Properties

Label 22.0.641...828.1
Degree $22$
Signature $[0, 11]$
Discriminant $-6.420\times 10^{25}$
Root discriminant \(14.90\)
Ramified primes $2,2029,7909899338257898069783$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_{22}$ (as 22T59)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 3*x^20 - 7*x^19 + 11*x^18 - 18*x^17 + 29*x^16 - 40*x^15 + 53*x^14 - 67*x^13 + 79*x^12 - 87*x^11 + 92*x^10 - 90*x^9 + 83*x^8 - 73*x^7 + 59*x^6 - 43*x^5 + 30*x^4 - 18*x^3 + 9*x^2 - 4*x + 1)
 
gp: K = bnfinit(y^22 - y^21 + 3*y^20 - 7*y^19 + 11*y^18 - 18*y^17 + 29*y^16 - 40*y^15 + 53*y^14 - 67*y^13 + 79*y^12 - 87*y^11 + 92*y^10 - 90*y^9 + 83*y^8 - 73*y^7 + 59*y^6 - 43*y^5 + 30*y^4 - 18*y^3 + 9*y^2 - 4*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - x^21 + 3*x^20 - 7*x^19 + 11*x^18 - 18*x^17 + 29*x^16 - 40*x^15 + 53*x^14 - 67*x^13 + 79*x^12 - 87*x^11 + 92*x^10 - 90*x^9 + 83*x^8 - 73*x^7 + 59*x^6 - 43*x^5 + 30*x^4 - 18*x^3 + 9*x^2 - 4*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - x^21 + 3*x^20 - 7*x^19 + 11*x^18 - 18*x^17 + 29*x^16 - 40*x^15 + 53*x^14 - 67*x^13 + 79*x^12 - 87*x^11 + 92*x^10 - 90*x^9 + 83*x^8 - 73*x^7 + 59*x^6 - 43*x^5 + 30*x^4 - 18*x^3 + 9*x^2 - 4*x + 1)
 

\( x^{22} - x^{21} + 3 x^{20} - 7 x^{19} + 11 x^{18} - 18 x^{17} + 29 x^{16} - 40 x^{15} + 53 x^{14} - 67 x^{13} + 79 x^{12} - 87 x^{11} + 92 x^{10} - 90 x^{9} + 83 x^{8} - 73 x^{7} + 59 x^{6} - 43 x^{5} + \cdots + 1 \) 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:   \(-64196743029301100734358828\) \(\medspace = -\,2^{2}\cdot 2029\cdot 7909899338257898069783\) 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:  \(14.90\)
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:  $2^{2/3}2029^{1/2}7909899338257898069783^{1/2}\approx 6359356409154.484$
Ramified primes:   \(2\), \(2029\), \(7909899338257898069783\) 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:  $\Q(\sqrt{-16049\!\cdots\!89707}$)
$\card{ \Aut(K/\Q) }$:  $1$
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:  $10$
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:   $a^{21}-a^{20}+3a^{19}-7a^{18}+11a^{17}-18a^{16}+29a^{15}-40a^{14}+53a^{13}-67a^{12}+79a^{11}-87a^{10}+92a^{9}-90a^{8}+83a^{7}-73a^{6}+59a^{5}-43a^{4}+30a^{3}-18a^{2}+9a-4$, $3a^{21}-a^{20}+9a^{19}-16a^{18}+24a^{17}-43a^{16}+66a^{15}-87a^{14}+119a^{13}-146a^{12}+169a^{11}-184a^{10}+194a^{9}-182a^{8}+168a^{7}-145a^{6}+112a^{5}-80a^{4}+57a^{3}-29a^{2}+15a-6$, $a^{21}-a^{20}+4a^{19}-7a^{18}+13a^{17}-22a^{16}+33a^{15}-47a^{14}+64a^{13}-78a^{12}+93a^{11}-102a^{10}+106a^{9}-102a^{8}+94a^{7}-79a^{6}+62a^{5}-45a^{4}+29a^{3}-15a^{2}+8a-2$, $a^{4}-a+1$, $a^{21}-a^{20}+3a^{19}-7a^{18}+11a^{17}-18a^{16}+29a^{15}-40a^{14}+53a^{13}-67a^{12}+79a^{11}-87a^{10}+92a^{9}-90a^{8}+83a^{7}-73a^{6}+59a^{5}-42a^{4}+30a^{3}-17a^{2}+8a-3$, $2a^{21}+5a^{19}-9a^{18}+11a^{17}-21a^{16}+33a^{15}-40a^{14}+55a^{13}-68a^{12}+77a^{11}-83a^{10}+89a^{9}-83a^{8}+78a^{7}-69a^{6}+54a^{5}-39a^{4}+30a^{3}-15a^{2}+8a-4$, $a^{21}-a^{20}+3a^{19}-7a^{18}+11a^{17}-18a^{16}+29a^{15}-40a^{14}+53a^{13}-67a^{12}+79a^{11}-87a^{10}+92a^{9}-90a^{8}+82a^{7}-73a^{6}+58a^{5}-41a^{4}+29a^{3}-16a^{2}+7a-3$, $3a^{21}+9a^{19}-13a^{18}+20a^{17}-36a^{16}+55a^{15}-70a^{14}+97a^{13}-117a^{12}+136a^{11}-145a^{10}+154a^{9}-142a^{8}+132a^{7}-111a^{6}+86a^{5}-59a^{4}+43a^{3}-19a^{2}+11a-3$, $4a^{21}-a^{20}+11a^{19}-20a^{18}+28a^{17}-50a^{16}+78a^{15}-99a^{14}+134a^{13}-165a^{12}+188a^{11}-201a^{10}+212a^{9}-196a^{8}+178a^{7}-153a^{6}+116a^{5}-80a^{4}+57a^{3}-27a^{2}+13a-6$, $2a^{21}+5a^{19}-9a^{18}+11a^{17}-21a^{16}+33a^{15}-40a^{14}+54a^{13}-68a^{12}+75a^{11}-80a^{10}+85a^{9}-78a^{8}+70a^{7}-61a^{6}+46a^{5}-31a^{4}+23a^{3}-11a^{2}+5a-3$ 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:  \( 9202.86473748 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 9202.86473748 \cdot 1}{2\cdot\sqrt{64196743029301100734358828}}\cr\approx \mathstrut & 0.346031071546 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 3*x^20 - 7*x^19 + 11*x^18 - 18*x^17 + 29*x^16 - 40*x^15 + 53*x^14 - 67*x^13 + 79*x^12 - 87*x^11 + 92*x^10 - 90*x^9 + 83*x^8 - 73*x^7 + 59*x^6 - 43*x^5 + 30*x^4 - 18*x^3 + 9*x^2 - 4*x + 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^21 + 3*x^20 - 7*x^19 + 11*x^18 - 18*x^17 + 29*x^16 - 40*x^15 + 53*x^14 - 67*x^13 + 79*x^12 - 87*x^11 + 92*x^10 - 90*x^9 + 83*x^8 - 73*x^7 + 59*x^6 - 43*x^5 + 30*x^4 - 18*x^3 + 9*x^2 - 4*x + 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^21 + 3*x^20 - 7*x^19 + 11*x^18 - 18*x^17 + 29*x^16 - 40*x^15 + 53*x^14 - 67*x^13 + 79*x^12 - 87*x^11 + 92*x^10 - 90*x^9 + 83*x^8 - 73*x^7 + 59*x^6 - 43*x^5 + 30*x^4 - 18*x^3 + 9*x^2 - 4*x + 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^21 + 3*x^20 - 7*x^19 + 11*x^18 - 18*x^17 + 29*x^16 - 40*x^15 + 53*x^14 - 67*x^13 + 79*x^12 - 87*x^11 + 92*x^10 - 90*x^9 + 83*x^8 - 73*x^7 + 59*x^6 - 43*x^5 + 30*x^4 - 18*x^3 + 9*x^2 - 4*x + 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

$S_{22}$ (as 22T59):

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 1124000727777607680000
The 1002 conjugacy class representatives for $S_{22}$ are not computed
Character table for $S_{22}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
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 44 sibling: 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 $19{,}\,{\href{/padicField/3.3.0.1}{3} }$ ${\href{/padicField/5.11.0.1}{11} }{,}\,{\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ $19{,}\,{\href{/padicField/7.3.0.1}{3} }$ ${\href{/padicField/11.11.0.1}{11} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }$ ${\href{/padicField/13.13.0.1}{13} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ $20{,}\,{\href{/padicField/17.2.0.1}{2} }$ $17{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ $20{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ $16{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ $18{,}\,{\href{/padicField/43.4.0.1}{4} }$ ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.5.0.1}{5} }$ ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$

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 2.3.2.1$x^{3} + 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.19.0.1$x^{19} + x^{5} + x^{2} + x + 1$$1$$19$$0$$C_{19}$$[\ ]^{19}$
\(2029\) Copy content Toggle raw display $\Q_{2029}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{2029}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$
Deg $13$$1$$13$$0$$C_{13}$$[\ ]^{13}$
\(790\!\cdots\!783\) 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$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $9$$1$$9$$0$$C_9$$[\ ]^{9}$