Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 + 44*x^20 - 55*x^19 - 111*x^18 + 372*x^17 - 61*x^16 - 838*x^15 + 563*x^14 + 1153*x^13 - 1030*x^12 - 1191*x^11 + 1050*x^10 + 987*x^9 - 634*x^8 - 630*x^7 + 192*x^6 + 264*x^5 - 3*x^4 - 53*x^3 - 10*x^2 + x - 1)
gp: K = bnfinit(y^22 - 11*y^21 + 44*y^20 - 55*y^19 - 111*y^18 + 372*y^17 - 61*y^16 - 838*y^15 + 563*y^14 + 1153*y^13 - 1030*y^12 - 1191*y^11 + 1050*y^10 + 987*y^9 - 634*y^8 - 630*y^7 + 192*y^6 + 264*y^5 - 3*y^4 - 53*y^3 - 10*y^2 + y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 11*x^21 + 44*x^20 - 55*x^19 - 111*x^18 + 372*x^17 - 61*x^16 - 838*x^15 + 563*x^14 + 1153*x^13 - 1030*x^12 - 1191*x^11 + 1050*x^10 + 987*x^9 - 634*x^8 - 630*x^7 + 192*x^6 + 264*x^5 - 3*x^4 - 53*x^3 - 10*x^2 + x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 11*x^21 + 44*x^20 - 55*x^19 - 111*x^18 + 372*x^17 - 61*x^16 - 838*x^15 + 563*x^14 + 1153*x^13 - 1030*x^12 - 1191*x^11 + 1050*x^10 + 987*x^9 - 634*x^8 - 630*x^7 + 192*x^6 + 264*x^5 - 3*x^4 - 53*x^3 - 10*x^2 + x - 1)
\( x^{22} - 11 x^{21} + 44 x^{20} - 55 x^{19} - 111 x^{18} + 372 x^{17} - 61 x^{16} - 838 x^{15} + 563 x^{14} + \cdots - 1 \)
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: | $[6, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(11179894991688395891652264149\)
\(\medspace = 83\cdot 199\cdot 937\cdot 1583^{2}\cdot 2731^{2}\cdot 6217^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(18.83\) | 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: | $83^{1/2}199^{1/2}937^{1/2}1583^{1/2}2731^{1/2}6217^{1/2}\approx 644951594.8448861$ | ||
| Ramified primes: |
\(83\), \(199\), \(937\), \(1583\), \(2731\), \(6217\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{15476429}$) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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: |
$a^{2}-a-1$, $a^{18}-9a^{17}+27a^{16}-12a^{15}-91a^{14}+133a^{13}+111a^{12}-289a^{11}-78a^{10}+346a^{9}+65a^{8}-263a^{7}-75a^{6}+117a^{5}+57a^{4}-21a^{3}-17a^{2}-2a$, $a^{10}-5a^{9}+5a^{8}+10a^{7}-16a^{6}-8a^{5}+15a^{4}+5a^{3}-6a^{2}-a+1$, $a^{16}-8a^{15}+20a^{14}-71a^{12}+62a^{11}+102a^{10}-125a^{9}-101a^{8}+120a^{7}+84a^{6}-59a^{5}-49a^{4}+6a^{3}+14a^{2}+4a+1$, $a^{8}-4a^{7}+2a^{6}+8a^{5}-6a^{4}-6a^{3}+2a^{2}+3a$, $a^{18}-9a^{17}+26a^{16}-4a^{15}-112a^{14}+140a^{13}+168a^{12}-358a^{11}-131a^{10}+457a^{9}+90a^{8}-358a^{7}-78a^{6}+164a^{5}+53a^{4}-34a^{3}-16a^{2}+a-1$, $a^{17}-9a^{16}+27a^{15}-12a^{14}-91a^{13}+133a^{12}+112a^{11}-295a^{10}-69a^{9}+356a^{8}+35a^{7}-269a^{6}-34a^{5}+123a^{4}+26a^{3}-29a^{2}-8a+3$, $a^{18}-9a^{17}+27a^{16}-12a^{15}-91a^{14}+134a^{13}+105a^{12}-281a^{11}-63a^{10}+311a^{9}+49a^{8}-207a^{7}-57a^{6}+68a^{5}+39a^{4}-a^{3}-8a^{2}-4a-2$, $a^{20}-10a^{19}+36a^{18}-38a^{17}-88a^{16}+252a^{15}-41a^{14}-478a^{13}+358a^{12}+472a^{11}-564a^{10}-276a^{9}+491a^{8}+86a^{7}-267a^{6}-4a^{5}+91a^{4}-6a^{3}-17a^{2}+2a$, $a^{18}-8a^{17}+19a^{16}+7a^{15}-83a^{14}+42a^{13}+174a^{12}-121a^{11}-262a^{10}+162a^{9}+289a^{8}-110a^{7}-221a^{6}+22a^{5}+105a^{4}+11a^{3}-24a^{2}-6a+2$, $a^{19}-8a^{18}+18a^{17}+14a^{16}-96a^{15}+29a^{14}+231a^{13}-120a^{12}-371a^{11}+161a^{10}+434a^{9}-79a^{8}-348a^{7}-40a^{6}+162a^{5}+64a^{4}-28a^{3}-22a^{2}-3a$, $a^{18}-9a^{17}+27a^{16}-12a^{15}-91a^{14}+133a^{13}+111a^{12}-289a^{11}-79a^{10}+351a^{9}+60a^{8}-272a^{7}-62a^{6}+124a^{5}+50a^{4}-27a^{3}-18a^{2}-a+2$, $a^{21}-11a^{20}+45a^{19}-65a^{18}-76a^{17}+342a^{16}-167a^{15}-602a^{14}+632a^{13}+619a^{12}-925a^{11}-481a^{10}+828a^{9}+334a^{8}-478a^{7}-212a^{6}+170a^{5}+97a^{4}-30a^{3}-24a^{2}+2$
| 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: | \( 231636.713172 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{8}\cdot 231636.713172 \cdot 1}{2\cdot\sqrt{11179894991688395891652264149}}\cr\approx \mathstrut & 0.170285416855 \end{aligned}\] (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 + 44*x^20 - 55*x^19 - 111*x^18 + 372*x^17 - 61*x^16 - 838*x^15 + 563*x^14 + 1153*x^13 - 1030*x^12 - 1191*x^11 + 1050*x^10 + 987*x^9 - 634*x^8 - 630*x^7 + 192*x^6 + 264*x^5 - 3*x^4 - 53*x^3 - 10*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 + 44*x^20 - 55*x^19 - 111*x^18 + 372*x^17 - 61*x^16 - 838*x^15 + 563*x^14 + 1153*x^13 - 1030*x^12 - 1191*x^11 + 1050*x^10 + 987*x^9 - 634*x^8 - 630*x^7 + 192*x^6 + 264*x^5 - 3*x^4 - 53*x^3 - 10*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 + 44*x^20 - 55*x^19 - 111*x^18 + 372*x^17 - 61*x^16 - 838*x^15 + 563*x^14 + 1153*x^13 - 1030*x^12 - 1191*x^11 + 1050*x^10 + 987*x^9 - 634*x^8 - 630*x^7 + 192*x^6 + 264*x^5 - 3*x^4 - 53*x^3 - 10*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 + 44*x^20 - 55*x^19 - 111*x^18 + 372*x^17 - 61*x^16 - 838*x^15 + 563*x^14 + 1153*x^13 - 1030*x^12 - 1191*x^11 + 1050*x^10 + 987*x^9 - 634*x^8 - 630*x^7 + 192*x^6 + 264*x^5 - 3*x^4 - 53*x^3 - 10*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 $C_2^{10}.(C_2\times S_{11})$ |
| Character table for $C_2^{10}.(C_2\times S_{11})$ |
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 | ${\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.6.0.1}{6} }^{2}$ | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.7.0.1}{7} }^{2}$ | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.11.0.1}{11} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | $22$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.10.0.1}{10} }$ | $18{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.14.0.1}{14} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.11.0.1}{11} }^{2}$ | ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.10.0.1}{10} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(83\)
| 83.2.1.2 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 83.10.0.1 | $x^{10} + 7 x^{5} + 73 x^{3} + 53 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
| 83.10.0.1 | $x^{10} + 7 x^{5} + 73 x^{3} + 53 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
|
\(199\)
| 199.2.0.1 | $x^{2} + 193 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 199.2.1.1 | $x^{2} + 597$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 199.2.0.1 | $x^{2} + 193 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 199.4.0.1 | $x^{4} + 7 x^{2} + 162 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 199.12.0.1 | $x^{12} + 33 x^{7} + 192 x^{6} + 197 x^{5} + 138 x^{4} + 69 x^{3} + 57 x^{2} + 151 x + 3$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
|
\(937\)
| $\Q_{937}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{937}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
|
\(1583\)
| $\Q_{1583}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{1583}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
|
\(2731\)
| $\Q_{2731}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{2731}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{2731}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{2731}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 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 $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
|
\(6217\)
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |