Properties

Label 22.12.127...624.2
Degree $22$
Signature $[12, 5]$
Discriminant $-1.270\times 10^{34}$
Root discriminant \(35.50\)
Ramified primes $2,55029067682009$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^{10}.(C_2\times S_{11})$ (as 22T53)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 4*x^20 - 4*x^18 + 31*x^16 - 8*x^14 - 81*x^12 + 48*x^10 + 88*x^8 - 60*x^6 - 35*x^4 + 22*x^2 + 1)
 
gp: K = bnfinit(y^22 - 4*y^20 - 4*y^18 + 31*y^16 - 8*y^14 - 81*y^12 + 48*y^10 + 88*y^8 - 60*y^6 - 35*y^4 + 22*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 4*x^20 - 4*x^18 + 31*x^16 - 8*x^14 - 81*x^12 + 48*x^10 + 88*x^8 - 60*x^6 - 35*x^4 + 22*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 4*x^20 - 4*x^18 + 31*x^16 - 8*x^14 - 81*x^12 + 48*x^10 + 88*x^8 - 60*x^6 - 35*x^4 + 22*x^2 + 1)
 

\( x^{22} - 4 x^{20} - 4 x^{18} + 31 x^{16} - 8 x^{14} - 81 x^{12} + 48 x^{10} + 88 x^{8} - 60 x^{6} - 35 x^{4} + 22 x^{2} + 1 \) 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:  $[12, 5]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-12701184200335173359101913935642624\) \(\medspace = -\,2^{22}\cdot 55029067682009^{2}\) 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:  \(35.50\)
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:   \(2\), \(55029067682009\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$ 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:  $16$
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:   $a$, $a^{16}-2a^{14}-7a^{12}+13a^{10}+15a^{8}-24a^{6}-11a^{4}+13a^{2}+1$, $a^{17}-3a^{15}-5a^{13}+20a^{11}+2a^{9}-39a^{7}+13a^{5}+23a^{3}-11a$, $a^{18}-3a^{16}-5a^{14}+19a^{12}+4a^{10}-34a^{8}+4a^{6}+18a^{4}-4a^{2}$, $a^{21}-3a^{19}-6a^{17}+22a^{15}+9a^{13}-53a^{11}-a^{9}+53a^{7}-4a^{5}-20a^{3}+a$, $a^{2}-1$, $a^{4}-2$, $a^{21}-2a^{19}-9a^{17}+16a^{15}+30a^{13}-42a^{11}-47a^{9}+41a^{7}+33a^{5}-11a^{3}-6a$, $a^{21}-3a^{19}-6a^{17}+21a^{15}+11a^{13}-46a^{11}-13a^{9}+38a^{7}+14a^{5}-10a^{3}-4a$, $2a^{20}-5a^{18}-16a^{16}+40a^{14}+45a^{12}-107a^{10}-56a^{8}+113a^{6}+27a^{4}-39a^{2}-1$, $a+1$, $2a^{21}-a^{20}-5a^{19}+3a^{18}-16a^{17}+6a^{16}+41a^{15}-22a^{14}+44a^{13}-9a^{12}-114a^{11}+52a^{10}-51a^{9}+2a^{8}+127a^{7}-47a^{6}+21a^{5}-46a^{3}+12a^{2}+a+1$, $a^{20}+a^{19}-3a^{18}-3a^{17}-7a^{16}-6a^{15}+24a^{14}+21a^{13}+16a^{12}+10a^{11}-65a^{10}-45a^{9}-17a^{8}-7a^{7}+71a^{6}+33a^{5}+11a^{4}+5a^{3}-25a^{2}-4a-3$, $a^{21}+4a^{20}-2a^{19}-9a^{18}-8a^{17}-33a^{16}+14a^{15}+71a^{14}+23a^{13}+96a^{12}-30a^{11}-186a^{10}-31a^{9}-122a^{8}+22a^{7}+192a^{6}+18a^{5}+60a^{4}-2a^{3}-65a^{2}-a-4$, $3a^{20}-8a^{18}+a^{17}-21a^{16}-2a^{15}+60a^{14}-8a^{13}+48a^{12}+13a^{11}-147a^{10}+22a^{9}-49a^{8}-23a^{7}+141a^{6}-24a^{5}+25a^{4}+10a^{3}-42a^{2}+7a-3$, $2a^{21}-a^{20}-5a^{19}+2a^{18}-15a^{17}+9a^{16}+38a^{15}-17a^{14}+38a^{13}-29a^{12}-94a^{11}+49a^{10}-41a^{9}+41a^{8}+89a^{7}-55a^{6}+15a^{5}-23a^{4}-26a^{3}+19a^{2}+2a+2$ 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:  \( 879851069.403 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{5}\cdot 879851069.403 \cdot 1}{2\cdot\sqrt{12701184200335173359101913935642624}}\cr\approx \mathstrut & 0.156572812593 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^22 - 4*x^20 - 4*x^18 + 31*x^16 - 8*x^14 - 81*x^12 + 48*x^10 + 88*x^8 - 60*x^6 - 35*x^4 + 22*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 - 4*x^20 - 4*x^18 + 31*x^16 - 8*x^14 - 81*x^12 + 48*x^10 + 88*x^8 - 60*x^6 - 35*x^4 + 22*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 - 4*x^20 - 4*x^18 + 31*x^16 - 8*x^14 - 81*x^12 + 48*x^10 + 88*x^8 - 60*x^6 - 35*x^4 + 22*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 - 4*x^20 - 4*x^18 + 31*x^16 - 8*x^14 - 81*x^12 + 48*x^10 + 88*x^8 - 60*x^6 - 35*x^4 + 22*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

$C_2^{10}.(C_2\times S_{11})$ (as 22T53):

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 81749606400
The 752 conjugacy class representatives for $C_2^{10}.(C_2\times S_{11})$ are not computed
Character table for $C_2^{10}.(C_2\times S_{11})$ is not computed

Intermediate fields

11.11.55029067682009.1

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 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 ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.7.0.1}{7} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ $22$ $22$ ${\href{/padicField/13.10.0.1}{10} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ $20{,}\,{\href{/padicField/17.2.0.1}{2} }$ $16{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ $22$ ${\href{/padicField/29.8.0.1}{8} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ ${\href{/padicField/41.6.0.1}{6} }^{3}{,}\,{\href{/padicField/41.4.0.1}{4} }$ ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ $22$ $20{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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$
\(55029067682009\) Copy content Toggle raw display $\Q_{55029067682009}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{55029067682009}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$2$$2$$2$
Deg $12$$1$$12$$0$$C_{12}$$[\ ]^{12}$