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Properties

Label 22.14.191...529.1
Degree $22$
Signature $[14, 4]$
Discriminant $1.913\times 10^{32}$
Root discriminant \(29.33\)
Ramified primes $7,251,33893,1792166448977$
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 - 146*x^18 + 402*x^17 + 46*x^16 - 1082*x^15 + 512*x^14 + 1700*x^13 - 1234*x^12 - 1826*x^11 + 1419*x^10 + 1403*x^9 - 797*x^8 - 807*x^7 + 55*x^6 + 331*x^5 + 160*x^4 - 51*x^3 - 59*x^2 - 14*x - 1)
 
gp: K = bnfinit(y^22 - 11*y^21 + 43*y^20 - 45*y^19 - 146*y^18 + 402*y^17 + 46*y^16 - 1082*y^15 + 512*y^14 + 1700*y^13 - 1234*y^12 - 1826*y^11 + 1419*y^10 + 1403*y^9 - 797*y^8 - 807*y^7 + 55*y^6 + 331*y^5 + 160*y^4 - 51*y^3 - 59*y^2 - 14*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 11*x^21 + 43*x^20 - 45*x^19 - 146*x^18 + 402*x^17 + 46*x^16 - 1082*x^15 + 512*x^14 + 1700*x^13 - 1234*x^12 - 1826*x^11 + 1419*x^10 + 1403*x^9 - 797*x^8 - 807*x^7 + 55*x^6 + 331*x^5 + 160*x^4 - 51*x^3 - 59*x^2 - 14*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 11*x^21 + 43*x^20 - 45*x^19 - 146*x^18 + 402*x^17 + 46*x^16 - 1082*x^15 + 512*x^14 + 1700*x^13 - 1234*x^12 - 1826*x^11 + 1419*x^10 + 1403*x^9 - 797*x^8 - 807*x^7 + 55*x^6 + 331*x^5 + 160*x^4 - 51*x^3 - 59*x^2 - 14*x - 1)
 

\( x^{22} - 11 x^{21} + 43 x^{20} - 45 x^{19} - 146 x^{18} + 402 x^{17} + 46 x^{16} - 1082 x^{15} + \cdots - 1 \) 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:  $[14, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(191266300800812904024665746296529\) \(\medspace = 7\cdot 251\cdot 33893\cdot 1792166448977^{2}\) 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:  \(29.33\)
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:  $7^{1/2}251^{1/2}33893^{1/2}1792166448977^{1/2}\approx 10330707324.70661$
Ramified primes:   \(7\), \(251\), \(33893\), \(1792166448977\) 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:  $\Q(\sqrt{59550001}$)
$\card{ \Aut(K/\Q) }$:  $2$
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:   $a^{20}-10a^{19}+34a^{18}-21a^{17}-133a^{16}+248a^{15}+161a^{14}-673a^{13}+1027a^{11}-207a^{10}-1006a^{9}+206a^{8}+603a^{7}+12a^{6}-192a^{5}-125a^{4}+14a^{3}+49a^{2}+12a+2$, $a^{20}-10a^{19}+37a^{18}-48a^{17}-53a^{16}+220a^{15}-130a^{14}-288a^{13}+435a^{12}+94a^{11}-621a^{10}+250a^{9}+510a^{8}-368a^{7}-253a^{6}+137a^{5}+125a^{4}+43a^{3}-45a^{2}-36a-5$, $a^{20}-10a^{19}+34a^{18}-21a^{17}-132a^{16}+240a^{15}+180a^{14}-666a^{13}-84a^{12}+1076a^{11}-46a^{10}-1140a^{9}+a^{8}+764a^{7}+190a^{6}-271a^{5}-221a^{4}-10a^{3}+75a^{2}+40a+6$, $4a^{20}-40a^{19}+134a^{18}-66a^{17}-582a^{16}+984a^{15}+903a^{14}-2937a^{13}-548a^{12}+4913a^{11}-108a^{10}-5303a^{9}+111a^{8}+3567a^{7}+615a^{6}-1236a^{5}-858a^{4}-8a^{3}+322a^{2}+133a+14$, $5a^{20}-50a^{19}+167a^{18}-78a^{17}-739a^{16}+1220a^{15}+1213a^{14}-3729a^{13}-898a^{12}+6389a^{11}+191a^{10}-7068a^{9}-233a^{8}+4891a^{7}+1085a^{6}-1731a^{5}-1264a^{4}-42a^{3}+460a^{2}+211a+25$, $a^{20}-10a^{19}+34a^{18}-21a^{17}-133a^{16}+248a^{15}+161a^{14}-673a^{13}+1027a^{11}-207a^{10}-1006a^{9}+206a^{8}+603a^{7}+12a^{6}-192a^{5}-125a^{4}+14a^{3}+50a^{2}+11a+1$, $a^{20}-10a^{19}+32a^{18}-3a^{17}-185a^{16}+256a^{15}+382a^{14}-932a^{13}-379a^{12}+1728a^{11}+200a^{10}-2007a^{9}-126a^{8}+1426a^{7}+267a^{6}-491a^{5}-327a^{4}-13a^{3}+130a^{2}+51a+4$, $a$, $3a^{20}-30a^{19}+97a^{18}-18a^{17}-528a^{16}+756a^{15}+1053a^{14}-2649a^{13}-1053a^{12}+4875a^{11}+626a^{10}-5678a^{9}-522a^{8}+4077a^{7}+953a^{6}-1446a^{5}-1022a^{4}-49a^{3}+381a^{2}+174a+18$, $a^{21}-5a^{20}-17a^{19}+157a^{18}-254a^{17}-443a^{16}+1502a^{15}+201a^{14}-3757a^{13}+914a^{12}+5887a^{11}-1913a^{10}-6289a^{9}+1393a^{8}+4440a^{7}+272a^{6}-1793a^{5}-1008a^{4}+129a^{3}+434a^{2}+162a+14$, $a^{2}-1$, $3a^{20}-30a^{19}+100a^{18}-45a^{17}-450a^{16}+744a^{15}+723a^{14}-2271a^{13}-464a^{12}+3837a^{11}-62a^{10}-4163a^{9}+110a^{8}+2803a^{7}+425a^{6}-965a^{5}-636a^{4}-a^{3}+248a^{2}+96a+7$, $9a^{21}-93a^{20}+332a^{19}-255a^{18}-1233a^{17}+2604a^{16}+1267a^{15}-7088a^{14}+839a^{13}+11148a^{12}-3551a^{11}-11542a^{10}+3726a^{9}+7765a^{8}-956a^{7}-3170a^{6}-1062a^{5}+570a^{4}+700a^{3}+77a^{2}-58a-8$, $5a^{20}-50a^{19}+171a^{18}-114a^{17}-633a^{16}+1188a^{15}+812a^{14}-3218a^{13}-267a^{12}+5112a^{11}-455a^{10}-5312a^{9}+293a^{8}+3502a^{7}+624a^{6}-1248a^{5}-868a^{4}+8a^{3}+316a^{2}+133a+15$, $a^{20}-10a^{19}+34a^{18}-21a^{17}-132a^{16}+240a^{15}+180a^{14}-666a^{13}-84a^{12}+1076a^{11}-46a^{10}-1140a^{9}+a^{8}+764a^{7}+190a^{6}-271a^{5}-221a^{4}-10a^{3}+75a^{2}+39a+6$, $5a^{21}-50a^{20}+165a^{19}-60a^{18}-793a^{17}+1244a^{16}+1394a^{15}-3988a^{14}-1137a^{13}+6978a^{12}+373a^{11}-7824a^{10}-315a^{9}+5444a^{8}+1155a^{7}-1909a^{6}-1369a^{5}-42a^{4}+504a^{3}+223a^{2}+27a$, $4a^{21}-46a^{20}+193a^{19}-257a^{18}-518a^{17}+1887a^{16}-467a^{15}-4528a^{14}+3791a^{13}+6268a^{12}-7593a^{11}-5849a^{10}+8339a^{9}+3999a^{8}-4947a^{7}-2459a^{6}+994a^{5}+1372a^{4}+403a^{3}-362a^{2}-204a-25$ 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:  \( 205753246.579 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{4}\cdot 205753246.579 \cdot 1}{2\cdot\sqrt{191266300800812904024665746296529}}\cr\approx \mathstrut & 0.189948944448 \end{aligned}\] (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 - 146*x^18 + 402*x^17 + 46*x^16 - 1082*x^15 + 512*x^14 + 1700*x^13 - 1234*x^12 - 1826*x^11 + 1419*x^10 + 1403*x^9 - 797*x^8 - 807*x^7 + 55*x^6 + 331*x^5 + 160*x^4 - 51*x^3 - 59*x^2 - 14*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 - 11*x^21 + 43*x^20 - 45*x^19 - 146*x^18 + 402*x^17 + 46*x^16 - 1082*x^15 + 512*x^14 + 1700*x^13 - 1234*x^12 - 1826*x^11 + 1419*x^10 + 1403*x^9 - 797*x^8 - 807*x^7 + 55*x^6 + 331*x^5 + 160*x^4 - 51*x^3 - 59*x^2 - 14*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 - 11*x^21 + 43*x^20 - 45*x^19 - 146*x^18 + 402*x^17 + 46*x^16 - 1082*x^15 + 512*x^14 + 1700*x^13 - 1234*x^12 - 1826*x^11 + 1419*x^10 + 1403*x^9 - 797*x^8 - 807*x^7 + 55*x^6 + 331*x^5 + 160*x^4 - 51*x^3 - 59*x^2 - 14*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 - 11*x^21 + 43*x^20 - 45*x^19 - 146*x^18 + 402*x^17 + 46*x^16 - 1082*x^15 + 512*x^14 + 1700*x^13 - 1234*x^12 - 1826*x^11 + 1419*x^10 + 1403*x^9 - 797*x^8 - 807*x^7 + 55*x^6 + 331*x^5 + 160*x^4 - 51*x^3 - 59*x^2 - 14*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

$C_2^{10}.(C_2\times S_{11})$ (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 $C_2^{10}.(C_2\times S_{11})$
Character table for $C_2^{10}.(C_2\times S_{11})$

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 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 ${\href{/padicField/2.11.0.1}{11} }^{2}$ ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.5.0.1}{5} }^{2}$ ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.10.0.1}{10} }$ R ${\href{/padicField/11.11.0.1}{11} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.5.0.1}{5} }^{2}$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.10.0.1}{10} }$ ${\href{/padicField/19.8.0.1}{8} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ $20{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.6.0.1}{6} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ $22$ ${\href{/padicField/47.10.0.1}{10} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.7.0.1}{7} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.8.0.1}{8} }$

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
\(7\) Copy content Toggle raw display 7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.6.0.1$x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
7.14.0.1$x^{14} + 5 x^{7} + 6 x^{5} + 2 x^{4} + 3 x^{2} + 6 x + 3$$1$$14$$0$$C_{14}$$[\ ]^{14}$
\(251\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $14$$1$$14$$0$$C_{14}$$[\ ]^{14}$
\(33893\) 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$$[\ ]_{2}$
Deg $16$$1$$16$$0$$C_{16}$$[\ ]^{16}$
\(1792166448977\) Copy content Toggle raw display $\Q_{1792166448977}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{1792166448977}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $16$$1$$16$$0$$C_{16}$$[\ ]^{16}$