Properties

Label 22.18.390...976.1
Degree $22$
Signature $[18, 2]$
Discriminant $3.903\times 10^{34}$
Root discriminant \(37.35\)
Ramified primes $2,137,293,11093,216649$
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 - 15*x^20 + 91*x^18 - 279*x^16 + 417*x^14 - 130*x^12 - 453*x^10 + 544*x^8 - 66*x^6 - 185*x^4 + 75*x^2 - 1)
 
gp: K = bnfinit(y^22 - 15*y^20 + 91*y^18 - 279*y^16 + 417*y^14 - 130*y^12 - 453*y^10 + 544*y^8 - 66*y^6 - 185*y^4 + 75*y^2 - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 15*x^20 + 91*x^18 - 279*x^16 + 417*x^14 - 130*x^12 - 453*x^10 + 544*x^8 - 66*x^6 - 185*x^4 + 75*x^2 - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 15*x^20 + 91*x^18 - 279*x^16 + 417*x^14 - 130*x^12 - 453*x^10 + 544*x^8 - 66*x^6 - 185*x^4 + 75*x^2 - 1)
 

\( x^{22} - 15 x^{20} + 91 x^{18} - 279 x^{16} + 417 x^{14} - 130 x^{12} - 453 x^{10} + 544 x^{8} - 66 x^{6} - 185 x^{4} + 75 x^{2} - 1 \) 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:  $[18, 2]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(39034415726392643532837005585022976\) \(\medspace = 2^{22}\cdot 137^{2}\cdot 293^{2}\cdot 11093^{2}\cdot 216649^{2}\) 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:  \(37.35\)
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:   \(2\), \(137\), \(293\), \(11093\), \(216649\) 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:  $19$
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:   $3a^{21}-40a^{19}+206a^{17}-489a^{15}+411a^{13}+356a^{11}-817a^{9}+222a^{7}+278a^{5}-118a^{3}-6a$, $a^{21}-14a^{19}+77a^{17}-202a^{15}+215a^{13}+85a^{11}-369a^{9}+182a^{7}+102a^{5}-80a^{3}+7a$, $a^{2}-1$, $3a^{21}-41a^{19}+218a^{17}-542a^{15}+507a^{13}+332a^{11}-941a^{9}+315a^{7}+313a^{5}-143a^{3}-10a$, $4a^{20}-55a^{18}+295a^{16}-744a^{14}+722a^{12}+418a^{10}-1317a^{8}+511a^{6}+412a^{4}-235a^{2}+3$, $a^{21}-15a^{19}+89a^{17}-255a^{15}+311a^{13}+61a^{11}-493a^{9}+275a^{7}+137a^{5}-106a^{3}+6a$, $7a^{20}-95a^{18}+501a^{16}-1234a^{14}+1142a^{12}+748a^{10}-2117a^{8}+767a^{6}+654a^{4}-364a^{2}+7$, $6a^{21}-81a^{19}+425a^{17}-1043a^{15}+970a^{13}+601a^{11}-1761a^{9}+686a^{7}+512a^{5}-320a^{3}+13a$, $a^{21}-14a^{19}+77a^{17}-202a^{15}+215a^{13}+85a^{11}-369a^{9}+182a^{7}+102a^{5}-80a^{3}+8a$, $8a^{20}-109a^{18}+578a^{16}-1436a^{14}+1357a^{12}+832a^{10}-2479a^{8}+935a^{6}+759a^{4}-431a^{2}+7$, $3a^{21}-40a^{19}+206a^{17}-489a^{15}+411a^{13}+356a^{11}-817a^{9}+222a^{7}+278a^{5}-117a^{3}-8a+1$, $a-1$, $5a^{21}-69a^{19}+372a^{17}-946a^{15}+937a^{13}+502a^{11}-1678a^{9}+673a^{7}+525a^{5}-298a^{3}-3a-1$, $a^{21}-14a^{19}+77a^{17}-202a^{15}+215a^{13}+85a^{11}-368a^{9}+176a^{7}+110a^{5}-75a^{3}-a+1$, $7a^{21}-96a^{19}+514a^{17}-1298a^{15}+1281a^{13}+662a^{11}-2254a^{9}+961a^{7}+649a^{5}-426a^{3}+18a-1$, $7a^{21}+a^{20}-95a^{19}-13a^{18}+501a^{17}+65a^{16}-1234a^{15}-149a^{14}+1142a^{13}+119a^{12}+748a^{11}+108a^{10}-2116a^{9}-237a^{8}+761a^{7}+70a^{6}+662a^{5}+74a^{4}-359a^{3}-37a^{2}-a+1$, $7a^{21}+a^{20}-96a^{19}-14a^{18}+514a^{17}+77a^{16}-1298a^{15}-202a^{14}+1281a^{13}+215a^{12}+662a^{11}+85a^{10}-2254a^{9}-368a^{8}+961a^{7}+176a^{6}+649a^{5}+110a^{4}-426a^{3}-75a^{2}+18a-1$, $5a^{21}-9a^{20}-69a^{19}+123a^{18}+373a^{17}-655a^{16}-958a^{15}+1638a^{14}+990a^{13}-1573a^{12}+407a^{11}-910a^{10}-1662a^{9}+2833a^{8}+817a^{7}-1109a^{6}+420a^{5}-853a^{4}-347a^{3}+500a^{2}+38a-11$, $5a^{21}+10a^{20}-69a^{19}-135a^{18}+373a^{17}+708a^{16}-959a^{15}-1734a^{14}+1000a^{13}+1596a^{12}+373a^{11}+1041a^{10}-1625a^{9}-2940a^{8}+840a^{7}+1077a^{6}+367a^{5}+891a^{4}-346a^{3}-505a^{2}+51a+12$ 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:  \( 7576192193.94 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{2}\cdot 7576192193.94 \cdot 1}{2\cdot\sqrt{39034415726392643532837005585022976}}\cr\approx \mathstrut & 0.198425189988 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^22 - 15*x^20 + 91*x^18 - 279*x^16 + 417*x^14 - 130*x^12 - 453*x^10 + 544*x^8 - 66*x^6 - 185*x^4 + 75*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 - 15*x^20 + 91*x^18 - 279*x^16 + 417*x^14 - 130*x^12 - 453*x^10 + 544*x^8 - 66*x^6 - 185*x^4 + 75*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 - 15*x^20 + 91*x^18 - 279*x^16 + 417*x^14 - 130*x^12 - 453*x^10 + 544*x^8 - 66*x^6 - 185*x^4 + 75*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 - 15*x^20 + 91*x^18 - 279*x^16 + 417*x^14 - 130*x^12 - 453*x^10 + 544*x^8 - 66*x^6 - 185*x^4 + 75*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

11.11.96470357797337.1

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: 22.18.28248324053221546961515534563512015217902284051423179842565864776423831827504888313734987193448503459155287522171306501220244138633403098320586348761480379236352.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.9.0.1}{9} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ $16{,}\,{\href{/padicField/5.6.0.1}{6} }$ ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.8.0.1}{8} }$ ${\href{/padicField/13.11.0.1}{11} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.8.0.1}{8} }$ ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.14.0.1}{14} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.4.0.1}{4} }$ ${\href{/padicField/47.7.0.1}{7} }^{2}{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}$ ${\href{/padicField/53.9.0.1}{9} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}$

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$
\(137\) Copy content Toggle raw display $\Q_{137}$$x + 134$$1$$1$$0$Trivial$[\ ]$
$\Q_{137}$$x + 134$$1$$1$$0$Trivial$[\ ]$
137.2.1.2$x^{2} + 411$$2$$1$$1$$C_2$$[\ ]_{2}$
137.2.1.2$x^{2} + 411$$2$$1$$1$$C_2$$[\ ]_{2}$
137.4.0.1$x^{4} + x^{2} + 95 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
137.12.0.1$x^{12} + x^{8} + 61 x^{7} + 40 x^{6} + 40 x^{5} + 12 x^{4} + 36 x^{3} + 135 x^{2} + 61 x + 3$$1$$12$$0$$C_{12}$$[\ ]^{12}$
\(293\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
\(11093\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$2$$2$$2$
\(216649\) Copy content Toggle raw display $\Q_{216649}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{216649}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $8$$1$$8$$0$$C_8$$[\ ]^{8}$
Deg $8$$1$$8$$0$$C_8$$[\ ]^{8}$