Properties

Label 22.6.156...136.1
Degree $22$
Signature $[6, 8]$
Discriminant $1.563\times 10^{30}$
Root discriminant \(23.57\)
Ramified primes $2,610429790897$
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 + x^20 - 4*x^18 - x^16 + 5*x^14 - 7*x^12 - 7*x^10 + 5*x^8 + 7*x^6 + 3*x^4 - x^2 - 1)
 
gp: K = bnfinit(y^22 + y^20 - 4*y^18 - y^16 + 5*y^14 - 7*y^12 - 7*y^10 + 5*y^8 + 7*y^6 + 3*y^4 - y^2 - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + x^20 - 4*x^18 - x^16 + 5*x^14 - 7*x^12 - 7*x^10 + 5*x^8 + 7*x^6 + 3*x^4 - x^2 - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + x^20 - 4*x^18 - x^16 + 5*x^14 - 7*x^12 - 7*x^10 + 5*x^8 + 7*x^6 + 3*x^4 - x^2 - 1)
 

\( x^{22} + x^{20} - 4x^{18} - x^{16} + 5x^{14} - 7x^{12} - 7x^{10} + 5x^{8} + 7x^{6} + 3x^{4} - x^{2} - 1 \) 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:  $[6, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(1562900555060447098970765787136\) \(\medspace = 2^{22}\cdot 610429790897^{2}\) 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:  \(23.57\)
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:   \(2\), \(610429790897\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{97}a^{20}-\frac{38}{97}a^{18}+\frac{23}{97}a^{16}-\frac{25}{97}a^{14}+\frac{10}{97}a^{12}-\frac{9}{97}a^{10}-\frac{44}{97}a^{8}-\frac{25}{97}a^{6}+\frac{12}{97}a^{4}+\frac{20}{97}a^{2}-\frac{5}{97}$, $\frac{1}{97}a^{21}-\frac{38}{97}a^{19}+\frac{23}{97}a^{17}-\frac{25}{97}a^{15}+\frac{10}{97}a^{13}-\frac{9}{97}a^{11}-\frac{44}{97}a^{9}-\frac{25}{97}a^{7}+\frac{12}{97}a^{5}+\frac{20}{97}a^{3}-\frac{5}{97}a$ 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:  $13$
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:   $a^{21}+a^{19}-4a^{17}-a^{15}+5a^{13}-7a^{11}-7a^{9}+5a^{7}+7a^{5}+3a^{3}-a$, $\frac{225}{97}a^{20}+\frac{83}{97}a^{18}-\frac{936}{97}a^{16}+\frac{389}{97}a^{14}+\frac{795}{97}a^{12}-\frac{2122}{97}a^{10}-\frac{103}{97}a^{8}+\frac{1068}{97}a^{6}+\frac{760}{97}a^{4}+\frac{329}{97}a^{2}-\frac{252}{97}$, $\frac{65}{97}a^{20}+\frac{52}{97}a^{18}-\frac{251}{97}a^{16}+\frac{24}{97}a^{14}+\frac{262}{97}a^{12}-\frac{585}{97}a^{10}-\frac{241}{97}a^{8}+\frac{315}{97}a^{6}+\frac{101}{97}a^{4}+\frac{136}{97}a^{2}+\frac{63}{97}$, $\frac{196}{97}a^{20}+\frac{21}{97}a^{18}-\frac{827}{97}a^{16}+\frac{532}{97}a^{14}+\frac{602}{97}a^{12}-\frac{1958}{97}a^{10}+\frac{300}{97}a^{8}+\frac{920}{97}a^{6}+\frac{509}{97}a^{4}+\frac{137}{97}a^{2}-\frac{301}{97}$, $\frac{358}{97}a^{21}+\frac{170}{97}a^{19}-\frac{1466}{97}a^{17}+\frac{459}{97}a^{15}+\frac{1349}{97}a^{13}-\frac{3222}{97}a^{11}-\frac{620}{97}a^{9}+\frac{1720}{97}a^{7}+\frac{1289}{97}a^{5}+\frac{467}{97}a^{3}-\frac{335}{97}a$, $\frac{89}{97}a^{20}+\frac{13}{97}a^{18}-\frac{378}{97}a^{16}+\frac{200}{97}a^{14}+\frac{308}{97}a^{12}-\frac{801}{97}a^{10}-\frac{36}{97}a^{8}+\frac{491}{97}a^{6}+\frac{486}{97}a^{4}-\frac{63}{97}a^{2}-\frac{154}{97}$, $\frac{37}{97}a^{21}+\frac{49}{97}a^{19}-\frac{119}{97}a^{17}-\frac{52}{97}a^{15}+\frac{79}{97}a^{13}-\frac{236}{97}a^{11}-\frac{173}{97}a^{9}-\frac{149}{97}a^{7}+\frac{250}{97}a^{5}+\frac{352}{97}a^{3}+\frac{106}{97}a$, $\frac{148}{97}a^{21}+\frac{2}{97}a^{19}-\frac{670}{97}a^{17}+\frac{471}{97}a^{15}+\frac{607}{97}a^{13}-\frac{1720}{97}a^{11}+\frac{278}{97}a^{9}+\frac{1247}{97}a^{7}+\frac{30}{97}a^{5}-\frac{144}{97}a^{3}-\frac{158}{97}a$, $\frac{113}{97}a^{21}-\frac{261}{97}a^{20}+\frac{71}{97}a^{19}-\frac{73}{97}a^{18}-\frac{408}{97}a^{17}+\frac{1078}{97}a^{16}+\frac{85}{97}a^{15}-\frac{556}{97}a^{14}+\frac{257}{97}a^{13}-\frac{864}{97}a^{12}-\frac{823}{97}a^{11}+\frac{2543}{97}a^{10}-\frac{219}{97}a^{9}-\frac{59}{97}a^{8}-\frac{12}{97}a^{7}-\frac{1235}{97}a^{6}+\frac{580}{97}a^{5}-\frac{610}{97}a^{4}+\frac{417}{97}a^{3}-\frac{176}{97}a^{2}-\frac{80}{97}a+\frac{238}{97}$, $\frac{154}{97}a^{21}+\frac{65}{97}a^{20}+\frac{65}{97}a^{19}+\frac{52}{97}a^{18}-\frac{629}{97}a^{17}-\frac{251}{97}a^{16}+\frac{224}{97}a^{15}+\frac{24}{97}a^{14}+\frac{570}{97}a^{13}+\frac{262}{97}a^{12}-\frac{1386}{97}a^{11}-\frac{585}{97}a^{10}-\frac{277}{97}a^{9}-\frac{241}{97}a^{8}+\frac{806}{97}a^{7}+\frac{315}{97}a^{6}+\frac{587}{97}a^{5}+\frac{101}{97}a^{4}-\frac{24}{97}a^{3}+\frac{136}{97}a^{2}-\frac{91}{97}a+\frac{63}{97}$, $\frac{27}{97}a^{20}-a^{19}-\frac{56}{97}a^{18}-a^{17}-\frac{155}{97}a^{16}+4a^{15}+\frac{295}{97}a^{14}+a^{13}+\frac{76}{97}a^{12}-5a^{11}-\frac{437}{97}a^{10}+7a^{9}+\frac{461}{97}a^{8}+7a^{7}+\frac{295}{97}a^{6}-5a^{5}-\frac{64}{97}a^{4}-7a^{3}-\frac{333}{97}a^{2}-3a-\frac{329}{97}$, $\frac{201}{97}a^{21}+\frac{105}{97}a^{20}+\frac{25}{97}a^{19}+\frac{84}{97}a^{18}-\frac{906}{97}a^{17}-\frac{398}{97}a^{16}+\frac{504}{97}a^{15}-\frac{6}{97}a^{14}+\frac{846}{97}a^{13}+\frac{371}{97}a^{12}-\frac{2100}{97}a^{11}-\frac{751}{97}a^{10}+\frac{80}{97}a^{9}-\frac{449}{97}a^{8}+\frac{1571}{97}a^{7}+\frac{285}{97}a^{6}+\frac{666}{97}a^{5}+\frac{678}{97}a^{4}-\frac{248}{97}a^{3}+\frac{257}{97}a^{2}-\frac{326}{97}a-\frac{137}{97}$, $\frac{126}{97}a^{21}+\frac{84}{97}a^{20}-\frac{35}{97}a^{19}+\frac{106}{97}a^{18}-\frac{594}{97}a^{17}-\frac{299}{97}a^{16}+\frac{536}{97}a^{15}-\frac{160}{97}a^{14}+\frac{484}{97}a^{13}+\frac{355}{97}a^{12}-\frac{1522}{97}a^{11}-\frac{465}{97}a^{10}+\frac{470}{97}a^{9}-\frac{689}{97}a^{8}+\frac{1021}{97}a^{7}+\frac{131}{97}a^{6}+\frac{251}{97}a^{5}+\frac{523}{97}a^{4}-\frac{390}{97}a^{3}+\frac{419}{97}a^{2}-\frac{436}{97}a+\frac{65}{97}$ 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:  \( 2599985.98276 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{8}\cdot 2599985.98276 \cdot 1}{2\cdot\sqrt{1562900555060447098970765787136}}\cr\approx \mathstrut & 0.161656904736 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^22 + x^20 - 4*x^18 - x^16 + 5*x^14 - 7*x^12 - 7*x^10 + 5*x^8 + 7*x^6 + 3*x^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 + x^20 - 4*x^18 - x^16 + 5*x^14 - 7*x^12 - 7*x^10 + 5*x^8 + 7*x^6 + 3*x^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 + x^20 - 4*x^18 - x^16 + 5*x^14 - 7*x^12 - 7*x^10 + 5*x^8 + 7*x^6 + 3*x^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 + x^20 - 4*x^18 - x^16 + 5*x^14 - 7*x^12 - 7*x^10 + 5*x^8 + 7*x^6 + 3*x^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 $C_2^{10}.S_{11}$ are not computed
Character table for $C_2^{10}.S_{11}$ is not computed

Intermediate fields

11.7.610429790897.1

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: 22.6.18393010934891211589809067376784369817321558276621681813550013756757903463071147752279226582555060637148121470574946816608031758679539712.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.8.0.1}{8} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ $18{,}\,{\href{/padicField/5.4.0.1}{4} }$ $16{,}\,{\href{/padicField/7.6.0.1}{6} }$ ${\href{/padicField/11.11.0.1}{11} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.5.0.1}{5} }^{2}$ ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.8.0.1}{8} }$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}$ ${\href{/padicField/23.11.0.1}{11} }^{2}$ ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.10.0.1}{10} }$ $16{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ ${\href{/padicField/37.11.0.1}{11} }^{2}$ ${\href{/padicField/41.7.0.1}{7} }^{2}{,}\,{\href{/padicField/41.4.0.1}{4} }^{2}$ $20{,}\,{\href{/padicField/43.2.0.1}{2} }$ $20{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.11.0.1}{11} }^{2}$ ${\href{/padicField/59.7.0.1}{7} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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$
\(610429790897\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $4$$2$$2$$2$
Deg $16$$1$$16$$0$$C_{16}$$[\ ]^{16}$