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Properties

Label 22.2.128...337.1
Degree $22$
Signature $[2, 10]$
Discriminant $1.282\times 10^{31}$
Root discriminant \(25.94\)
Ramified primes $31,43,547,632447,34374601$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^{10}.(C_2\times S_{11})$ (as 22T53)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 + 75*x^20 - 365*x^19 + 1409*x^18 - 4473*x^17 + 12052*x^16 - 27974*x^15 + 56686*x^14 - 100938*x^13 + 158847*x^12 - 221312*x^11 + 273270*x^10 - 298381*x^9 + 287150*x^8 - 241819*x^7 + 176511*x^6 - 109912*x^5 + 57096*x^4 - 23845*x^3 + 7532*x^2 - 1599*x + 163)
 
Copy content gp:K = bnfinit(y^22 - 11*y^21 + 75*y^20 - 365*y^19 + 1409*y^18 - 4473*y^17 + 12052*y^16 - 27974*y^15 + 56686*y^14 - 100938*y^13 + 158847*y^12 - 221312*y^11 + 273270*y^10 - 298381*y^9 + 287150*y^8 - 241819*y^7 + 176511*y^6 - 109912*y^5 + 57096*y^4 - 23845*y^3 + 7532*y^2 - 1599*y + 163, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 11*x^21 + 75*x^20 - 365*x^19 + 1409*x^18 - 4473*x^17 + 12052*x^16 - 27974*x^15 + 56686*x^14 - 100938*x^13 + 158847*x^12 - 221312*x^11 + 273270*x^10 - 298381*x^9 + 287150*x^8 - 241819*x^7 + 176511*x^6 - 109912*x^5 + 57096*x^4 - 23845*x^3 + 7532*x^2 - 1599*x + 163);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^22 - 11*x^21 + 75*x^20 - 365*x^19 + 1409*x^18 - 4473*x^17 + 12052*x^16 - 27974*x^15 + 56686*x^14 - 100938*x^13 + 158847*x^12 - 221312*x^11 + 273270*x^10 - 298381*x^9 + 287150*x^8 - 241819*x^7 + 176511*x^6 - 109912*x^5 + 57096*x^4 - 23845*x^3 + 7532*x^2 - 1599*x + 163)
 

\( x^{22} - 11 x^{21} + 75 x^{20} - 365 x^{19} + 1409 x^{18} - 4473 x^{17} + 12052 x^{16} - 27974 x^{15} + \cdots + 163 \) 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:   \(12816596924190768698410425412337\) \(\medspace = 31\cdot 43^{2}\cdot 547^{2}\cdot 632447\cdot 34374601^{2}\) 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:  \(25.94\)
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:  $31^{1/2}43^{1/2}547^{1/2}632447^{1/2}34374601^{1/2}\approx 3981435085.1469045$
Ramified primes:   \(31\), \(43\), \(547\), \(632447\), \(34374601\) 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:  $\Q(\sqrt{19605857}$)
$\Aut(K/\Q)$:   $C_2$
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:   $a^{2}-a+2$, $a^{20}-10a^{19}+63a^{18}-282a^{17}+1001a^{16}-2908a^{15}+7142a^{14}-15016a^{13}+27386a^{12}-43520a^{11}+60555a^{10}-73717a^{9}+78443a^{8}-72504a^{7}+57760a^{6}-39051a^{5}+21940a^{4}-9870a^{3}+3347a^{2}-760a+83$, $3a^{18}-27a^{17}+155a^{16}-628a^{15}+2014a^{14}-5250a^{13}+11504a^{12}-21392a^{11}+34199a^{10}-47069a^{9}+55963a^{8}-57176a^{7}+49950a^{6}-36756a^{5}+22342a^{4}-10802a^{3}+3901a^{2}-931a+101$, $7a^{20}-70a^{19}+440a^{18}-1965a^{17}+6958a^{16}-20168a^{15}+49434a^{14}-103768a^{13}+189038a^{12}-300226a^{11}+417745a^{10}-508859a^{9}+542185a^{8}-502116a^{7}+401084a^{6}-272089a^{5}+153510a^{4}-69401a^{3}+23653a^{2}-5392a+581$, $a^{20}-10a^{19}+71a^{18}-354a^{17}+1416a^{16}-4596a^{15}+12583a^{14}-29275a^{13}+58821a^{12}-102351a^{11}+155278a^{10}-205078a^{9}+235936a^{8}-234865a^{7}+201041a^{6}-145668a^{5}+87605a^{4}-42111a^{3}+15225a^{2}-3669a+412$, $8a^{20}-80a^{19}+503a^{18}-2247a^{17}+7959a^{16}-23076a^{15}+56576a^{14}-118784a^{13}+216424a^{12}-343746a^{11}+478300a^{10}-582576a^{9}+620628a^{8}-574620a^{7}+458844a^{6}-311140a^{5}+175450a^{4}-79271a^{3}+26999a^{2}-6151a+662$, $8a^{20}-80a^{19}+499a^{18}-2211a^{17}+7752a^{16}-22236a^{15}+53876a^{14}-111728a^{13}+200914a^{12}-314802a^{11}+431831a^{10}-518311a^{9}+543786a^{8}-495606a^{7}+389289a^{6}-259507a^{5}+143726a^{4}-63733a^{3}+21292a^{2}-4759a+506$, $13a^{20}-130a^{19}+816a^{18}-3639a^{17}+12863a^{16}-37216a^{15}+91041a^{14}-190721a^{13}+346718a^{12}-549493a^{11}+762966a^{10}-927441a^{9}+986171a^{8}-911523a^{7}+726805a^{6}-492269a^{5}+277387a^{4}-125312a^{3}+42723a^{2}-9759a+1057$, $a^{21}-13a^{20}+96a^{19}-501a^{18}+2030a^{17}-6704a^{16}+18566a^{15}-43944a^{14}+89943a^{13}-160513a^{12}+250834a^{11}-344062a^{10}+413923a^{9}-435670a^{8}+398668a^{7}-314316a^{6}+210179a^{5}-116566a^{4}+51584a^{3}-17110a^{2}+3748a-386$, $a^{21}-13a^{20}+93a^{19}-471a^{18}+1849a^{17}-5928a^{16}+15960a^{15}-36807a^{14}+73559a^{13}-128485a^{12}+196997a^{11}-265789a^{10}+315374a^{9}-328285a^{8}+298038a^{7}-233859a^{6}+156314a^{5}-87035a^{4}+38980a^{3}-13190a^{2}+3012a-329$, $a^{21}-12a^{20}+83a^{19}-408a^{18}+1566a^{17}-4920a^{16}+13016a^{15}-29539a^{14}+58180a^{13}-100265a^{12}+151837a^{11}-202543a^{10}+237823a^{9}-245235a^{8}+220727a^{7}-171923a^{6}+114161a^{5}-63270a^{4}+28234a^{3}-9562a^{2}+2190a-246$ 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:  \( 3786540.2671 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 3786540.2671 \cdot 1}{2\cdot\sqrt{12816596924190768698410425412337}}\cr\approx \mathstrut & 0.20285461653 \end{aligned}\] (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 - 11*x^21 + 75*x^20 - 365*x^19 + 1409*x^18 - 4473*x^17 + 12052*x^16 - 27974*x^15 + 56686*x^14 - 100938*x^13 + 158847*x^12 - 221312*x^11 + 273270*x^10 - 298381*x^9 + 287150*x^8 - 241819*x^7 + 176511*x^6 - 109912*x^5 + 57096*x^4 - 23845*x^3 + 7532*x^2 - 1599*x + 163) 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 - 11*x^21 + 75*x^20 - 365*x^19 + 1409*x^18 - 4473*x^17 + 12052*x^16 - 27974*x^15 + 56686*x^14 - 100938*x^13 + 158847*x^12 - 221312*x^11 + 273270*x^10 - 298381*x^9 + 287150*x^8 - 241819*x^7 + 176511*x^6 - 109912*x^5 + 57096*x^4 - 23845*x^3 + 7532*x^2 - 1599*x + 163, 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 - 11*x^21 + 75*x^20 - 365*x^19 + 1409*x^18 - 4473*x^17 + 12052*x^16 - 27974*x^15 + 56686*x^14 - 100938*x^13 + 158847*x^12 - 221312*x^11 + 273270*x^10 - 298381*x^9 + 287150*x^8 - 241819*x^7 + 176511*x^6 - 109912*x^5 + 57096*x^4 - 23845*x^3 + 7532*x^2 - 1599*x + 163); 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 - 11*x^21 + 75*x^20 - 365*x^19 + 1409*x^18 - 4473*x^17 + 12052*x^16 - 27974*x^15 + 56686*x^14 - 100938*x^13 + 158847*x^12 - 221312*x^11 + 273270*x^10 - 298381*x^9 + 287150*x^8 - 241819*x^7 + 176511*x^6 - 109912*x^5 + 57096*x^4 - 23845*x^3 + 7532*x^2 - 1599*x + 163); 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

$C_2^{10}.(C_2\times S_{11})$ (as 22T53):

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 81749606400
The 752 conjugacy class representatives for $C_2^{10}.(C_2\times S_{11})$
Character table for $C_2^{10}.(C_2\times S_{11})$

Intermediate fields

11.7.808524990121.1

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 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 ${\href{/padicField/2.11.0.1}{11} }^{2}$ $22$ $22$ $22$ $18{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ ${\href{/padicField/13.7.0.1}{7} }^{2}{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.6.0.1}{6} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.11.0.1}{11} }^{2}$ R ${\href{/padicField/37.11.0.1}{11} }^{2}$ $20{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ R ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ $20{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.5.0.1}{5} }^{2}$

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
\(31\) Copy content Toggle raw display 31.1.2.1a1.1$x^{2} + 31$$2$$1$$1$$C_2$$$[\ ]_{2}$$
31.5.1.0a1.1$x^{5} + 7 x + 28$$1$$5$$0$$C_5$$$[\ ]^{5}$$
31.5.1.0a1.1$x^{5} + 7 x + 28$$1$$5$$0$$C_5$$$[\ ]^{5}$$
31.10.1.0a1.1$x^{10} + 30 x^{5} + 26 x^{4} + 13 x^{3} + 13 x^{2} + 13 x + 3$$1$$10$$0$$C_{10}$$$[\ ]^{10}$$
\(43\) Copy content Toggle raw display 43.1.2.1a1.1$x^{2} + 43$$2$$1$$1$$C_2$$$[\ ]_{2}$$
43.1.2.1a1.1$x^{2} + 43$$2$$1$$1$$C_2$$$[\ ]_{2}$$
43.4.1.0a1.1$x^{4} + 5 x^{2} + 42 x + 3$$1$$4$$0$$C_4$$$[\ ]^{4}$$
43.7.1.0a1.1$x^{7} + 42 x^{2} + 7 x + 40$$1$$7$$0$$C_7$$$[\ ]^{7}$$
43.7.1.0a1.1$x^{7} + 42 x^{2} + 7 x + 40$$1$$7$$0$$C_7$$$[\ ]^{7}$$
\(547\) Copy content Toggle raw display $\Q_{547}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{547}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $4$$1$$4$$0$$C_4$$$[\ ]^{4}$$
Deg $6$$1$$6$$0$$C_6$$$[\ ]^{6}$$
Deg $6$$1$$6$$0$$C_6$$$[\ ]^{6}$$
\(632447\) Copy content Toggle raw display $\Q_{632447}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{632447}$$x$$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 $4$$1$$4$$0$$C_4$$$[\ ]^{4}$$
Deg $5$$1$$5$$0$$C_5$$$[\ ]^{5}$$
Deg $5$$1$$5$$0$$C_5$$$[\ ]^{5}$$
\(34374601\) Copy content Toggle raw display $\Q_{34374601}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{34374601}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $4$$2$$2$$2$
Deg $5$$1$$5$$0$$C_5$$$[\ ]^{5}$$
Deg $5$$1$$5$$0$$C_5$$$[\ ]^{5}$$
Deg $6$$1$$6$$0$$C_6$$$[\ ]^{6}$$

Spectrum of ring of integers

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