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Properties

Label 22.2.529...664.1
Degree $22$
Signature $[2, 10]$
Discriminant $5.299\times 10^{32}$
Root discriminant \(30.72\)
Ramified primes $2,1831$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^{10}.\PSL(2,11)$ (as 22T39)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 5*x^20 - 11*x^18 - 34*x^16 + 29*x^14 + 127*x^12 - 268*x^10 + 239*x^8 - 119*x^6 + 29*x^4 - 1)
 
gp: K = bnfinit(y^22 + 5*y^20 - 11*y^18 - 34*y^16 + 29*y^14 + 127*y^12 - 268*y^10 + 239*y^8 - 119*y^6 + 29*y^4 - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + 5*x^20 - 11*x^18 - 34*x^16 + 29*x^14 + 127*x^12 - 268*x^10 + 239*x^8 - 119*x^6 + 29*x^4 - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 5*x^20 - 11*x^18 - 34*x^16 + 29*x^14 + 127*x^12 - 268*x^10 + 239*x^8 - 119*x^6 + 29*x^4 - 1)
 

\( x^{22} + 5x^{20} - 11x^{18} - 34x^{16} + 29x^{14} + 127x^{12} - 268x^{10} + 239x^{8} - 119x^{6} + 29x^{4} - 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $22$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[2, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(529866739430089519756710630129664\) \(\medspace = 2^{22}\cdot 1831^{8}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(30.72\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  not computed
Ramified primes:   \(2\), \(1831\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $2$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{1013417}a^{20}-\frac{157127}{1013417}a^{18}-\frac{198618}{1013417}a^{16}+\frac{53610}{1013417}a^{14}-\frac{324387}{1013417}a^{12}-\frac{256638}{1013417}a^{10}+\frac{152684}{1013417}a^{8}+\frac{92009}{1013417}a^{6}-\frac{151385}{1013417}a^{4}+\frac{504025}{1013417}a^{2}+\frac{82250}{1013417}$, $\frac{1}{1013417}a^{21}-\frac{157127}{1013417}a^{19}-\frac{198618}{1013417}a^{17}+\frac{53610}{1013417}a^{15}-\frac{324387}{1013417}a^{13}-\frac{256638}{1013417}a^{11}+\frac{152684}{1013417}a^{9}+\frac{92009}{1013417}a^{7}-\frac{151385}{1013417}a^{5}+\frac{504025}{1013417}a^{3}+\frac{82250}{1013417}a$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $a^{21}+5a^{19}-11a^{17}-34a^{15}+29a^{13}+127a^{11}-268a^{9}+239a^{7}-119a^{5}+29a^{3}$, $\frac{14570928}{1013417}a^{20}+\frac{80207062}{1013417}a^{18}-\frac{119791049}{1013417}a^{16}-\frac{555806538}{1013417}a^{14}+\frac{141708758}{1013417}a^{12}+\frac{1921624050}{1013417}a^{10}-\frac{2933517644}{1013417}a^{8}+\frac{2004609959}{1013417}a^{6}-\frac{730952065}{1013417}a^{4}+\frac{63428094}{1013417}a^{2}+\frac{26339978}{1013417}$, $\frac{28877382}{1013417}a^{20}+\frac{159229403}{1013417}a^{18}-\frac{235688276}{1013417}a^{16}-\frac{1102226216}{1013417}a^{14}+\frac{270340981}{1013417}a^{12}+\frac{3801159423}{1013417}a^{10}-\frac{5787309779}{1013417}a^{8}+\frac{3943528551}{1013417}a^{6}-\frac{1423063213}{1013417}a^{4}+\frac{109952178}{1013417}a^{2}+\frac{56743029}{1013417}$, $\frac{27880906}{1013417}a^{21}+\frac{153506853}{1013417}a^{19}-\frac{228837455}{1013417}a^{17}-\frac{1062517575}{1013417}a^{15}+\frac{269647788}{1013417}a^{13}+\frac{3669269785}{1013417}a^{11}-\frac{5615662846}{1013417}a^{9}+\frac{3849369893}{1013417}a^{7}-\frac{1408530233}{1013417}a^{5}+\frac{121749065}{1013417}a^{3}+\frac{51624819}{1013417}a$, $\frac{86692489}{1013417}a^{21}+\frac{477703100}{1013417}a^{19}-\frac{709628194}{1013417}a^{17}-\frac{3308544316}{1013417}a^{15}+\frac{823861978}{1013417}a^{13}+\frac{11422451535}{1013417}a^{11}-\frac{17402029352}{1013417}a^{9}+\frac{11866656009}{1013417}a^{7}-\frac{4303730034}{1013417}a^{5}+\frac{349834445}{1013417}a^{3}+\frac{164657286}{1013417}a$, $\frac{12832600}{1013417}a^{20}+\frac{71173291}{1013417}a^{18}-\frac{102396820}{1013417}a^{16}-\frac{492938278}{1013417}a^{14}+\frac{103649623}{1013417}a^{12}+\frac{1691235000}{1013417}a^{10}-\frac{2514923058}{1013417}a^{8}+\frac{1676979924}{1013417}a^{6}-\frac{592362297}{1013417}a^{4}+\frac{39150823}{1013417}a^{2}+\frac{24772589}{1013417}$, $\frac{73275972}{1013417}a^{21}-\frac{45560151}{1013417}a^{20}+\frac{403921339}{1013417}a^{19}-\frac{250525223}{1013417}a^{18}-\frac{599152229}{1013417}a^{17}+\frac{376140771}{1013417}a^{16}-\frac{2798730606}{1013417}a^{15}+\frac{1736176842}{1013417}a^{14}+\frac{691175922}{1013417}a^{13}-\frac{455537531}{1013417}a^{12}+\frac{9662610018}{1013417}a^{11}-\frac{6010395194}{1013417}a^{10}-\frac{14684858964}{1013417}a^{9}+\frac{9217024484}{1013417}a^{8}+\frac{9983622019}{1013417}a^{7}-\frac{6297624032}{1013417}a^{6}-\frac{3603981732}{1013417}a^{5}+\frac{2277054364}{1013417}a^{4}+\frac{273892489}{1013417}a^{3}-\frac{179870023}{1013417}a^{2}+\frac{147663830}{1013417}a-\frac{89425376}{1013417}$, $\frac{39234838}{1013417}a^{21}-\frac{35967127}{1013417}a^{20}+\frac{217231390}{1013417}a^{19}-\frac{198919965}{1013417}a^{18}-\frac{315212213}{1013417}a^{17}+\frac{290356368}{1013417}a^{16}-\frac{1504671909}{1013417}a^{15}+\frac{1378692040}{1013417}a^{14}+\frac{329996726}{1013417}a^{13}-\frac{311957304}{1013417}a^{12}+\frac{5169264170}{1013417}a^{11}-\frac{4744675893}{1013417}a^{10}-\frac{7729178424}{1013417}a^{9}+\frac{7112722938}{1013417}a^{8}+\frac{5207591700}{1013417}a^{7}-\frac{4791798057}{1013417}a^{6}-\frac{1869607185}{1013417}a^{5}+\frac{1718458778}{1013417}a^{4}+\frac{139082737}{1013417}a^{3}-\frac{130273860}{1013417}a^{2}+\frac{75115161}{1013417}a-\frac{68127479}{1013417}$, $\frac{439547399}{1013417}a^{21}-\frac{316711817}{1013417}a^{20}+\frac{2423713421}{1013417}a^{19}-\frac{1747044027}{1013417}a^{18}-\frac{3589225362}{1013417}a^{17}+\frac{2581843972}{1013417}a^{16}-\frac{16791503437}{1013417}a^{15}+\frac{12099851820}{1013417}a^{14}+\frac{4115204523}{1013417}a^{13}-\frac{2938205461}{1013417}a^{12}+\frac{57948908700}{1013417}a^{11}-\frac{41731797793}{1013417}a^{10}-\frac{88001522225}{1013417}a^{9}+\frac{63341332544}{1013417}a^{8}+\frac{59778373708}{1013417}a^{7}-\frac{43018472240}{1013417}a^{6}-\frac{21542002476}{1013417}a^{5}+\frac{15498598259}{1013417}a^{4}+\frac{1655138506}{1013417}a^{3}-\frac{1190183895}{1013417}a^{2}+\frac{854022403}{1013417}a-\frac{614561813}{1013417}$, $\frac{145469564}{1013417}a^{21}-\frac{98390012}{1013417}a^{20}+\frac{801243496}{1013417}a^{19}-\frac{541768421}{1013417}a^{18}-\frac{1193288411}{1013417}a^{17}+\frac{808020414}{1013417}a^{16}-\frac{5553032078}{1013417}a^{15}+\frac{3754706200}{1013417}a^{14}+\frac{1398086922}{1013417}a^{13}-\frac{952271215}{1013417}a^{12}+\frac{19190647481}{1013417}a^{11}-\frac{12980130218}{1013417}a^{10}-\frac{29233050695}{1013417}a^{9}+\frac{19794944200}{1013417}a^{8}+\frac{19901159103}{1013417}a^{7}-\frac{13485246989}{1013417}a^{6}-\frac{7187953303}{1013417}a^{5}+\frac{4876165449}{1013417}a^{4}+\frac{561073007}{1013417}a^{3}-\frac{383059102}{1013417}a^{2}+\frac{286095106}{1013417}a-\frac{193428001}{1013417}$, $\frac{64795504}{1013417}a^{21}+\frac{60367729}{1013417}a^{20}+\frac{357202220}{1013417}a^{19}+\frac{330924299}{1013417}a^{18}-\frac{528666673}{1013417}a^{17}-\frac{505721813}{1013417}a^{16}-\frac{2468125021}{1013417}a^{15}-\frac{2303291890}{1013417}a^{14}+\frac{611813251}{1013417}a^{13}+\frac{644017671}{1013417}a^{12}+\frac{8505338856}{1013417}a^{11}+\frac{8022911467}{1013417}a^{10}-\frac{13033012053}{1013417}a^{9}-\frac{12284014290}{1013417}a^{8}+\frac{8898277933}{1013417}a^{7}+\frac{8383972956}{1013417}a^{6}-\frac{3232320036}{1013417}a^{5}-\frac{3031270903}{1013417}a^{4}+\frac{266842296}{1013417}a^{3}+\frac{239606633}{1013417}a^{2}+\frac{127608918}{1013417}a+\frac{120594620}{1013417}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 90508363.0226 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 90508363.0226 \cdot 1}{2\cdot\sqrt{529866739430089519756710630129664}}\cr\approx \mathstrut & 0.754108888674 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^22 + 5*x^20 - 11*x^18 - 34*x^16 + 29*x^14 + 127*x^12 - 268*x^10 + 239*x^8 - 119*x^6 + 29*x^4 - 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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^22 + 5*x^20 - 11*x^18 - 34*x^16 + 29*x^14 + 127*x^12 - 268*x^10 + 239*x^8 - 119*x^6 + 29*x^4 - 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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^22 + 5*x^20 - 11*x^18 - 34*x^16 + 29*x^14 + 127*x^12 - 268*x^10 + 239*x^8 - 119*x^6 + 29*x^4 - 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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 5*x^20 - 11*x^18 - 34*x^16 + 29*x^14 + 127*x^12 - 268*x^10 + 239*x^8 - 119*x^6 + 29*x^4 - 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

$C_2^{10}.\PSL(2,11)$ (as 22T39):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A non-solvable group of order 675840
The 56 conjugacy class representatives for $C_2^{10}.\PSL(2,11)$
Character table for $C_2^{10}.\PSL(2,11)$

Intermediate fields

11.3.11239665258721.1

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Degree 22 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 ${\href{/padicField/3.10.0.1}{10} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ ${\href{/padicField/5.11.0.1}{11} }^{2}$ ${\href{/padicField/7.10.0.1}{10} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ ${\href{/padicField/11.11.0.1}{11} }^{2}$ ${\href{/padicField/13.5.0.1}{5} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ ${\href{/padicField/31.11.0.1}{11} }^{2}$ ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{7}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ ${\href{/padicField/59.11.0.1}{11} }^{2}$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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 Deg $22$$2$$11$$22$
\(1831\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $6$$2$$3$$3$
Deg $6$$2$$3$$3$