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Properties

Label 22.0.707...496.2
Degree $22$
Signature $[0, 11]$
Discriminant $-7.077\times 10^{28}$
Root discriminant \(20.48\)
Ramified primes $2,167$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^{10}.D_{22}$ (as 22T32)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 3*x^20 + 4*x^18 + 4*x^16 + 11*x^14 + 10*x^12 + 4*x^10 + 12*x^8 + 7*x^6 + 6*x^4 + 4*x^2 + 1)
 
gp: K = bnfinit(y^22 + 3*y^20 + 4*y^18 + 4*y^16 + 11*y^14 + 10*y^12 + 4*y^10 + 12*y^8 + 7*y^6 + 6*y^4 + 4*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + 3*x^20 + 4*x^18 + 4*x^16 + 11*x^14 + 10*x^12 + 4*x^10 + 12*x^8 + 7*x^6 + 6*x^4 + 4*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 3*x^20 + 4*x^18 + 4*x^16 + 11*x^14 + 10*x^12 + 4*x^10 + 12*x^8 + 7*x^6 + 6*x^4 + 4*x^2 + 1)
 

\( x^{22} + 3x^{20} + 4x^{18} + 4x^{16} + 11x^{14} + 10x^{12} + 4x^{10} + 12x^{8} + 7x^{6} + 6x^{4} + 4x^{2} + 1 \) 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:  $[0, 11]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-70765994783241803539463274496\) \(\medspace = -\,2^{22}\cdot 167^{10}\) 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:  \(20.48\)
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:   \(2\), \(167\) 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:  \(\Q(\sqrt{-1}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5}a^{14}-\frac{1}{5}a^{12}-\frac{1}{5}a^{10}-\frac{1}{5}a^{8}-\frac{2}{5}a^{6}-\frac{1}{5}a^{4}-\frac{2}{5}a^{2}-\frac{1}{5}$, $\frac{1}{5}a^{15}-\frac{1}{5}a^{13}-\frac{1}{5}a^{11}-\frac{1}{5}a^{9}-\frac{2}{5}a^{7}-\frac{1}{5}a^{5}-\frac{2}{5}a^{3}-\frac{1}{5}a$, $\frac{1}{5}a^{16}-\frac{2}{5}a^{12}-\frac{2}{5}a^{10}+\frac{2}{5}a^{8}+\frac{2}{5}a^{6}+\frac{2}{5}a^{4}+\frac{2}{5}a^{2}-\frac{1}{5}$, $\frac{1}{5}a^{17}-\frac{2}{5}a^{13}-\frac{2}{5}a^{11}+\frac{2}{5}a^{9}+\frac{2}{5}a^{7}+\frac{2}{5}a^{5}+\frac{2}{5}a^{3}-\frac{1}{5}a$, $\frac{1}{5}a^{18}+\frac{1}{5}a^{12}-\frac{2}{5}a^{6}-\frac{2}{5}$, $\frac{1}{5}a^{19}+\frac{1}{5}a^{13}-\frac{2}{5}a^{7}-\frac{2}{5}a$, $\frac{1}{65}a^{20}-\frac{3}{65}a^{18}-\frac{4}{65}a^{16}+\frac{2}{65}a^{14}-\frac{1}{65}a^{12}-\frac{23}{65}a^{10}-\frac{1}{65}a^{8}+\frac{31}{65}a^{6}+\frac{16}{65}a^{4}-\frac{12}{65}a^{2}+\frac{24}{65}$, $\frac{1}{65}a^{21}-\frac{3}{65}a^{19}-\frac{4}{65}a^{17}+\frac{2}{65}a^{15}-\frac{1}{65}a^{13}-\frac{23}{65}a^{11}-\frac{1}{65}a^{9}+\frac{31}{65}a^{7}+\frac{16}{65}a^{5}-\frac{12}{65}a^{3}+\frac{24}{65}a$ 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:  $10$
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:   $a^{21}+3a^{19}+4a^{17}+4a^{15}+11a^{13}+10a^{11}+4a^{9}+12a^{7}+7a^{5}+6a^{3}+4a$, $\frac{58}{65}a^{20}+\frac{164}{65}a^{18}+\frac{197}{65}a^{16}+\frac{181}{65}a^{14}+\frac{592}{65}a^{12}+\frac{473}{65}a^{10}+\frac{17}{13}a^{8}+\frac{123}{13}a^{6}+\frac{291}{65}a^{4}+\frac{227}{65}a^{2}+\frac{92}{65}$, $\frac{1}{13}a^{20}-\frac{2}{65}a^{18}-\frac{4}{13}a^{16}-\frac{16}{65}a^{14}+\frac{34}{65}a^{12}-\frac{89}{65}a^{10}-\frac{44}{65}a^{8}+\frac{51}{65}a^{6}-\frac{154}{65}a^{4}-\frac{8}{65}a^{2}-\frac{15}{13}$, $\frac{27}{65}a^{21}+\frac{49}{65}a^{19}+\frac{7}{13}a^{17}+\frac{54}{65}a^{15}+\frac{272}{65}a^{13}+\frac{3}{65}a^{11}-\frac{1}{65}a^{9}+\frac{408}{65}a^{7}-\frac{127}{65}a^{5}+\frac{92}{65}a^{3}+\frac{10}{13}a$, $\frac{28}{65}a^{21}+\frac{72}{65}a^{19}+\frac{83}{65}a^{17}+\frac{69}{65}a^{15}+\frac{49}{13}a^{13}+\frac{123}{65}a^{11}+\frac{24}{65}a^{9}+\frac{41}{13}a^{7}-\frac{4}{13}a^{5}+\frac{158}{65}a^{3}+\frac{22}{65}a$, $\frac{14}{13}a^{20}+\frac{193}{65}a^{18}+\frac{48}{13}a^{16}+\frac{244}{65}a^{14}+\frac{749}{65}a^{12}+\frac{561}{65}a^{10}+\frac{216}{65}a^{8}+\frac{831}{65}a^{6}+\frac{301}{65}a^{4}+\frac{382}{65}a^{2}+a+\frac{37}{13}$, $\frac{27}{65}a^{21}+\frac{4}{65}a^{20}+\frac{15}{13}a^{19}+\frac{27}{65}a^{18}+\frac{74}{65}a^{17}+\frac{62}{65}a^{16}+\frac{41}{65}a^{15}+\frac{73}{65}a^{14}+\frac{233}{65}a^{13}+\frac{74}{65}a^{12}+\frac{198}{65}a^{11}+\frac{142}{65}a^{10}-\frac{8}{13}a^{9}+\frac{87}{65}a^{8}+\frac{53}{13}a^{7}+\frac{7}{65}a^{6}+\frac{224}{65}a^{5}+\frac{5}{13}a^{4}+\frac{66}{65}a^{3}+\frac{43}{65}a^{2}+\frac{102}{65}a+\frac{1}{13}$, $\frac{69}{65}a^{21}+\frac{21}{65}a^{20}+\frac{144}{65}a^{19}+\frac{16}{13}a^{18}+\frac{101}{65}a^{17}+\frac{124}{65}a^{16}+\frac{86}{65}a^{15}+\frac{107}{65}a^{14}+\frac{124}{13}a^{13}+\frac{226}{65}a^{12}+\frac{116}{65}a^{11}+\frac{336}{65}a^{10}-\frac{173}{65}a^{9}+\frac{27}{13}a^{8}+\frac{173}{13}a^{7}+\frac{131}{65}a^{6}-\frac{21}{13}a^{5}+\frac{297}{65}a^{4}+\frac{19}{13}a^{3}+\frac{229}{65}a^{2}+\frac{44}{65}a+\frac{2}{13}$, $\frac{11}{13}a^{21}+\frac{53}{65}a^{20}+\frac{134}{65}a^{19}+\frac{153}{65}a^{18}+\frac{131}{65}a^{17}+\frac{204}{65}a^{16}+\frac{97}{65}a^{15}+\frac{223}{65}a^{14}+\frac{93}{13}a^{13}+\frac{122}{13}a^{12}+\frac{191}{65}a^{11}+\frac{497}{65}a^{10}-\frac{11}{13}a^{9}+\frac{207}{65}a^{8}+\frac{107}{13}a^{7}+\frac{642}{65}a^{6}+\frac{7}{13}a^{5}+\frac{263}{65}a^{4}+\frac{263}{65}a^{3}+\frac{222}{65}a^{2}+\frac{124}{65}a+\frac{23}{13}$, $\frac{341}{65}a^{21}+\frac{87}{65}a^{20}+\frac{888}{65}a^{19}+\frac{194}{65}a^{18}+\frac{1002}{65}a^{17}+\frac{37}{13}a^{16}+\frac{191}{13}a^{15}+\frac{174}{65}a^{14}+\frac{678}{13}a^{13}+\frac{797}{65}a^{12}+\frac{2102}{65}a^{11}+\frac{248}{65}a^{10}+\frac{478}{65}a^{9}+\frac{69}{65}a^{8}+\frac{796}{13}a^{7}+\frac{968}{65}a^{6}+\frac{189}{13}a^{5}-\frac{142}{65}a^{4}+\frac{1589}{65}a^{3}+\frac{477}{65}a^{2}+\frac{813}{65}a-\frac{1}{13}$ 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:  \( 290901.068764 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 290901.068764 \cdot 1}{2\cdot\sqrt{70765994783241803539463274496}}\cr\approx \mathstrut & 0.329443971710 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^22 + 3*x^20 + 4*x^18 + 4*x^16 + 11*x^14 + 10*x^12 + 4*x^10 + 12*x^8 + 7*x^6 + 6*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 + 3*x^20 + 4*x^18 + 4*x^16 + 11*x^14 + 10*x^12 + 4*x^10 + 12*x^8 + 7*x^6 + 6*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 + 3*x^20 + 4*x^18 + 4*x^16 + 11*x^14 + 10*x^12 + 4*x^10 + 12*x^8 + 7*x^6 + 6*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 + 3*x^20 + 4*x^18 + 4*x^16 + 11*x^14 + 10*x^12 + 4*x^10 + 12*x^8 + 7*x^6 + 6*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

$C_2^{10}.D_{22}$ (as 22T32):

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 solvable group of order 45056
The 200 conjugacy class representatives for $C_2^{10}.D_{22}$
Character table for $C_2^{10}.D_{22}$

Intermediate fields

11.1.129891985607.1

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 siblings: data not computed
Degree 44 siblings: data not computed
Minimal sibling: 22.0.70765994783241803539463274496.4

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $22$ ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ $22$ $22$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }^{5}$ $22$ ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{8}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.11.0.1}{11} }^{2}$ $22$ ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }^{5}$ ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{8}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ $22$ ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }^{5}$ ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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$
\(167\) Copy content Toggle raw display 167.2.0.1$x^{2} + 166 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.1.2$x^{2} + 167$$2$$1$$1$$C_2$$[\ ]_{2}$
167.2.1.2$x^{2} + 167$$2$$1$$1$$C_2$$[\ ]_{2}$
167.2.1.2$x^{2} + 167$$2$$1$$1$$C_2$$[\ ]_{2}$
167.2.1.2$x^{2} + 167$$2$$1$$1$$C_2$$[\ ]_{2}$
167.2.1.2$x^{2} + 167$$2$$1$$1$$C_2$$[\ ]_{2}$
167.2.1.2$x^{2} + 167$$2$$1$$1$$C_2$$[\ ]_{2}$
167.4.2.1$x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
167.4.2.1$x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$