Properties

Label 22.16.289...759.1
Degree $22$
Signature $[16, 3]$
Discriminant $-2.897\times 10^{32}$
Root discriminant \(29.89\)
Ramified primes $809,4759,8674315276967$
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 - 11*x^21 + 43*x^20 - 45*x^19 - 152*x^18 + 456*x^17 - 114*x^16 - 1026*x^15 + 1114*x^14 + 790*x^13 - 1814*x^12 + 120*x^11 + 1304*x^10 - 464*x^9 - 493*x^8 + 217*x^7 + 170*x^6 - 46*x^5 - 77*x^4 + 9*x^3 + 18*x^2 - 1)
 
gp: K = bnfinit(y^22 - 11*y^21 + 43*y^20 - 45*y^19 - 152*y^18 + 456*y^17 - 114*y^16 - 1026*y^15 + 1114*y^14 + 790*y^13 - 1814*y^12 + 120*y^11 + 1304*y^10 - 464*y^9 - 493*y^8 + 217*y^7 + 170*y^6 - 46*y^5 - 77*y^4 + 9*y^3 + 18*y^2 - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 11*x^21 + 43*x^20 - 45*x^19 - 152*x^18 + 456*x^17 - 114*x^16 - 1026*x^15 + 1114*x^14 + 790*x^13 - 1814*x^12 + 120*x^11 + 1304*x^10 - 464*x^9 - 493*x^8 + 217*x^7 + 170*x^6 - 46*x^5 - 77*x^4 + 9*x^3 + 18*x^2 - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 11*x^21 + 43*x^20 - 45*x^19 - 152*x^18 + 456*x^17 - 114*x^16 - 1026*x^15 + 1114*x^14 + 790*x^13 - 1814*x^12 + 120*x^11 + 1304*x^10 - 464*x^9 - 493*x^8 + 217*x^7 + 170*x^6 - 46*x^5 - 77*x^4 + 9*x^3 + 18*x^2 - 1)
 

\( x^{22} - 11 x^{21} + 43 x^{20} - 45 x^{19} - 152 x^{18} + 456 x^{17} - 114 x^{16} - 1026 x^{15} + 1114 x^{14} + 790 x^{13} - 1814 x^{12} + 120 x^{11} + 1304 x^{10} - 464 x^{9} - 493 x^{8} + \cdots - 1 \) 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:  $[16, 3]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-289690752824370116310308034941759\) \(\medspace = -\,809\cdot 4759\cdot 8674315276967^{2}\) 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:  \(29.89\)
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:  $809^{1/2}4759^{1/2}8674315276967^{1/2}\approx 5778960349.413772$
Ramified primes:   \(809\), \(4759\), \(8674315276967\) 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:  $\Q(\sqrt{-3850031}$)
$\card{ \Aut(K/\Q) }$:  $2$
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:  $18$
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:   $a^{20}-10a^{19}+33a^{18}-12a^{17}-165a^{16}+300a^{15}+160a^{14}-862a^{13}+369a^{12}+984a^{11}-930a^{10}-405a^{9}+749a^{8}-32a^{7}-265a^{6}+17a^{5}+77a^{4}+26a^{3}-30a^{2}-5a+3$, $a^{2}-a-1$, $a$, $a^{21}-10a^{20}+33a^{19}-12a^{18}-164a^{17}+292a^{16}+178a^{15}-848a^{14}+266a^{13}+1056a^{12}-758a^{11}-638a^{10}+666a^{9}+202a^{8}-291a^{7}-74a^{6}+96a^{5}+50a^{4}-27a^{3}-18a^{2}$, $a^{20}-10a^{19}+34a^{18}-21a^{17}-139a^{16}+296a^{15}+43a^{14}-687a^{13}+470a^{12}+573a^{11}-771a^{10}-78a^{9}+455a^{8}-87a^{7}-125a^{6}+5a^{5}+50a^{4}+9a^{3}-19a^{2}+a+1$, $2a^{20}-20a^{19}+67a^{18}-33a^{17}-303a^{16}+588a^{15}+222a^{14}-1542a^{13}+749a^{12}+1642a^{11}-1592a^{10}-687a^{9}+1205a^{8}+34a^{7}-446a^{6}-17a^{5}+142a^{4}+51a^{3}-48a^{2}-14a+3$, $2a^{20}-20a^{19}+68a^{18}-42a^{17}-277a^{16}+584a^{15}+105a^{14}-1367a^{13}+850a^{12}+1231a^{11}-1433a^{10}-360a^{9}+912a^{8}-25a^{7}-304a^{6}-21a^{5}+107a^{4}+32a^{3}-33a^{2}-9a+1$, $4a^{20}-40a^{19}+135a^{18}-75a^{17}-581a^{16}+1180a^{15}+308a^{14}-2916a^{13}+1688a^{12}+2794a^{11}-3143a^{10}-853a^{9}+2150a^{8}-154a^{7}-725a^{6}+21a^{5}+241a^{4}+61a^{3}-84a^{2}-11a+8$, $a^{20}-10a^{19}+33a^{18}-12a^{17}-165a^{16}+300a^{15}+160a^{14}-862a^{13}+369a^{12}+984a^{11}-930a^{10}-405a^{9}+749a^{8}-32a^{7}-265a^{6}+17a^{5}+78a^{4}+24a^{3}-32a^{2}-2a+5$, $3a^{20}-30a^{19}+101a^{18}-54a^{17}-442a^{16}+884a^{15}+265a^{14}-2229a^{13}+1219a^{12}+2215a^{11}-2363a^{10}-765a^{9}+1661a^{8}-57a^{7}-569a^{6}-4a^{5}+185a^{4}+56a^{3}-65a^{2}-11a+5$, $4a^{21}-45a^{20}+183a^{19}-224a^{18}-554a^{17}+1951a^{16}-917a^{15}-3848a^{14}+5310a^{13}+1815a^{12}-7473a^{11}+2298a^{10}+4335a^{9}-2800a^{8}-1055a^{7}+991a^{6}+358a^{5}-213a^{4}-231a^{3}+82a^{2}+35a-7$, $a^{21}-11a^{20}+43a^{19}-45a^{18}-152a^{17}+456a^{16}-114a^{15}-1026a^{14}+1114a^{13}+790a^{12}-1814a^{11}+120a^{10}+1304a^{9}-464a^{8}-493a^{7}+217a^{6}+170a^{5}-46a^{4}-77a^{3}+9a^{2}+17a$, $2a^{21}-21a^{20}+77a^{19}-67a^{18}-283a^{17}+735a^{16}-93a^{15}-1592a^{14}+1525a^{13}+1094a^{12}-2289a^{11}+287a^{10}+1326a^{9}-618a^{8}-324a^{7}+198a^{6}+110a^{5}-30a^{4}-55a^{3}+20a^{2}+5a-3$, $a^{21}-11a^{20}+43a^{19}-45a^{18}-153a^{17}+465a^{16}-140a^{15}-1022a^{14}+1231a^{13}+615a^{12}-1914a^{11}+525a^{10}+1154a^{9}-781a^{8}-233a^{7}+282a^{6}+61a^{5}-54a^{4}-56a^{3}+30a^{2}+6a-3$, $a^{21}-7a^{20}+3a^{19}+89a^{18}-218a^{17}-151a^{16}+1070a^{15}-602a^{14}-1970a^{13}+2364a^{12}+1378a^{11}-3119a^{10}+95a^{9}+1896a^{8}-511a^{7}-618a^{6}+151a^{5}+226a^{4}+9a^{3}-83a^{2}-3a+7$, $4a^{20}-40a^{19}+135a^{18}-75a^{17}-581a^{16}+1180a^{15}+308a^{14}-2916a^{13}+1689a^{12}+2788a^{11}-3134a^{10}-843a^{9}+2116a^{8}-144a^{7}-694a^{6}+a^{5}+234a^{4}+67a^{3}-82a^{2}-12a+5$, $3a^{21}-34a^{20}+140a^{19}-179a^{18}-401a^{17}+1487a^{16}-785a^{15}-2807a^{14}+4086a^{13}+1111a^{12}-5480a^{11}+1891a^{10}+2987a^{9}-2052a^{8}-659a^{7}+684a^{6}+238a^{5}-152a^{4}-151a^{3}+55a^{2}+16a-5$, $2a^{21}-23a^{20}+97a^{19}-133a^{18}-259a^{17}+1064a^{16}-685a^{15}-1930a^{14}+3235a^{13}+459a^{12}-4329a^{11}+1975a^{10}+2370a^{9}-2038a^{8}-488a^{7}+760a^{6}+150a^{5}-196a^{4}-122a^{3}+73a^{2}+26a-8$ 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:  \( 334750790.993 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{3}\cdot 334750790.993 \cdot 1}{2\cdot\sqrt{289690752824370116310308034941759}}\cr\approx \mathstrut & 0.159861482349 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 + 43*x^20 - 45*x^19 - 152*x^18 + 456*x^17 - 114*x^16 - 1026*x^15 + 1114*x^14 + 790*x^13 - 1814*x^12 + 120*x^11 + 1304*x^10 - 464*x^9 - 493*x^8 + 217*x^7 + 170*x^6 - 46*x^5 - 77*x^4 + 9*x^3 + 18*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 - 11*x^21 + 43*x^20 - 45*x^19 - 152*x^18 + 456*x^17 - 114*x^16 - 1026*x^15 + 1114*x^14 + 790*x^13 - 1814*x^12 + 120*x^11 + 1304*x^10 - 464*x^9 - 493*x^8 + 217*x^7 + 170*x^6 - 46*x^5 - 77*x^4 + 9*x^3 + 18*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 - 11*x^21 + 43*x^20 - 45*x^19 - 152*x^18 + 456*x^17 - 114*x^16 - 1026*x^15 + 1114*x^14 + 790*x^13 - 1814*x^12 + 120*x^11 + 1304*x^10 - 464*x^9 - 493*x^8 + 217*x^7 + 170*x^6 - 46*x^5 - 77*x^4 + 9*x^3 + 18*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 - 11*x^21 + 43*x^20 - 45*x^19 - 152*x^18 + 456*x^17 - 114*x^16 - 1026*x^15 + 1114*x^14 + 790*x^13 - 1814*x^12 + 120*x^11 + 1304*x^10 - 464*x^9 - 493*x^8 + 217*x^7 + 170*x^6 - 46*x^5 - 77*x^4 + 9*x^3 + 18*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

11.9.8674315276967.1

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 ${\href{/padicField/2.11.0.1}{11} }^{2}$ ${\href{/padicField/3.11.0.1}{11} }^{2}$ ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ ${\href{/padicField/7.9.0.1}{9} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ $22$ $16{,}\,{\href{/padicField/13.6.0.1}{6} }$ ${\href{/padicField/17.9.0.1}{9} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }$ ${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.11.0.1}{11} }^{2}$ ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ $20{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.11.0.1}{11} }^{2}$ ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ $20{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }$

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
\(809\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $10$$1$$10$$0$$C_{10}$$[\ ]^{10}$
\(4759\) Copy content Toggle raw display $\Q_{4759}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{4759}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{4759}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{4759}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$
\(8674315276967\) Copy content Toggle raw display Deg $4$$2$$2$$2$
Deg $9$$1$$9$$0$$C_9$$[\ ]^{9}$
Deg $9$$1$$9$$0$$C_9$$[\ ]^{9}$