Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 + 66*x^20 - 275*x^19 + 879*x^18 - 2268*x^17 + 4871*x^16 - 8878*x^15 + 13903*x^14 - 18847*x^13 + 22192*x^12 - 22691*x^11 + 20056*x^10 - 15173*x^9 + 9644*x^8 - 4974*x^7 + 1932*x^6 - 448*x^5 - 27*x^4 + 71*x^3 - 24*x^2 + x + 1)
gp: K = bnfinit(y^22 - 11*y^21 + 66*y^20 - 275*y^19 + 879*y^18 - 2268*y^17 + 4871*y^16 - 8878*y^15 + 13903*y^14 - 18847*y^13 + 22192*y^12 - 22691*y^11 + 20056*y^10 - 15173*y^9 + 9644*y^8 - 4974*y^7 + 1932*y^6 - 448*y^5 - 27*y^4 + 71*y^3 - 24*y^2 + y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 11*x^21 + 66*x^20 - 275*x^19 + 879*x^18 - 2268*x^17 + 4871*x^16 - 8878*x^15 + 13903*x^14 - 18847*x^13 + 22192*x^12 - 22691*x^11 + 20056*x^10 - 15173*x^9 + 9644*x^8 - 4974*x^7 + 1932*x^6 - 448*x^5 - 27*x^4 + 71*x^3 - 24*x^2 + x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 11*x^21 + 66*x^20 - 275*x^19 + 879*x^18 - 2268*x^17 + 4871*x^16 - 8878*x^15 + 13903*x^14 - 18847*x^13 + 22192*x^12 - 22691*x^11 + 20056*x^10 - 15173*x^9 + 9644*x^8 - 4974*x^7 + 1932*x^6 - 448*x^5 - 27*x^4 + 71*x^3 - 24*x^2 + x + 1)
\( x^{22} - 11 x^{21} + 66 x^{20} - 275 x^{19} + 879 x^{18} - 2268 x^{17} + 4871 x^{16} - 8878 x^{15} + \cdots + 1 \)
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: | $[4, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-951194438369517944038992707\)
\(\medspace = -\,139\cdot 1583^{2}\cdot 2731^{2}\cdot 6217^{2}\cdot 9473\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(16.84\) | 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: | $139^{1/2}1583^{1/2}2731^{1/2}6217^{1/2}9473^{1/2}\approx 188123439.29282743$ | ||
| Ramified primes: |
\(139\), \(1583\), \(2731\), \(6217\), \(9473\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1316747}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$
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: | $12$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: |
$a^{2}-a+1$, $a^{18}-9a^{17}+45a^{16}-156a^{15}+413a^{14}-875a^{13}+1525a^{12}-2221a^{11}+2724a^{10}-2818a^{9}+2445a^{8}-1755a^{7}+1011a^{6}-439a^{5}+119a^{4}-a^{3}-15a^{2}+6a$, $a^{20}-10a^{19}+55a^{18}-210a^{17}+613a^{16}-1436a^{15}+2777a^{14}-4509a^{13}+6204a^{12}-7259a^{11}+7203a^{10}-6002a^{9}+4106a^{8}-2197a^{7}+806a^{6}-90a^{5}-116a^{4}+91a^{3}-28a^{2}+a+3$, $a^{16}-8a^{15}+36a^{14}-112a^{13}+265a^{12}-498a^{11}+762a^{10}-961a^{9}+1001a^{8}-856a^{7}+588a^{6}-311a^{5}+113a^{4}-18a^{3}-6a^{2}+4a+1$, $a^{8}-4a^{7}+10a^{6}-16a^{5}+18a^{4}-14a^{3}+6a^{2}-a-2$, $a^{19}-10a^{18}+54a^{17}-201a^{16}+569a^{15}-1288a^{14}+2401a^{13}-3753a^{12}+4972a^{11}-5613a^{10}+5403a^{9}-4416a^{8}+3033a^{7}-1717a^{6}+773a^{5}-257a^{4}+51a^{3}+a^{2}-5a+2$, $a^{18}-9a^{17}+44a^{16}-148a^{15}+376a^{14}-756a^{13}+1232a^{12}-1646a^{11}+1801a^{10}-1591a^{9}+1088a^{8}-510a^{7}+78a^{6}+116a^{5}-123a^{4}+62a^{3}-12a^{2}-3a+1$, $a^{21}-10a^{20}+56a^{19}-220a^{18}+669a^{17}-1653a^{16}+3419a^{15}-6028a^{14}+9162a^{13}-12078a^{12}+13833a^{11}-13731a^{10}+11722a^{9}-8483a^{8}+5064a^{7}-2365a^{6}+755a^{5}-79a^{4}-66a^{3}+37a^{2}-2a-2$, $a^{21}-10a^{20}+56a^{19}-219a^{18}+660a^{17}-1609a^{16}+3270a^{15}-5644a^{14}+8372a^{13}-10747a^{12}+11971a^{11}-11559a^{10}+9618a^{9}-6817a^{8}+4025a^{7}-1900a^{6}+654a^{5}-119a^{4}-18a^{3}+19a^{2}-2a-1$, $a^{21}-11a^{20}+65a^{19}-265a^{18}+824a^{17}-2058a^{16}+4257a^{15}-7434a^{14}+11090a^{13}-14226a^{12}+15722a^{11}-14928a^{10}+12069a^{9}-8155a^{8}+4433a^{7}-1769a^{6}+363a^{5}+111a^{4}-132a^{3}+51a^{2}-3a-4$, $a^{21}-10a^{20}+55a^{19}-210a^{18}+613a^{17}-1435a^{16}+2769a^{15}-4473a^{14}+6092a^{13}-6994a^{12}+6705a^{11}-5239a^{10}+3140a^{9}-1181a^{8}-80a^{7}+542a^{6}-475a^{5}+242a^{4}-67a^{3}+8a-2$, $a^{20}-11a^{19}+65a^{18}-264a^{17}+814a^{16}-2005a^{15}+4065a^{14}-6910a^{13}+9957a^{12}-12231a^{11}+12816a^{10}-11405a^{9}+8522a^{8}-5230a^{7}+2523a^{6}-863a^{5}+141a^{4}+40a^{3}-29a^{2}+6a+1$
| 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: | \( 43108.9874663 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{9}\cdot 43108.9874663 \cdot 1}{2\cdot\sqrt{951194438369517944038992707}}\cr\approx \mathstrut & 0.170664020859 \end{aligned}\] (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 + 66*x^20 - 275*x^19 + 879*x^18 - 2268*x^17 + 4871*x^16 - 8878*x^15 + 13903*x^14 - 18847*x^13 + 22192*x^12 - 22691*x^11 + 20056*x^10 - 15173*x^9 + 9644*x^8 - 4974*x^7 + 1932*x^6 - 448*x^5 - 27*x^4 + 71*x^3 - 24*x^2 + 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 + 66*x^20 - 275*x^19 + 879*x^18 - 2268*x^17 + 4871*x^16 - 8878*x^15 + 13903*x^14 - 18847*x^13 + 22192*x^12 - 22691*x^11 + 20056*x^10 - 15173*x^9 + 9644*x^8 - 4974*x^7 + 1932*x^6 - 448*x^5 - 27*x^4 + 71*x^3 - 24*x^2 + 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 + 66*x^20 - 275*x^19 + 879*x^18 - 2268*x^17 + 4871*x^16 - 8878*x^15 + 13903*x^14 - 18847*x^13 + 22192*x^12 - 22691*x^11 + 20056*x^10 - 15173*x^9 + 9644*x^8 - 4974*x^7 + 1932*x^6 - 448*x^5 - 27*x^4 + 71*x^3 - 24*x^2 + 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 + 66*x^20 - 275*x^19 + 879*x^18 - 2268*x^17 + 4871*x^16 - 8878*x^15 + 13903*x^14 - 18847*x^13 + 22192*x^12 - 22691*x^11 + 20056*x^10 - 15173*x^9 + 9644*x^8 - 4974*x^7 + 1932*x^6 - 448*x^5 - 27*x^4 + 71*x^3 - 24*x^2 + 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 $C_2^{10}.(C_2\times S_{11})$ are not computed |
| Character table for $C_2^{10}.(C_2\times S_{11})$ is not computed |
Intermediate fields
| 11.3.26877166541.1 |
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 | ${\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.6.0.1}{6} }^{2}$ | ${\href{/padicField/3.7.0.1}{7} }^{2}{,}\,{\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}$ | $22$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | $22$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.6.0.1}{6} }^{2}$ | $18{,}\,{\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.14.0.1}{14} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.11.0.1}{11} }^{2}$ | ${\href{/padicField/37.7.0.1}{7} }^{2}{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(139\)
| 139.2.1.1 | $x^{2} + 278$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 139.10.0.1 | $x^{10} + 110 x^{5} + 48 x^{4} + 130 x^{3} + 66 x^{2} + 106 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
| 139.10.0.1 | $x^{10} + 110 x^{5} + 48 x^{4} + 130 x^{3} + 66 x^{2} + 106 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
|
\(1583\)
| 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 $4$ | $2$ | $2$ | $2$ | ||||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
|
\(2731\)
| $\Q_{2731}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{2731}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 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 $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
|
\(6217\)
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
|
\(9473\)
| 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 $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |