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Properties

Label 22.2.158...992.2
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 + 13*x^20 + 78*x^18 + 280*x^16 + 657*x^14 + 1027*x^12 + 1021*x^10 + 513*x^8 - 88*x^6 - 299*x^4 - 163*x^2 - 27)
 
Copy content gp:K = bnfinit(y^22 + 13*y^20 + 78*y^18 + 280*y^16 + 657*y^14 + 1027*y^12 + 1021*y^10 + 513*y^8 - 88*y^6 - 299*y^4 - 163*y^2 - 27, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + 13*x^20 + 78*x^18 + 280*x^16 + 657*x^14 + 1027*x^12 + 1021*x^10 + 513*x^8 - 88*x^6 - 299*x^4 - 163*x^2 - 27);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^22 + 13*x^20 + 78*x^18 + 280*x^16 + 657*x^14 + 1027*x^12 + 1021*x^10 + 513*x^8 - 88*x^6 - 299*x^4 - 163*x^2 - 27)
 

\( x^{22} + 13 x^{20} + 78 x^{18} + 280 x^{16} + 657 x^{14} + 1027 x^{12} + 1021 x^{10} + 513 x^{8} + \cdots - 27 \) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{47}a^{20}-\frac{2}{47}a^{18}+\frac{14}{47}a^{16}+\frac{23}{47}a^{14}-\frac{17}{47}a^{12}+\frac{13}{47}a^{10}-\frac{20}{47}a^{8}+\frac{14}{47}a^{6}-\frac{16}{47}a^{4}-\frac{12}{47}a^{2}+\frac{17}{47}$, $\frac{1}{141}a^{21}-\frac{2}{141}a^{19}-\frac{11}{47}a^{17}+\frac{70}{141}a^{15}+\frac{10}{47}a^{13}+\frac{13}{141}a^{11}-\frac{20}{141}a^{9}-\frac{11}{47}a^{7}-\frac{16}{141}a^{5}-\frac{59}{141}a^{3}+\frac{17}{141}a$ 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:   $a^{2}+1$, $\frac{38}{47}a^{20}+\frac{488}{47}a^{18}+\frac{2835}{47}a^{16}+\frac{9616}{47}a^{14}+\frac{20598}{47}a^{12}+\frac{27801}{47}a^{10}+\frac{21048}{47}a^{8}+\frac{3634}{47}a^{6}-\frac{7846}{47}a^{4}-\frac{6284}{47}a^{2}-\frac{1328}{47}$, $\frac{3}{47}a^{20}+\frac{88}{47}a^{18}+\frac{747}{47}a^{16}+\frac{3218}{47}a^{14}+\frac{8127}{47}a^{12}+\frac{12400}{47}a^{10}+\frac{10421}{47}a^{8}+\frac{2345}{47}a^{6}-\frac{3620}{47}a^{4}-\frac{3044}{47}a^{2}-\frac{607}{47}$, $\frac{51}{47}a^{20}+\frac{603}{47}a^{18}+\frac{3299}{47}a^{16}+\frac{10714}{47}a^{14}+\frac{22351}{47}a^{12}+\frac{29803}{47}a^{10}+\frac{22621}{47}a^{8}+\frac{4051}{47}a^{6}-\frac{8477}{47}a^{4}-\frac{6910}{47}a^{2}-\frac{1342}{47}$, $\frac{12}{47}a^{20}+\frac{164}{47}a^{18}+\frac{1014}{47}a^{16}+\frac{3660}{47}a^{14}+\frac{8350}{47}a^{12}+\frac{12047}{47}a^{10}+\frac{9912}{47}a^{8}+\frac{2283}{47}a^{6}-\frac{3482}{47}a^{4}-\frac{3058}{47}a^{2}-\frac{595}{47}$, $\frac{21}{47}a^{20}+\frac{287}{47}a^{18}+\frac{1704}{47}a^{16}+\frac{5747}{47}a^{14}+\frac{11910}{47}a^{12}+\frac{15078}{47}a^{10}+\frac{10014}{47}a^{8}+\frac{341}{47}a^{6}-\frac{4566}{47}a^{4}-\frac{2790}{47}a^{2}-\frac{442}{47}$, $\frac{34}{141}a^{21}+\frac{355}{141}a^{19}+\frac{566}{47}a^{17}+\frac{4777}{141}a^{15}+\frac{2878}{47}a^{13}+\frac{10030}{141}a^{11}+\frac{6652}{141}a^{9}+\frac{284}{47}a^{7}-\frac{2518}{141}a^{5}-\frac{2006}{141}a^{3}-a^{2}-\frac{409}{141}a-1$, $\frac{5}{141}a^{21}+\frac{131}{141}a^{19}+\frac{368}{47}a^{17}+\frac{4862}{141}a^{15}+\frac{4280}{47}a^{13}+\frac{21074}{141}a^{11}+\frac{20204}{141}a^{9}+\frac{2530}{47}a^{7}-\frac{5015}{141}a^{5}-\frac{6781}{141}a^{3}-a^{2}-\frac{2171}{141}a-2$, $\frac{92}{141}a^{21}-\frac{64}{47}a^{20}+\frac{1085}{141}a^{19}-\frac{671}{47}a^{18}+\frac{1902}{47}a^{17}-\frac{3152}{47}a^{16}+\frac{17156}{141}a^{15}-\frac{8428}{47}a^{14}+\frac{10555}{47}a^{13}-\frac{13717}{47}a^{12}+\frac{35177}{141}a^{11}-\frac{13005}{47}a^{10}+\frac{19310}{141}a^{9}-\frac{5112}{47}a^{8}-\frac{730}{47}a^{7}+\frac{2582}{47}a^{6}-\frac{10073}{141}a^{5}+\frac{3891}{47}a^{4}-\frac{5710}{141}a^{3}+\frac{1567}{47}a^{2}-\frac{1256}{141}a+\frac{181}{47}$, $\frac{146}{141}a^{21}+\frac{51}{47}a^{20}+\frac{1541}{141}a^{19}+\frac{603}{47}a^{18}+\frac{2436}{47}a^{17}+\frac{3252}{47}a^{16}+\frac{19667}{141}a^{15}+\frac{10291}{47}a^{14}+\frac{10578}{47}a^{13}+\frac{20659}{47}a^{12}+\frac{28265}{141}a^{11}+\frac{26137}{47}a^{10}+\frac{7091}{141}a^{9}+\frac{18297}{47}a^{8}-\frac{3627}{47}a^{7}+\frac{2218}{47}a^{6}-\frac{10373}{141}a^{5}-\frac{7114}{47}a^{4}-\frac{2833}{141}a^{3}-\frac{5077}{47}a^{2}-\frac{338}{141}a-\frac{919}{47}$, $\frac{371}{141}a^{21}+\frac{267}{47}a^{20}+\frac{3770}{141}a^{19}+\frac{3273}{47}a^{18}+\frac{5742}{47}a^{17}+\frac{18120}{47}a^{16}+\frac{44582}{141}a^{15}+\frac{58452}{47}a^{14}+\frac{22980}{47}a^{13}+\frac{118789}{47}a^{12}+\frac{57698}{141}a^{11}+\frac{151239}{47}a^{10}+\frac{10628}{141}a^{9}+\frac{105392}{47}a^{8}-\frac{8217}{47}a^{7}+\frac{10506}{47}a^{6}-\frac{19190}{141}a^{5}-\frac{44128}{47}a^{4}-\frac{1726}{141}a^{3}-\frac{31216}{47}a^{2}+\frac{1231}{141}a-\frac{5801}{47}$ 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:  \( 109823350.649 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 109823350.649 \cdot 1}{2\cdot\sqrt{1589600218290268559270131890388992}}\cr\approx \mathstrut & 0.528298541891 \end{aligned}\] (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 + 13*x^20 + 78*x^18 + 280*x^16 + 657*x^14 + 1027*x^12 + 1021*x^10 + 513*x^8 - 88*x^6 - 299*x^4 - 163*x^2 - 27) 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 + 13*x^20 + 78*x^18 + 280*x^16 + 657*x^14 + 1027*x^12 + 1021*x^10 + 513*x^8 - 88*x^6 - 299*x^4 - 163*x^2 - 27, 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 + 13*x^20 + 78*x^18 + 280*x^16 + 657*x^14 + 1027*x^12 + 1021*x^10 + 513*x^8 - 88*x^6 - 299*x^4 - 163*x^2 - 27); 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 + 13*x^20 + 78*x^18 + 280*x^16 + 657*x^14 + 1027*x^12 + 1021*x^10 + 513*x^8 - 88*x^6 - 299*x^4 - 163*x^2 - 27); 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

11.3.11239665258721.1

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 R R $22$ ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ ${\href{/padicField/11.11.0.1}{11} }^{2}$ ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ $22$ ${\href{/padicField/37.6.0.1}{6} }^{3}{,}\,{\href{/padicField/37.4.0.1}{4} }$ ${\href{/padicField/41.5.0.1}{5} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{6}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ ${\href{/padicField/59.11.0.1}{11} }^{2}$

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

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 2.11.2.22a151.2$x^{22} + 2 x^{21} + 2 x^{20} + 2 x^{15} + 4 x^{13} + 4 x^{12} + 6 x^{11} + 2 x^{10} + 2 x^{9} + 2 x^{6} + 5 x^{4} + 2 x^{3} + 6 x^{2} + 2 x + 9$$2$$11$$22$22T28not computed
\(3\) Copy content Toggle raw display 3.1.2.1a1.1$x^{2} + 3$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.5.1.0a1.1$x^{5} + 2 x + 1$$1$$5$$0$$C_5$$$[\ ]^{5}$$
3.5.1.0a1.1$x^{5} + 2 x + 1$$1$$5$$0$$C_5$$$[\ ]^{5}$$
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 $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_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)