Properties

Label 22.2.104...544.1
Degree $22$
Signature $[2, 10]$
Discriminant $1.041\times 10^{29}$
Root discriminant \(20.84\)
Ramified primes $2,19457,8095783$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^{10}.S_{11}$ (as 22T51)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 2*x^20 - x^18 + 2*x^16 - 18*x^12 - 24*x^10 + 3*x^8 + 36*x^6 + 27*x^4 + 4*x^2 - 1)
 
gp: K = bnfinit(y^22 + 2*y^20 - y^18 + 2*y^16 - 18*y^12 - 24*y^10 + 3*y^8 + 36*y^6 + 27*y^4 + 4*y^2 - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + 2*x^20 - x^18 + 2*x^16 - 18*x^12 - 24*x^10 + 3*x^8 + 36*x^6 + 27*x^4 + 4*x^2 - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 2*x^20 - x^18 + 2*x^16 - 18*x^12 - 24*x^10 + 3*x^8 + 36*x^6 + 27*x^4 + 4*x^2 - 1)
 

\( x^{22} + 2x^{20} - x^{18} + 2x^{16} - 18x^{12} - 24x^{10} + 3x^{8} + 36x^{6} + 27x^{4} + 4x^{2} - 1 \) 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:   \(104070916689387515610916716544\) \(\medspace = 2^{22}\cdot 19457^{2}\cdot 8095783^{2}\) 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:  \(20.84\)
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:   \(2\), \(19457\), \(8095783\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{131749}a^{20}-\frac{5480}{131749}a^{18}+\frac{2587}{131749}a^{16}+\frac{46960}{131749}a^{14}+\frac{2826}{131749}a^{12}+\frac{54232}{131749}a^{10}+\frac{57645}{131749}a^{8}+\frac{55964}{131749}a^{6}+\frac{48809}{131749}a^{4}+\frac{11308}{131749}a^{2}+\frac{63327}{131749}$, $\frac{1}{131749}a^{21}-\frac{5480}{131749}a^{19}+\frac{2587}{131749}a^{17}+\frac{46960}{131749}a^{15}+\frac{2826}{131749}a^{13}+\frac{54232}{131749}a^{11}+\frac{57645}{131749}a^{9}+\frac{55964}{131749}a^{7}+\frac{48809}{131749}a^{5}+\frac{11308}{131749}a^{3}+\frac{63327}{131749}a$ 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:   $\frac{9568}{131749}a^{20}+\frac{3462}{131749}a^{18}-\frac{16396}{131749}a^{16}+\frac{49190}{131749}a^{14}-\frac{101126}{131749}a^{12}-\frac{67535}{131749}a^{10}-\frac{85703}{131749}a^{8}+\frac{35616}{131749}a^{6}+\frac{217805}{131749}a^{4}+\frac{160764}{131749}a^{2}+\frac{130834}{131749}$, $\frac{227019}{131749}a^{20}+\frac{305185}{131749}a^{18}-\frac{434136}{131749}a^{16}+\frac{737152}{131749}a^{14}-\frac{457183}{131749}a^{12}-\frac{3806865}{131749}a^{10}-\frac{2884644}{131749}a^{8}+\frac{2706728}{131749}a^{6}+\frac{6408176}{131749}a^{4}+\frac{1714324}{131749}a^{2}-\frac{413914}{131749}$, $a$, $\frac{641126}{131749}a^{20}+\frac{902346}{131749}a^{18}-\frac{1180940}{131749}a^{16}+\frac{1971715}{131749}a^{14}-\frac{1175913}{131749}a^{12}-\frac{10873278}{131749}a^{10}-\frac{8953146}{131749}a^{8}+\frac{7225995}{131749}a^{6}+\frac{18931808}{131749}a^{4}+\frac{6161039}{131749}a^{2}-\frac{1161873}{131749}$, $\frac{251539}{131749}a^{21}+\frac{319565}{131749}a^{19}-\frac{503914}{131749}a^{17}+\frac{841841}{131749}a^{15}-\frac{595386}{131749}a^{13}-\frac{4176129}{131749}a^{11}-\frac{2964265}{131749}a^{9}+\frac{3173420}{131749}a^{7}+\frac{7055685}{131749}a^{5}+\frac{1654839}{131749}a^{3}-\frac{561337}{131749}a$, $\frac{235865}{131749}a^{21}+\frac{312737}{131749}a^{19}-\frac{473860}{131749}a^{17}+\frac{740715}{131749}a^{15}-\frac{490697}{131749}a^{13}-\frac{4032200}{131749}a^{11}-\frac{2957353}{131749}a^{9}+\frac{3046777}{131749}a^{7}+\frac{6958113}{131749}a^{5}+\frac{1747401}{131749}a^{3}-\frac{683518}{131749}a$, $\frac{551453}{131749}a^{20}+\frac{754867}{131749}a^{18}-\frac{1023253}{131749}a^{16}+\frac{1757424}{131749}a^{14}-\frac{1106735}{131749}a^{12}-\frac{9187589}{131749}a^{10}-\frac{7400228}{131749}a^{8}+\frac{6295139}{131749}a^{6}+\frac{15885653}{131749}a^{4}+\frac{4893318}{131749}a^{2}-\frac{975048}{131749}$, $\frac{161540}{131749}a^{21}-\frac{130438}{131749}a^{20}+\frac{245829}{131749}a^{19}-\frac{201583}{131749}a^{18}-\frac{267346}{131749}a^{17}+\frac{229581}{131749}a^{16}+\frac{469725}{131749}a^{15}-\frac{357470}{131749}a^{14}-\frac{261743}{131749}a^{13}+\frac{147663}{131749}a^{12}-\frac{2779204}{131749}a^{11}+\frac{2325174}{131749}a^{10}-\frac{2549251}{131749}a^{9}+\frac{2056653}{131749}a^{8}+\frac{1389168}{131749}a^{7}-\frac{1464628}{131749}a^{6}+\frac{4961668}{131749}a^{5}-\frac{3993885}{131749}a^{4}+\frac{2365917}{131749}a^{3}-\frac{1643837}{131749}a^{2}+\frac{60726}{131749}a+\frac{151576}{131749}$, $\frac{529400}{131749}a^{21}+\frac{382892}{131749}a^{20}+\frac{791474}{131749}a^{19}+\frac{513410}{131749}a^{18}-\frac{895298}{131749}a^{17}-\frac{737872}{131749}a^{16}+\frac{1563935}{131749}a^{15}+\frac{1217537}{131749}a^{14}-\frac{847738}{131749}a^{13}-\frac{792239}{131749}a^{12}-\frac{9016714}{131749}a^{11}-\frac{6416647}{131749}a^{10}-\frac{8189806}{131749}a^{9}-\frac{4975343}{131749}a^{8}+\frac{5159938}{131749}a^{7}+\frac{4726496}{131749}a^{6}+\frac{15890106}{131749}a^{5}+\frac{11046894}{131749}a^{4}+\frac{6631588}{131749}a^{3}+\frac{2973827}{131749}a^{2}-\frac{63736}{131749}a-\frac{870017}{131749}$, $\frac{123365}{131749}a^{21}-\frac{254199}{131749}a^{20}+\frac{95668}{131749}a^{19}-\frac{366904}{131749}a^{18}-\frac{346070}{131749}a^{17}+\frac{473442}{131749}a^{16}+\frac{480368}{131749}a^{15}-\frac{725640}{131749}a^{14}-\frac{505360}{131749}a^{13}+\frac{456170}{131749}a^{12}-\frac{1991524}{131749}a^{11}+\frac{4447662}{131749}a^{10}-\frac{567344}{131749}a^{9}+\frac{3774895}{131749}a^{8}+\frac{2590993}{131749}a^{7}-\frac{2897792}{131749}a^{6}+\frac{3159714}{131749}a^{5}-\frac{7905354}{131749}a^{4}-\frac{737486}{131749}a^{3}-\frac{3012837}{131749}a^{2}-\frac{907341}{131749}a+\frac{223241}{131749}$, $\frac{656404}{131749}a^{21}+\frac{566405}{131749}a^{20}+\frac{839521}{131749}a^{19}+\frac{765785}{131749}a^{18}-\frac{1313203}{131749}a^{17}-\frac{1076635}{131749}a^{16}+\frac{2185039}{131749}a^{15}+\frac{1812923}{131749}a^{14}-\frac{1477455}{131749}a^{13}-\frac{1143812}{131749}a^{12}-\frac{10886243}{131749}a^{11}-\frac{9489318}{131749}a^{10}-\frac{7809160}{131749}a^{9}-\frac{7394146}{131749}a^{8}+\frac{8510467}{131749}a^{7}+\frac{6726215}{131749}a^{6}+\frac{18277625}{131749}a^{5}+\frac{16183608}{131749}a^{4}+\frac{3830242}{131749}a^{3}+\frac{4541320}{131749}a^{2}-\frac{1875368}{131749}a-\frac{1253305}{131749}$ 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:  \( 243223.959653 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 243223.959653 \cdot 1}{2\cdot\sqrt{104070916689387515610916716544}}\cr\approx \mathstrut & 0.144600682806 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^22 + 2*x^20 - x^18 + 2*x^16 - 18*x^12 - 24*x^10 + 3*x^8 + 36*x^6 + 27*x^4 + 4*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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^22 + 2*x^20 - x^18 + 2*x^16 - 18*x^12 - 24*x^10 + 3*x^8 + 36*x^6 + 27*x^4 + 4*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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^22 + 2*x^20 - x^18 + 2*x^16 - 18*x^12 - 24*x^10 + 3*x^8 + 36*x^6 + 27*x^4 + 4*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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 2*x^20 - x^18 + 2*x^16 - 18*x^12 - 24*x^10 + 3*x^8 + 36*x^6 + 27*x^4 + 4*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):

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

Intermediate fields

11.5.157519649831.1

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
Degree 44 sibling: data not computed
Minimal sibling: 22.8.6213581786996136997545952595085659731356637711243979839317721535086212076464978309381089131358842015486585577154540390549316173824.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.11.0.1}{11} }^{2}$ ${\href{/padicField/5.11.0.1}{11} }^{2}$ ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{3}$ ${\href{/padicField/11.8.0.1}{8} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ $20{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}$ ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{3}$ ${\href{/padicField/29.11.0.1}{11} }^{2}$ ${\href{/padicField/31.10.0.1}{10} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.9.0.1}{9} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.11.0.1}{11} }^{2}$ ${\href{/padicField/43.9.0.1}{9} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.9.0.1}{9} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ $18{,}\,{\href{/padicField/53.4.0.1}{4} }$ ${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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$
\(19457\) Copy content Toggle raw display $\Q_{19457}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{19457}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $4$$2$$2$$2$
Deg $8$$1$$8$$0$$C_8$$[\ ]^{8}$
Deg $8$$1$$8$$0$$C_8$$[\ ]^{8}$
\(8095783\) 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$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$