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Properties

Label 22.2.158...992.1
Degree $22$
Signature $[2, 10]$
Discriminant $1.590\times 10^{33}$
Root discriminant \(32.30\)
Ramified primes $2,3,1831$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^{11}.\PSL(2,11)$ (as 22T42)

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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^22 + 9*x^20 + 36*x^18 + 79*x^16 + 99*x^14 + 68*x^12 + 24*x^10 + 3*x^8 - 9*x^6 - 15*x^4 - 11*x^2 - 3)
 
Copy content gp:K = bnfinit(y^22 + 9*y^20 + 36*y^18 + 79*y^16 + 99*y^14 + 68*y^12 + 24*y^10 + 3*y^8 - 9*y^6 - 15*y^4 - 11*y^2 - 3, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + 9*x^20 + 36*x^18 + 79*x^16 + 99*x^14 + 68*x^12 + 24*x^10 + 3*x^8 - 9*x^6 - 15*x^4 - 11*x^2 - 3);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^22 + 9*x^20 + 36*x^18 + 79*x^16 + 99*x^14 + 68*x^12 + 24*x^10 + 3*x^8 - 9*x^6 - 15*x^4 - 11*x^2 - 3)
 

\( x^{22} + 9 x^{20} + 36 x^{18} + 79 x^{16} + 99 x^{14} + 68 x^{12} + 24 x^{10} + 3 x^{8} - 9 x^{6} + \cdots - 3 \) 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:   \(1589600218290268559270131890388992\) \(\medspace = 2^{22}\cdot 3\cdot 1831^{8}\) 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:  \(32.30\)
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:   \(2\), \(3\), \(1831\) 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:  \(\Q(\sqrt{3}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{839}a^{20}+\frac{415}{839}a^{18}-\frac{113}{839}a^{16}+\frac{346}{839}a^{14}-\frac{377}{839}a^{12}-\frac{296}{839}a^{10}-\frac{175}{839}a^{8}+\frac{268}{839}a^{6}-\frac{271}{839}a^{4}-\frac{132}{839}a^{2}+\frac{93}{839}$, $\frac{1}{839}a^{21}+\frac{415}{839}a^{19}-\frac{113}{839}a^{17}+\frac{346}{839}a^{15}-\frac{377}{839}a^{13}-\frac{296}{839}a^{11}-\frac{175}{839}a^{9}+\frac{268}{839}a^{7}-\frac{271}{839}a^{5}-\frac{132}{839}a^{3}+\frac{93}{839}a$ 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:  $11$
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:   $a^{20}+8a^{18}+28a^{16}+51a^{14}+48a^{12}+20a^{10}+4a^{8}-a^{6}-8a^{4}-7a^{2}-4$, $\frac{334}{839}a^{20}+\frac{2692}{839}a^{18}+\frac{9242}{839}a^{16}+\frac{15723}{839}a^{14}+\frac{11678}{839}a^{12}+\frac{977}{839}a^{10}+\frac{280}{839}a^{8}+\frac{3095}{839}a^{6}-\frac{1580}{839}a^{4}-\frac{2138}{839}a^{2}-\frac{820}{839}$, $\frac{265}{839}a^{20}+\frac{1744}{839}a^{18}+\frac{4454}{839}a^{16}+\frac{3595}{839}a^{14}-\frac{3420}{839}a^{12}-\frac{5447}{839}a^{10}+\frac{1448}{839}a^{8}-\frac{295}{839}a^{6}-\frac{3856}{839}a^{4}+\frac{1936}{839}a^{2}+\frac{314}{839}$, $\frac{883}{839}a^{20}+\frac{7353}{839}a^{18}+\frac{26910}{839}a^{16}+\frac{52140}{839}a^{14}+\frac{53888}{839}a^{12}+\frac{26409}{839}a^{10}+\frac{5724}{839}a^{8}-\frac{793}{839}a^{6}-\frac{8568}{839}a^{4}-\frac{8325}{839}a^{2}-\frac{4298}{839}$, $\frac{1711}{839}a^{20}+\frac{14534}{839}a^{18}+\frac{54162}{839}a^{16}+\frac{107064}{839}a^{14}+\frac{112570}{839}a^{12}+\frac{53996}{839}a^{10}+\frac{7649}{839}a^{8}-\frac{2063}{839}a^{6}-\frac{13977}{839}a^{4}-\frac{16102}{839}a^{2}-\frac{8677}{839}$, $\frac{60}{839}a^{20}+\frac{569}{839}a^{18}+\frac{2449}{839}a^{16}+\frac{5658}{839}a^{14}+\frac{6745}{839}a^{12}+\frac{2376}{839}a^{10}-\frac{2110}{839}a^{8}-\frac{700}{839}a^{6}-\frac{319}{839}a^{4}-\frac{1208}{839}a^{2}-\frac{293}{839}$, $\frac{42}{839}a^{21}+\frac{1199}{839}a^{20}+\frac{650}{839}a^{19}+\frac{10126}{839}a^{18}+\frac{3644}{839}a^{17}+\frac{37347}{839}a^{16}+\frac{10337}{839}a^{15}+\frac{72542}{839}a^{14}+\frac{15209}{839}a^{13}+\frac{74030}{839}a^{12}+\frac{9382}{839}a^{11}+\frac{34392}{839}a^{10}-\frac{638}{839}a^{9}+\frac{7476}{839}a^{8}-\frac{1329}{839}a^{7}+\frac{1673}{839}a^{6}-\frac{475}{839}a^{5}-\frac{11143}{839}a^{4}-\frac{3027}{839}a^{3}-\frac{12282}{839}a^{2}-\frac{289}{839}a-\frac{5114}{839}$, $\frac{1604}{839}a^{21}+\frac{146}{839}a^{20}+\frac{13757}{839}a^{19}+\frac{1021}{839}a^{18}+\frac{51990}{839}a^{17}+\frac{2799}{839}a^{16}+\frac{105280}{839}a^{15}+\frac{2693}{839}a^{14}+\frac{115993}{839}a^{13}-\frac{1346}{839}a^{12}+\frac{61337}{839}a^{11}-\frac{1266}{839}a^{10}+\frac{7077}{839}a^{9}+\frac{8849}{839}a^{8}-\frac{13959}{839}a^{7}+\frac{12280}{839}a^{6}-\frac{23574}{839}a^{5}+\frac{706}{839}a^{4}-\frac{15402}{839}a^{3}-\frac{5848}{839}a^{2}-\frac{5204}{839}a-\frac{2363}{839}$, $\frac{951}{839}a^{21}-\frac{761}{839}a^{20}+\frac{7886}{839}a^{19}-\frac{6224}{839}a^{18}+\frac{28455}{839}a^{17}-\frac{22238}{839}a^{16}+\frac{53015}{839}a^{15}-\frac{41810}{839}a^{14}+\frac{48388}{839}a^{13}-\frac{41991}{839}a^{12}+\frac{12154}{839}a^{11}-\frac{22249}{839}a^{10}-\frac{7854}{839}a^{9}-\frac{10294}{839}a^{8}-\frac{6061}{839}a^{7}-\frac{5105}{839}a^{6}-\frac{11894}{839}a^{5}+\frac{4032}{839}a^{4}-\frac{11428}{839}a^{3}+\frac{3128}{839}a^{2}-\frac{3847}{839}a+\frac{1381}{839}$, $\frac{1510}{839}a^{21}-\frac{1182}{839}a^{20}+\frac{13341}{839}a^{19}-\frac{9783}{839}a^{18}+\frac{51705}{839}a^{17}-\frac{35073}{839}a^{16}+\frac{107155}{839}a^{15}-\frac{64982}{839}a^{14}+\frac{119549}{839}a^{13}-\frac{60303}{839}a^{12}+\frac{62313}{839}a^{11}-\frac{21805}{839}a^{10}+\frac{10942}{839}a^{9}-\frac{3739}{839}a^{8}+\frac{2799}{839}a^{7}-\frac{2990}{839}a^{6}-\frac{11524}{839}a^{5}+\frac{9892}{839}a^{4}-\frac{20613}{839}a^{3}+\frac{9199}{839}a^{2}-\frac{8073}{839}a+\frac{1661}{839}$, $\frac{2401}{839}a^{21}-\frac{1759}{839}a^{20}+\frac{20658}{839}a^{19}-\frac{14318}{839}a^{18}+\frac{77711}{839}a^{17}-\frac{50416}{839}a^{16}+\frac{154512}{839}a^{15}-\frac{90951}{839}a^{14}+\frac{161192}{839}a^{13}-\frac{80211}{839}a^{12}+\frac{72091}{839}a^{11}-\frac{23847}{839}a^{10}+\frac{5198}{839}a^{9}-\frac{88}{839}a^{8}-\frac{2562}{839}a^{7}+\frac{945}{839}a^{6}-\frac{23099}{839}a^{5}+\frac{18595}{839}a^{4}-\frac{29994}{839}a^{3}+\frac{14048}{839}a^{2}-\frac{10788}{839}a+\frac{3374}{839}$ 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:  \( 74452581.4489 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 74452581.4489 \cdot 1}{2\cdot\sqrt{1589600218290268559270131890388992}}\cr\approx \mathstrut & 0.358149610143 \end{aligned}\] (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 + 9*x^20 + 36*x^18 + 79*x^16 + 99*x^14 + 68*x^12 + 24*x^10 + 3*x^8 - 9*x^6 - 15*x^4 - 11*x^2 - 3) 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 + 9*x^20 + 36*x^18 + 79*x^16 + 99*x^14 + 68*x^12 + 24*x^10 + 3*x^8 - 9*x^6 - 15*x^4 - 11*x^2 - 3, 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 + 9*x^20 + 36*x^18 + 79*x^16 + 99*x^14 + 68*x^12 + 24*x^10 + 3*x^8 - 9*x^6 - 15*x^4 - 11*x^2 - 3); 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 + 9*x^20 + 36*x^18 + 79*x^16 + 99*x^14 + 68*x^12 + 24*x^10 + 3*x^8 - 9*x^6 - 15*x^4 - 11*x^2 - 3); 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

$C_2^{11}.\PSL(2,11)$ (as 22T42):

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 1351680
The 112 conjugacy class representatives for $C_2^{11}.\PSL(2,11)$
Character table for $C_2^{11}.\PSL(2,11)$

Intermediate fields

11.3.11239665258721.2

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 siblings: 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 R $22$ ${\href{/padicField/7.5.0.1}{5} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.11.0.1}{11} }^{2}$ ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.10.0.1}{10} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.3.0.1}{3} }^{6}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ $22$ ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{7}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }^{6}$ ${\href{/padicField/59.11.0.1}{11} }^{2}$

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 2.11.2.22a39.2$x^{22} + 2 x^{19} + 2 x^{17} + 2 x^{13} + 2 x^{11} + 2 x^{10} + 4 x^{8} + 2 x^{6} + x^{4} + 2 x^{2} + 7$$2$$11$$22$22T28not computed
\(3\) Copy content Toggle raw display 3.1.2.1a1.2$x^{2} + 6$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.10.1.0a1.1$x^{10} + 2 x^{6} + 2 x^{5} + 2 x^{4} + x + 2$$1$$10$$0$$C_{10}$$$[\ ]^{10}$$
3.10.1.0a1.1$x^{10} + 2 x^{6} + 2 x^{5} + 2 x^{4} + x + 2$$1$$10$$0$$C_{10}$$$[\ ]^{10}$$
\(1831\) Copy content Toggle raw display Deg $4$$2$$2$$2$
Deg $6$$1$$6$$0$$C_6$$$[\ ]^{6}$$
Deg $6$$2$$3$$3$
Deg $6$$2$$3$$3$

Spectrum of ring of integers

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