Properties

Label 22.14.134...816.1
Degree $22$
Signature $[14, 4]$
Discriminant $1.347\times 10^{31}$
Root discriminant \(26.00\)
Ramified primes $2,1792166448977$
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 - 12*x^20 + 64*x^18 - 200*x^16 + 403*x^14 - 544*x^12 + 510*x^10 - 358*x^8 + 197*x^6 - 73*x^4 + 14*x^2 - 1)
 
gp: K = bnfinit(y^22 - 12*y^20 + 64*y^18 - 200*y^16 + 403*y^14 - 544*y^12 + 510*y^10 - 358*y^8 + 197*y^6 - 73*y^4 + 14*y^2 - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 12*x^20 + 64*x^18 - 200*x^16 + 403*x^14 - 544*x^12 + 510*x^10 - 358*x^8 + 197*x^6 - 73*x^4 + 14*x^2 - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 12*x^20 + 64*x^18 - 200*x^16 + 403*x^14 - 544*x^12 + 510*x^10 - 358*x^8 + 197*x^6 - 73*x^4 + 14*x^2 - 1)
 

\( x^{22} - 12 x^{20} + 64 x^{18} - 200 x^{16} + 403 x^{14} - 544 x^{12} + 510 x^{10} - 358 x^{8} + 197 x^{6} - 73 x^{4} + 14 x^{2} - 1 \) 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:  $[14, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(13471519681654627790892423970816\) \(\medspace = 2^{22}\cdot 1792166448977^{2}\) 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:  \(26.00\)
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:   \(2\), \(1792166448977\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$ Copy content Toggle raw display

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

Monogenic:  Yes
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:  $17$
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:   $a$, $a^{20}-11a^{18}+53a^{16}-147a^{14}+256a^{12}-288a^{10}+222a^{8}-136a^{6}+61a^{4}-12a^{2}+2$, $4a^{21}-49a^{19}+264a^{17}-825a^{15}+1644a^{13}-2160a^{11}+1930a^{9}-1281a^{7}+669a^{5}-212a^{3}+23a$, $a^{20}-11a^{18}+54a^{16}-156a^{14}+291a^{12}-365a^{10}+324a^{8}-220a^{6}+115a^{4}-40a^{2}+6$, $4a^{20}-46a^{18}+234a^{16}-693a^{14}+1309a^{12}-1630a^{10}+1393a^{8}-902a^{6}+455a^{4}-133a^{2}+14$, $a^{3}-2a$, $6a^{20}-71a^{18}+371a^{16}-1128a^{14}+2191a^{12}-2813a^{10}+2476a^{8}-1637a^{6}+845a^{4}-264a^{2}+31$, $3a^{20}-38a^{18}+210a^{16}-669a^{14}+1353a^{12}-1795a^{10}+1606a^{8}-1061a^{6}+555a^{4}-174a^{2}+18$, $a^{21}+a^{20}-11a^{19}-11a^{18}+53a^{17}+53a^{16}-147a^{15}-147a^{14}+256a^{13}+256a^{12}-288a^{11}-288a^{10}+222a^{9}+222a^{8}-136a^{7}-136a^{6}+61a^{5}+61a^{4}-12a^{3}-12a^{2}+2a+2$, $8a^{21}-4a^{20}-94a^{19}+48a^{18}+489a^{17}-253a^{16}-1482a^{15}+772a^{14}+2869a^{13}-1497a^{12}-3666a^{11}+1904a^{10}+3199a^{9}-1642a^{8}-2086a^{7}+1059a^{6}+1065a^{5}-534a^{4}-324a^{3}+154a^{2}+33a-14$, $a^{2}-a-1$, $a^{21}+5a^{20}-12a^{19}-60a^{18}+64a^{17}+317a^{16}-200a^{15}-972a^{14}+403a^{13}+1900a^{12}-544a^{11}-2448a^{10}+510a^{9}+2152a^{8}-358a^{7}-1417a^{6}+197a^{5}+730a^{4}-73a^{3}-223a^{2}+13a+24$, $6a^{21}+a^{20}-72a^{19}-11a^{18}+381a^{17}+54a^{16}-1172a^{15}-156a^{14}+2303a^{13}+291a^{12}-2992a^{11}-365a^{10}+2662a^{9}+324a^{8}-1775a^{7}-220a^{6}+927a^{5}+115a^{4}-296a^{3}-40a^{2}+37a+6$, $5a^{21}-4a^{20}-55a^{19}+46a^{18}+269a^{17}-234a^{16}-769a^{15}+693a^{14}+1405a^{13}-1309a^{12}-1699a^{11}+1630a^{10}+1428a^{9}-1393a^{8}-921a^{7}+902a^{6}+456a^{5}-455a^{4}-129a^{3}+132a^{2}+15a-13$, $a^{20}-11a^{18}+54a^{16}-156a^{14}+291a^{12}-365a^{10}+324a^{8}-220a^{6}+115a^{4}-40a^{2}-a+6$, $2a^{21}-25a^{19}+137a^{17}-435a^{15}+882a^{13}-1183a^{11}+1083a^{9}-735a^{7}+390a^{5}-131a^{3}+17a+1$, $8a^{21}-a^{20}-93a^{19}+11a^{18}+479a^{17}-53a^{16}-1438a^{15}+147a^{14}+2758a^{13}-256a^{12}-3494a^{11}+288a^{10}+3034a^{9}-222a^{8}-1982a^{7}+136a^{6}+1011a^{5}-61a^{4}-304a^{3}+12a^{2}+34a-2$ 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:  \( 45766346.4516 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{4}\cdot 45766346.4516 \cdot 1}{2\cdot\sqrt{13471519681654627790892423970816}}\cr\approx \mathstrut & 0.159201577209 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^22 - 12*x^20 + 64*x^18 - 200*x^16 + 403*x^14 - 544*x^12 + 510*x^10 - 358*x^8 + 197*x^6 - 73*x^4 + 14*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 - 12*x^20 + 64*x^18 - 200*x^16 + 403*x^14 - 544*x^12 + 510*x^10 - 358*x^8 + 197*x^6 - 73*x^4 + 14*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 - 12*x^20 + 64*x^18 - 200*x^16 + 403*x^14 - 544*x^12 + 510*x^10 - 358*x^8 + 197*x^6 - 73*x^4 + 14*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 - 12*x^20 + 64*x^18 - 200*x^16 + 403*x^14 - 544*x^12 + 510*x^10 - 358*x^8 + 197*x^6 - 73*x^4 + 14*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

11.7.1792166448977.1

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: 22.14.2569340566868698646254256448691034157941557089947947038410673463363632924132693018179259270357029410958376322699590917878586229919830027272192.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.5.0.1}{5} }^{2}$ ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.10.0.1}{10} }$ ${\href{/padicField/7.7.0.1}{7} }^{2}{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.11.0.1}{11} }^{2}$ ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.10.0.1}{10} }$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.10.0.1}{10} }$ ${\href{/padicField/19.8.0.1}{8} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ $20{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.10.0.1}{10} }$ ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.11.0.1}{11} }^{2}$ ${\href{/padicField/47.10.0.1}{10} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.8.0.1}{8} }$

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$
\(1792166448977\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $16$$1$$16$$0$$C_{16}$$[\ ]^{16}$