Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^20 - x^18 - 2*x^16 + 18*x^12 - 24*x^10 - 3*x^8 + 36*x^6 - 27*x^4 + 4*x^2 + 1)
gp: K = bnfinit(y^22 - 2*y^20 - y^18 - 2*y^16 + 18*y^12 - 24*y^10 - 3*y^8 + 36*y^6 - 27*y^4 + 4*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 2*x^20 - x^18 - 2*x^16 + 18*x^12 - 24*x^10 - 3*x^8 + 36*x^6 - 27*x^4 + 4*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 2*x^20 - x^18 - 2*x^16 + 18*x^12 - 24*x^10 - 3*x^8 + 36*x^6 - 27*x^4 + 4*x^2 + 1)
\( x^{22} - 2x^{20} - x^{18} - 2x^{16} + 18x^{12} - 24x^{10} - 3x^{8} + 36x^{6} - 27x^{4} + 4x^{2} + 1 \)
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: | $[8, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-104070916689387515610916716544\)
\(\medspace = -\,2^{22}\cdot 19457^{2}\cdot 8095783^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(20.84\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(19457\), \(8095783\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{131749}a^{20}+\frac{5480}{131749}a^{18}+\frac{2587}{131749}a^{16}-\frac{46960}{131749}a^{14}+\frac{2826}{131749}a^{12}-\frac{54232}{131749}a^{10}+\frac{57645}{131749}a^{8}-\frac{55964}{131749}a^{6}+\frac{48809}{131749}a^{4}-\frac{11308}{131749}a^{2}+\frac{63327}{131749}$, $\frac{1}{131749}a^{21}+\frac{5480}{131749}a^{19}+\frac{2587}{131749}a^{17}-\frac{46960}{131749}a^{15}+\frac{2826}{131749}a^{13}-\frac{54232}{131749}a^{11}+\frac{57645}{131749}a^{9}-\frac{55964}{131749}a^{7}+\frac{48809}{131749}a^{5}-\frac{11308}{131749}a^{3}+\frac{63327}{131749}a$
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: | $14$ | 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: |
$\frac{9568}{131749}a^{20}-\frac{3462}{131749}a^{18}-\frac{16396}{131749}a^{16}-\frac{49190}{131749}a^{14}-\frac{101126}{131749}a^{12}+\frac{67535}{131749}a^{10}-\frac{85703}{131749}a^{8}-\frac{35616}{131749}a^{6}+\frac{217805}{131749}a^{4}-\frac{160764}{131749}a^{2}+\frac{130834}{131749}$, $a$, $\frac{298999}{131749}a^{21}-\frac{443040}{131749}a^{19}-\frac{514962}{131749}a^{17}-\frac{897357}{131749}a^{15}-\frac{462159}{131749}a^{13}+\frac{5096116}{131749}a^{11}-\frac{4481538}{131749}a^{9}-\frac{3033271}{131749}a^{7}+\frac{8832644}{131749}a^{5}-\frac{3431579}{131749}a^{3}-\frac{388356}{131749}a$, $\frac{641126}{131749}a^{20}-\frac{902346}{131749}a^{18}-\frac{1180940}{131749}a^{16}-\frac{1971715}{131749}a^{14}-\frac{1175913}{131749}a^{12}+\frac{10873278}{131749}a^{10}-\frac{8953146}{131749}a^{8}-\frac{7225995}{131749}a^{6}+\frac{18931808}{131749}a^{4}-\frac{6161039}{131749}a^{2}-\frac{1161873}{131749}$, $\frac{242886}{131749}a^{21}-\frac{308365}{131749}a^{19}-\frac{491895}{131749}a^{17}-\frac{810877}{131749}a^{15}-\frac{543450}{131749}a^{13}+\frac{4023938}{131749}a^{11}-\frac{2832987}{131749}a^{9}-\frac{3226252}{131749}a^{7}+\frac{6835204}{131749}a^{5}-\frac{1300975}{131749}a^{3}-\frac{849275}{131749}a$, $\frac{235865}{131749}a^{20}-\frac{312737}{131749}a^{18}-\frac{473860}{131749}a^{16}-\frac{740715}{131749}a^{14}-\frac{490697}{131749}a^{12}+\frac{4032200}{131749}a^{10}-\frac{2957353}{131749}a^{8}-\frac{3046777}{131749}a^{6}+\frac{6958113}{131749}a^{4}-\frac{1747401}{131749}a^{2}-\frac{683518}{131749}$, $\frac{551453}{131749}a^{20}-\frac{754867}{131749}a^{18}-\frac{1023253}{131749}a^{16}-\frac{1757424}{131749}a^{14}-\frac{1106735}{131749}a^{12}+\frac{9187589}{131749}a^{10}-\frac{7400228}{131749}a^{8}-\frac{6295139}{131749}a^{6}+\frac{15885653}{131749}a^{4}-\frac{4893318}{131749}a^{2}-\frac{975048}{131749}$, $\frac{15674}{131749}a^{20}-\frac{6828}{131749}a^{18}-\frac{30054}{131749}a^{16}-\frac{101126}{131749}a^{14}-\frac{104689}{131749}a^{12}+\frac{143929}{131749}a^{10}-\frac{6912}{131749}a^{8}-\frac{126643}{131749}a^{6}+\frac{97572}{131749}a^{4}+\frac{92562}{131749}a^{2}+a+\frac{122181}{131749}$, $\frac{251539}{131749}a^{21}+\frac{15674}{131749}a^{20}-\frac{319565}{131749}a^{19}-\frac{6828}{131749}a^{18}-\frac{503914}{131749}a^{17}-\frac{30054}{131749}a^{16}-\frac{841841}{131749}a^{15}-\frac{101126}{131749}a^{14}-\frac{595386}{131749}a^{13}-\frac{104689}{131749}a^{12}+\frac{4176129}{131749}a^{11}+\frac{143929}{131749}a^{10}-\frac{2964265}{131749}a^{9}-\frac{6912}{131749}a^{8}-\frac{3173420}{131749}a^{7}-\frac{126643}{131749}a^{6}+\frac{7055685}{131749}a^{5}+\frac{97572}{131749}a^{4}-\frac{1654839}{131749}a^{3}+\frac{92562}{131749}a^{2}-\frac{561337}{131749}a+\frac{122181}{131749}$, $\frac{227019}{131749}a^{20}-\frac{305185}{131749}a^{18}-\frac{434136}{131749}a^{16}-\frac{737152}{131749}a^{14}-\frac{457183}{131749}a^{12}+\frac{3806865}{131749}a^{10}-\frac{2884644}{131749}a^{8}-\frac{2706728}{131749}a^{6}+\frac{6408176}{131749}a^{4}-\frac{1714324}{131749}a^{2}-a-\frac{413914}{131749}$, $\frac{1030124}{131749}a^{21}-\frac{641126}{131749}a^{20}-\frac{1419122}{131749}a^{19}+\frac{902346}{131749}a^{18}-\frac{1932470}{131749}a^{17}+\frac{1180940}{131749}a^{16}-\frac{3241188}{131749}a^{15}+\frac{1971715}{131749}a^{14}-\frac{1971715}{131749}a^{13}+\frac{1175913}{131749}a^{12}+\frac{17366319}{131749}a^{11}-\frac{10873278}{131749}a^{10}-\frac{13849698}{131749}a^{9}+\frac{8953146}{131749}a^{8}-\frac{12043518}{131749}a^{7}+\frac{7225995}{131749}a^{6}+\frac{29858469}{131749}a^{5}-\frac{18931808}{131749}a^{4}-\frac{8881540}{131749}a^{3}+\frac{6161039}{131749}a^{2}-\frac{2040543}{131749}a+\frac{1030124}{131749}$, $\frac{178049}{131749}a^{21}+\frac{306020}{131749}a^{20}-\frac{288072}{131749}a^{19}-\frac{438668}{131749}a^{18}-\frac{245239}{131749}a^{17}-\frac{532997}{131749}a^{16}-\frac{521249}{131749}a^{15}-\frac{967519}{131749}a^{14}-\frac{246455}{131749}a^{13}-\frac{514912}{131749}a^{12}+\frac{2961069}{131749}a^{11}+\frac{5087854}{131749}a^{10}-\frac{3169718}{131749}a^{9}-\frac{4357172}{131749}a^{8}-\frac{1079609}{131749}a^{7}-\frac{3212746}{131749}a^{6}+\frac{5236063}{131749}a^{5}+\frac{8709735}{131749}a^{4}-\frac{2888352}{131749}a^{3}-\frac{2985153}{131749}a^{2}+\frac{493101}{131749}a-\frac{422364}{131749}$, $\frac{551453}{131749}a^{21}+\frac{6106}{131749}a^{20}-\frac{754867}{131749}a^{19}-\frac{3366}{131749}a^{18}-\frac{1023253}{131749}a^{17}-\frac{13658}{131749}a^{16}-\frac{1757424}{131749}a^{15}-\frac{51936}{131749}a^{14}-\frac{1106735}{131749}a^{13}-\frac{3563}{131749}a^{12}+\frac{9187589}{131749}a^{11}+\frac{76394}{131749}a^{10}-\frac{7400228}{131749}a^{9}+\frac{78791}{131749}a^{8}-\frac{6295139}{131749}a^{7}-\frac{91027}{131749}a^{6}+\frac{15885653}{131749}a^{5}-\frac{120233}{131749}a^{4}-\frac{4893318}{131749}a^{3}+\frac{385075}{131749}a^{2}-\frac{975048}{131749}a-\frac{8653}{131749}$, $\frac{943111}{131749}a^{21}-\frac{560219}{131749}a^{20}-\frac{1318982}{131749}a^{19}+\frac{805572}{131749}a^{18}-\frac{1744311}{131749}a^{17}+\frac{1006439}{131749}a^{16}-\frac{2910696}{131749}a^{15}+\frac{1693159}{131749}a^{14}-\frac{1763321}{131749}a^{13}+\frac{971082}{131749}a^{12}+\frac{15952163}{131749}a^{11}-\frac{9535516}{131749}a^{10}-\frac{13107410}{131749}a^{9}+\frac{8132067}{131749}a^{8}-\frac{10837034}{131749}a^{7}+\frac{6110538}{131749}a^{6}+\frac{27661983}{131749}a^{5}-\frac{16615089}{131749}a^{4}-\frac{8971817}{131749}a^{3}+\frac{5866241}{131749}a^{2}-\frac{1647371}{131749}a+\frac{777354}{131749}$
| 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: | \( 967898.03203 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{7}\cdot 967898.03203 \cdot 1}{2\cdot\sqrt{104070916689387515610916716544}}\cr\approx \mathstrut & 0.14846837836 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^20 - x^18 - 2*x^16 + 18*x^12 - 24*x^10 - 3*x^8 + 36*x^6 - 27*x^4 + 4*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 - 2*x^20 - x^18 - 2*x^16 + 18*x^12 - 24*x^10 - 3*x^8 + 36*x^6 - 27*x^4 + 4*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 - 2*x^20 - x^18 - 2*x^16 + 18*x^12 - 24*x^10 - 3*x^8 + 36*x^6 - 27*x^4 + 4*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 - 2*x^20 - x^18 - 2*x^16 + 18*x^12 - 24*x^10 - 3*x^8 + 36*x^6 - 27*x^4 + 4*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 $C_2^{10}.(C_2\times S_{11})$ |
| Character table for $C_2^{10}.(C_2\times S_{11})$ |
Intermediate fields
| 11.5.157519649831.1 |
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 | R | $22$ | ${\href{/padicField/5.11.0.1}{11} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.4.0.1}{4} }^{3}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}{,}\,{\href{/padicField/11.6.0.1}{6} }$ | $20{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.11.0.1}{11} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.9.0.1}{9} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.11.0.1}{11} }^{2}$ | $18{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | $18{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | $18{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $22$ | $2$ | $11$ | $22$ | |||
|
\(19457\)
| $\Q_{19457}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{19457}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
|
\(8095783\)
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |