href{/padicField/13.14.0.1}{14} }{,}\,{\href{/padicField/13.8.0.1}{8} }$ ${\href{/padicField/17.9.0.1}{9} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}$ $18{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.10.0.1}{10} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ $16{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ $18{,}\,{\href{/padicField/59.4.0.1}{4} }$

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 2.11.2.22a135.2$x^{22} + 2 x^{21} + 2 x^{19} + 2 x^{17} + 2 x^{16} + 2 x^{15} + 2 x^{14} + 2 x^{13} + 4 x^{12} + 2 x^{11} + 4 x^{10} + 4 x^{8} + 2 x^{7} + 4 x^{6} + 4 x^{5} + 3 x^{4} + 4 x^{3} + 2 x^{2} + 6 x + 3$$2$$11$$22$22T23not computed
\(200601609583\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $4$$2$$2$$2$
Deg $16$$1$$16$$0$$C_{16}$$$[\ ]^{16}$$

Spectrum of ring of integers

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