Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 + 64*x^20 - 255*x^19 + 770*x^18 - 1857*x^17 + 3688*x^16 - 6146*x^15 + 8690*x^14 - 10480*x^13 + 10777*x^12 - 9386*x^11 + 6807*x^10 - 3960*x^9 + 1680*x^8 - 341*x^7 - 170*x^6 + 212*x^5 - 109*x^4 + 28*x^3 + 3*x^2 - 5*x + 1)
gp: K = bnfinit(y^22 - 11*y^21 + 64*y^20 - 255*y^19 + 770*y^18 - 1857*y^17 + 3688*y^16 - 6146*y^15 + 8690*y^14 - 10480*y^13 + 10777*y^12 - 9386*y^11 + 6807*y^10 - 3960*y^9 + 1680*y^8 - 341*y^7 - 170*y^6 + 212*y^5 - 109*y^4 + 28*y^3 + 3*y^2 - 5*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 11*x^21 + 64*x^20 - 255*x^19 + 770*x^18 - 1857*x^17 + 3688*x^16 - 6146*x^15 + 8690*x^14 - 10480*x^13 + 10777*x^12 - 9386*x^11 + 6807*x^10 - 3960*x^9 + 1680*x^8 - 341*x^7 - 170*x^6 + 212*x^5 - 109*x^4 + 28*x^3 + 3*x^2 - 5*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 11*x^21 + 64*x^20 - 255*x^19 + 770*x^18 - 1857*x^17 + 3688*x^16 - 6146*x^15 + 8690*x^14 - 10480*x^13 + 10777*x^12 - 9386*x^11 + 6807*x^10 - 3960*x^9 + 1680*x^8 - 341*x^7 - 170*x^6 + 212*x^5 - 109*x^4 + 28*x^3 + 3*x^2 - 5*x + 1)
\( x^{22} - 11 x^{21} + 64 x^{20} - 255 x^{19} + 770 x^{18} - 1857 x^{17} + 3688 x^{16} - 6146 x^{15} + \cdots + 1 \)
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: | $[6, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(440358787157109245022232573\)
\(\medspace = 37\cdot 163^{2}\cdot 14281\cdot 177106931^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(16.26\) | 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: | $37^{1/2}163^{1/2}14281^{1/2}177106931^{1/2}\approx 123507051.11934274$ | ||
| Ramified primes: |
\(37\), \(163\), \(14281\), \(177106931\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{528397}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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: | $13$ | 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: |
$a^{20}-10a^{19}+53a^{18}-192a^{17}+525a^{16}-1140a^{15}+2023a^{14}-2983a^{13}+3684a^{12}-3813a^{11}+3280a^{10}-2293a^{9}+1234a^{8}-433a^{7}+13a^{6}+105a^{5}-78a^{4}+29a^{3}-2a^{2}-3a+2$, $a^{18}-9a^{17}+43a^{16}-140a^{15}+343a^{14}-665a^{13}+1049a^{12}-1367a^{11}+1479a^{10}-1323a^{9}+961a^{8}-543a^{7}+212a^{6}-30a^{5}-28a^{4}+25a^{3}-10a^{2}+2a+1$, $a^{20}-10a^{19}+54a^{18}-201a^{17}+569a^{16}-1288a^{15}+2400a^{14}-3746a^{13}+4945a^{12}-5542a^{11}+5262a^{10}-4195a^{9}+2751a^{8}-1420a^{7}+513a^{6}-69a^{5}-59a^{4}+51a^{3}-19a^{2}+3a+2$, $a^{17}-8a^{16}+35a^{15}-105a^{14}+238a^{13}-427a^{12}+623a^{11}-750a^{10}+749a^{9}-619a^{8}+417a^{7}-222a^{6}+85a^{5}-17a^{4}-6a^{3}+6a^{2}-4a$, $a^{17}-9a^{16}+43a^{15}-140a^{14}+343a^{13}-665a^{12}+1050a^{11}-1373a^{10}+1499a^{9}-1368a^{8}+1036a^{7}-639a^{6}+307a^{5}-102a^{4}+11a^{3}+12a^{2}-10a+4$, $a^{20}-10a^{19}+53a^{18}-192a^{17}+526a^{16}-1148a^{15}+2058a^{14}-3088a^{13}+3922a^{12}-4240a^{11}+3903a^{10}-3043a^{9}+1983a^{8}-1052a^{7}+429a^{6}-114a^{5}+2a^{4}+17a^{3}-10a^{2}+3a-1$, $a^{20}-10a^{19}+54a^{18}-201a^{17}+569a^{16}-1288a^{15}+2400a^{14}-3746a^{13}+4945a^{12}-5542a^{11}+5262a^{10}-4195a^{9}+2752a^{8}-1424a^{7}+522a^{6}-82a^{5}-46a^{4}+42a^{3}-16a^{2}+3a+1$, $a^{21}-10a^{20}+53a^{19}-192a^{18}+525a^{17}-1139a^{16}+2014a^{15}-2941a^{14}+3552a^{13}-3504a^{12}+2712a^{11}-1449a^{10}+203a^{9}+609a^{8}-856a^{7}+694a^{6}-390a^{5}+146a^{4}-21a^{3}-12a^{2}+11a-3$, $2a^{21}-21a^{20}+118a^{19}-456a^{18}+1339a^{17}-3145a^{16}+6088a^{15}-9892a^{14}+13634a^{13}-16016a^{12}+16019a^{11}-13537a^{10}+9489a^{9}-5303a^{8}+2137a^{7}-395a^{6}-207a^{5}+228a^{4}-101a^{3}+20a^{2}+7a-3$, $a^{21}-11a^{20}+64a^{19}-255a^{18}+769a^{17}-1849a^{16}+3653a^{15}-6041a^{14}+8452a^{13}-10053a^{12}+10155a^{11}-8642a^{10}+6078a^{9}-3386a^{8}+1338a^{7}-214a^{6}-163a^{5}+162a^{4}-69a^{3}+12a^{2}+6a-3$, $3a^{21}-32a^{20}+181a^{19}-701a^{18}+2056a^{17}-4810a^{16}+9250a^{15}-14890a^{14}+20267a^{13}-23415a^{12}+22900a^{11}-18748a^{10}+12514a^{9}-6395a^{8}+2036a^{7}+128a^{6}-664a^{5}+468a^{4}-172a^{3}+18a^{2}+20a-10$, $2a^{21}-21a^{20}+117a^{19}-447a^{18}+1295a^{17}-2996a^{16}+5703a^{15}-9094a^{14}+12269a^{13}-14055a^{12}+13630a^{11}-11057a^{10}+7296a^{9}-3659a^{8}+1104a^{7}+136a^{6}-419a^{5}+282a^{4}-100a^{3}+9a^{2}+12a-5$, $a^{21}-10a^{20}+54a^{19}-202a^{18}+579a^{17}-1340a^{16}+2583a^{15}-4228a^{14}+5944a^{13}-7216a^{12}+7560a^{11}-6786a^{10}+5129a^{9}-3149a^{8}+1442a^{7}-358a^{6}-99a^{5}+159a^{4}-79a^{3}+14a^{2}+7a-3$
| 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: | \( 42752.2517797 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{8}\cdot 42752.2517797 \cdot 1}{2\cdot\sqrt{440358787157109245022232573}}\cr\approx \mathstrut & 0.158359669199 \end{aligned}\] (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 + 64*x^20 - 255*x^19 + 770*x^18 - 1857*x^17 + 3688*x^16 - 6146*x^15 + 8690*x^14 - 10480*x^13 + 10777*x^12 - 9386*x^11 + 6807*x^10 - 3960*x^9 + 1680*x^8 - 341*x^7 - 170*x^6 + 212*x^5 - 109*x^4 + 28*x^3 + 3*x^2 - 5*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 + 64*x^20 - 255*x^19 + 770*x^18 - 1857*x^17 + 3688*x^16 - 6146*x^15 + 8690*x^14 - 10480*x^13 + 10777*x^12 - 9386*x^11 + 6807*x^10 - 3960*x^9 + 1680*x^8 - 341*x^7 - 170*x^6 + 212*x^5 - 109*x^4 + 28*x^3 + 3*x^2 - 5*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 + 64*x^20 - 255*x^19 + 770*x^18 - 1857*x^17 + 3688*x^16 - 6146*x^15 + 8690*x^14 - 10480*x^13 + 10777*x^12 - 9386*x^11 + 6807*x^10 - 3960*x^9 + 1680*x^8 - 341*x^7 - 170*x^6 + 212*x^5 - 109*x^4 + 28*x^3 + 3*x^2 - 5*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 + 64*x^20 - 255*x^19 + 770*x^18 - 1857*x^17 + 3688*x^16 - 6146*x^15 + 8690*x^14 - 10480*x^13 + 10777*x^12 - 9386*x^11 + 6807*x^10 - 3960*x^9 + 1680*x^8 - 341*x^7 - 170*x^6 + 212*x^5 - 109*x^4 + 28*x^3 + 3*x^2 - 5*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
| 11.3.28868429753.1 |
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 | $22$ | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.5.0.1}{5} }^{2}$ | $18{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.11.0.1}{11} }^{2}$ | ${\href{/padicField/11.11.0.1}{11} }^{2}$ | ${\href{/padicField/13.9.0.1}{9} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | $16{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | $16{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.11.0.1}{11} }^{2}$ | R | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.9.0.1}{9} }^{2}{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(37\)
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.2.1.2 | $x^{2} + 74$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.6.0.1 | $x^{6} + 35 x^{3} + 4 x^{2} + 30 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 37.6.0.1 | $x^{6} + 35 x^{3} + 4 x^{2} + 30 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(163\)
| 163.2.1.2 | $x^{2} + 163$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 163.2.1.2 | $x^{2} + 163$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 163.4.0.1 | $x^{4} + 8 x^{2} + 91 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 163.4.0.1 | $x^{4} + 8 x^{2} + 91 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 163.10.0.1 | $x^{10} + 3 x^{6} + 111 x^{5} + 120 x^{4} + 125 x^{3} + 15 x^{2} + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
|
\(14281\)
| 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 $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
|
\(177106931\)
| $\Q_{177106931}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{177106931}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |