Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 + 6*x^20 - 10*x^19 + 17*x^18 - 21*x^17 + 22*x^16 - 15*x^15 - 5*x^14 + 38*x^13 - 83*x^12 + 135*x^11 - 185*x^10 + 221*x^9 - 234*x^8 + 222*x^7 - 187*x^6 + 138*x^5 - 88*x^4 + 47*x^3 - 20*x^2 + 6*x - 1)
gp: K = bnfinit(y^22 - 2*y^21 + 6*y^20 - 10*y^19 + 17*y^18 - 21*y^17 + 22*y^16 - 15*y^15 - 5*y^14 + 38*y^13 - 83*y^12 + 135*y^11 - 185*y^10 + 221*y^9 - 234*y^8 + 222*y^7 - 187*y^6 + 138*y^5 - 88*y^4 + 47*y^3 - 20*y^2 + 6*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 2*x^21 + 6*x^20 - 10*x^19 + 17*x^18 - 21*x^17 + 22*x^16 - 15*x^15 - 5*x^14 + 38*x^13 - 83*x^12 + 135*x^11 - 185*x^10 + 221*x^9 - 234*x^8 + 222*x^7 - 187*x^6 + 138*x^5 - 88*x^4 + 47*x^3 - 20*x^2 + 6*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 2*x^21 + 6*x^20 - 10*x^19 + 17*x^18 - 21*x^17 + 22*x^16 - 15*x^15 - 5*x^14 + 38*x^13 - 83*x^12 + 135*x^11 - 185*x^10 + 221*x^9 - 234*x^8 + 222*x^7 - 187*x^6 + 138*x^5 - 88*x^4 + 47*x^3 - 20*x^2 + 6*x - 1)
\( x^{22} - 2 x^{21} + 6 x^{20} - 10 x^{19} + 17 x^{18} - 21 x^{17} + 22 x^{16} - 15 x^{15} - 5 x^{14} + \cdots - 1 \)
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: | $[2, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(890830078698256823295734989\)
\(\medspace = 34092855421\cdot 26129523845912209\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(16.79\) | 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: | $34092855421^{1/2}26129523845912209^{1/2}\approx 29846776688584.93$ | ||
| Ramified primes: |
\(34092855421\), \(26129523845912209\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{89083\!\cdots\!34989}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{17}a^{21}-\frac{7}{17}a^{20}+\frac{7}{17}a^{19}+\frac{6}{17}a^{18}+\frac{4}{17}a^{17}-\frac{7}{17}a^{16}+\frac{6}{17}a^{15}+\frac{6}{17}a^{14}-\frac{1}{17}a^{13}-\frac{8}{17}a^{12}+\frac{8}{17}a^{11}-\frac{7}{17}a^{10}+\frac{3}{17}a^{9}+\frac{2}{17}a^{8}-\frac{6}{17}a^{7}-\frac{3}{17}a^{6}-\frac{2}{17}a^{5}-\frac{5}{17}a^{4}+\frac{5}{17}a^{3}+\frac{5}{17}a^{2}+\frac{6}{17}a-\frac{7}{17}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| 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: | $11$ | 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: |
$a$, $\frac{29}{17}a^{21}+\frac{35}{17}a^{20}+\frac{67}{17}a^{19}+\frac{140}{17}a^{18}-\frac{20}{17}a^{17}+\frac{358}{17}a^{16}-\frac{251}{17}a^{15}+\frac{446}{17}a^{14}-\frac{369}{17}a^{13}+\frac{74}{17}a^{12}+\frac{368}{17}a^{11}-\frac{985}{17}a^{10}+\frac{1787}{17}a^{9}-\frac{2543}{17}a^{8}+\frac{2971}{17}a^{7}-\frac{3079}{17}a^{6}+\frac{2832}{17}a^{5}-\frac{2219}{17}a^{4}+\frac{1454}{17}a^{3}-\frac{807}{17}a^{2}+\frac{327}{17}a-\frac{84}{17}$, $\frac{82}{17}a^{21}-\frac{81}{17}a^{20}+\frac{370}{17}a^{19}-\frac{409}{17}a^{18}+\frac{804}{17}a^{17}-\frac{727}{17}a^{16}+\frac{696}{17}a^{15}-\frac{222}{17}a^{14}-\frac{932}{17}a^{13}+\frac{2217}{17}a^{12}-\frac{4087}{17}a^{11}+\frac{5869}{17}a^{10}-\frac{7353}{17}a^{9}+\frac{8052}{17}a^{8}-\frac{7802}{17}a^{7}+\frac{6843}{17}a^{6}-\frac{5128}{17}a^{5}+\frac{3313}{17}a^{4}-\frac{1817}{17}a^{3}+\frac{750}{17}a^{2}-\frac{222}{17}a+\frac{21}{17}$, $5a^{21}-9a^{20}+28a^{19}-44a^{18}+75a^{17}-88a^{16}+89a^{15}-53a^{14}-40a^{13}+185a^{12}-377a^{11}+592a^{10}-790a^{9}+920a^{8}-949a^{7}+876a^{6}-713a^{5}+503a^{4}-302a^{3}+147a^{2}-53a+11$, $\frac{7}{17}a^{21}+\frac{2}{17}a^{20}+\frac{49}{17}a^{19}-\frac{9}{17}a^{18}+\frac{130}{17}a^{17}-\frac{66}{17}a^{16}+\frac{178}{17}a^{15}-\frac{94}{17}a^{14}+\frac{27}{17}a^{13}+\frac{80}{17}a^{12}-\frac{420}{17}a^{11}+\frac{648}{17}a^{10}-\frac{982}{17}a^{9}+\frac{1306}{17}a^{8}-\frac{1436}{17}a^{7}+\frac{1458}{17}a^{6}-\frac{1289}{17}a^{5}+\frac{1053}{17}a^{4}-\frac{696}{17}a^{3}+\frac{392}{17}a^{2}-\frac{196}{17}a+\frac{53}{17}$, $\frac{88}{17}a^{21}-\frac{89}{17}a^{20}+\frac{412}{17}a^{19}-\frac{458}{17}a^{18}+\frac{930}{17}a^{17}-\frac{837}{17}a^{16}+\frac{885}{17}a^{15}-\frac{288}{17}a^{14}-\frac{904}{17}a^{13}+\frac{2475}{17}a^{12}-\frac{4583}{17}a^{11}+\frac{6745}{17}a^{10}-\frac{8559}{17}a^{9}+\frac{9458}{17}a^{8}-\frac{9368}{17}a^{7}+\frac{8270}{17}a^{6}-\frac{6330}{17}a^{5}+\frac{4184}{17}a^{4}-\frac{2348}{17}a^{3}+\frac{1001}{17}a^{2}-\frac{305}{17}a+\frac{30}{17}$, $2a^{21}-3a^{20}+11a^{19}-15a^{18}+29a^{17}-30a^{16}+34a^{15}-17a^{14}-15a^{13}+69a^{12}-139a^{11}+216a^{10}-288a^{9}+334a^{8}-342a^{7}+315a^{6}-255a^{5}+178a^{4}-106a^{3}+52a^{2}-18a+3$, $\frac{38}{17}a^{21}-\frac{113}{17}a^{20}+\frac{232}{17}a^{19}-\frac{503}{17}a^{18}+\frac{679}{17}a^{17}-\frac{946}{17}a^{16}+\frac{823}{17}a^{15}-\frac{503}{17}a^{14}-\frac{395}{17}a^{13}+\frac{1991}{17}a^{12}-\frac{3742}{17}a^{11}+\frac{5922}{17}a^{10}-\frac{7808}{17}a^{9}+\frac{8950}{17}a^{8}-\frac{9102}{17}a^{7}+\frac{8233}{17}a^{6}-\frac{6570}{17}a^{5}+\frac{4451}{17}a^{4}-\frac{2564}{17}a^{3}+\frac{1176}{17}a^{2}-\frac{367}{17}a+\frac{57}{17}$, $\frac{7}{17}a^{21}-\frac{66}{17}a^{20}+\frac{100}{17}a^{19}-\frac{281}{17}a^{18}+\frac{402}{17}a^{17}-\frac{576}{17}a^{16}+\frac{620}{17}a^{15}-\frac{417}{17}a^{14}+\frac{61}{17}a^{13}+\frac{930}{17}a^{12}-\frac{2052}{17}a^{11}+\frac{3368}{17}a^{10}-\frac{4841}{17}a^{9}+\frac{5777}{17}a^{8}-\frac{6128}{17}a^{7}+\frac{5793}{17}a^{6}-\frac{4876}{17}a^{5}+\frac{3501}{17}a^{4}-\frac{2090}{17}a^{3}+\frac{1072}{17}a^{2}-\frac{366}{17}a+\frac{53}{17}$, $\frac{47}{17}a^{21}-\frac{6}{17}a^{20}+\frac{193}{17}a^{19}-\frac{58}{17}a^{18}+\frac{341}{17}a^{17}-\frac{57}{17}a^{16}+\frac{197}{17}a^{15}+\frac{180}{17}a^{14}-\frac{523}{17}a^{13}+\frac{882}{17}a^{12}-\frac{1426}{17}a^{11}+\frac{1762}{17}a^{10}-\frac{1950}{17}a^{9}+\frac{1828}{17}a^{8}-\frac{1540}{17}a^{7}+\frac{1066}{17}a^{6}-\frac{519}{17}a^{5}+\frac{156}{17}a^{4}+\frac{48}{17}a^{3}-\frac{139}{17}a^{2}+\frac{78}{17}a-\frac{23}{17}$, $a^{21}-5a^{20}+9a^{19}-24a^{18}+32a^{17}-52a^{16}+48a^{15}-43a^{14}+71a^{12}-170a^{11}+288a^{10}-407a^{9}+496a^{8}-532a^{7}+510a^{6}-434a^{5}+319a^{4}-202a^{3}+105a^{2}-41a+10$
| 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: | \( 58031.388562 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 58031.388562 \cdot 1}{2\cdot\sqrt{890830078698256823295734989}}\cr\approx \mathstrut & 0.37290156446 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 + 6*x^20 - 10*x^19 + 17*x^18 - 21*x^17 + 22*x^16 - 15*x^15 - 5*x^14 + 38*x^13 - 83*x^12 + 135*x^11 - 185*x^10 + 221*x^9 - 234*x^8 + 222*x^7 - 187*x^6 + 138*x^5 - 88*x^4 + 47*x^3 - 20*x^2 + 6*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 - 2*x^21 + 6*x^20 - 10*x^19 + 17*x^18 - 21*x^17 + 22*x^16 - 15*x^15 - 5*x^14 + 38*x^13 - 83*x^12 + 135*x^11 - 185*x^10 + 221*x^9 - 234*x^8 + 222*x^7 - 187*x^6 + 138*x^5 - 88*x^4 + 47*x^3 - 20*x^2 + 6*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 - 2*x^21 + 6*x^20 - 10*x^19 + 17*x^18 - 21*x^17 + 22*x^16 - 15*x^15 - 5*x^14 + 38*x^13 - 83*x^12 + 135*x^11 - 185*x^10 + 221*x^9 - 234*x^8 + 222*x^7 - 187*x^6 + 138*x^5 - 88*x^4 + 47*x^3 - 20*x^2 + 6*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 - 2*x^21 + 6*x^20 - 10*x^19 + 17*x^18 - 21*x^17 + 22*x^16 - 15*x^15 - 5*x^14 + 38*x^13 - 83*x^12 + 135*x^11 - 185*x^10 + 221*x^9 - 234*x^8 + 222*x^7 - 187*x^6 + 138*x^5 - 88*x^4 + 47*x^3 - 20*x^2 + 6*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
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 | ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.9.0.1}{9} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.14.0.1}{14} }{,}\,{\href{/padicField/3.8.0.1}{8} }$ | $17{,}\,{\href{/padicField/5.5.0.1}{5} }$ | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.10.0.1}{10} }$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.11.0.1}{11} }{,}\,{\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.13.0.1}{13} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.13.0.1}{13} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.13.0.1}{13} }{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.13.0.1}{13} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{5}$ | $18{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(34092855421\)
| $\Q_{34092855421}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{34092855421}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{34092855421}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 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 $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
|
\(26129523845912209\)
| $\Q_{26129523845912209}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |