Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^20 - x^18 + 4*x^16 - x^14 - 5*x^12 - x^10 + 8*x^8 + 3*x^6 - 5*x^4 - x^2 + 1)
gp: K = bnfinit(y^22 - 2*y^20 - y^18 + 4*y^16 - y^14 - 5*y^12 - y^10 + 8*y^8 + 3*y^6 - 5*y^4 - y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 2*x^20 - x^18 + 4*x^16 - x^14 - 5*x^12 - x^10 + 8*x^8 + 3*x^6 - 5*x^4 - x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 2*x^20 - x^18 + 4*x^16 - x^14 - 5*x^12 - x^10 + 8*x^8 + 3*x^6 - 5*x^4 - x^2 + 1)
\( x^{22} - 2x^{20} - x^{18} + 4x^{16} - x^{14} - 5x^{12} - x^{10} + 8x^{8} + 3x^{6} - 5x^{4} - x^{2} + 1 \)
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: |
\(-2741869623221317890608793124864\)
\(\medspace = -\,2^{22}\cdot 43^{2}\cdot 547^{2}\cdot 34374601^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(24.19\) | 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: |
\(2\), \(43\), \(547\), \(34374601\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$
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: | $14$ | 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: |
$a$, $a^{20}-2a^{18}-a^{16}+4a^{14}-a^{12}-5a^{10}-a^{8}+8a^{6}+3a^{4}-4a^{2}-1$, $3a^{18}-7a^{16}+10a^{12}-6a^{10}-10a^{8}-2a^{6}+22a^{4}+3a^{2}-9$, $13a^{20}-29a^{18}-a^{16}+41a^{14}-24a^{12}-43a^{10}-11a^{8}+88a^{6}+12a^{4}-33a^{2}+3$, $9a^{21}-21a^{19}+a^{17}+29a^{15}-19a^{13}-29a^{11}-5a^{9}+64a^{7}+3a^{5}-26a^{3}+3a$, $4a^{20}-3a^{18}-13a^{16}+11a^{14}+11a^{12}-22a^{10}-24a^{8}+20a^{6}+43a^{4}-a^{2}-13$, $a^{21}+6a^{19}-18a^{17}+a^{15}+25a^{13}-17a^{11}-30a^{9}-a^{7}+58a^{5}+9a^{3}-22a$, $8a^{21}-24a^{19}+13a^{17}+26a^{15}-34a^{13}-16a^{11}+14a^{9}+60a^{7}-35a^{5}-27a^{3}+17a$, $a-1$, $8a^{21}-24a^{19}+12a^{17}+28a^{15}-34a^{13}-18a^{11}+15a^{9}+63a^{7}-32a^{5}-33a^{3}+16a-1$, $14a^{21}-8a^{20}-26a^{19}+16a^{18}-13a^{17}+5a^{16}+44a^{15}-25a^{14}-9a^{13}+8a^{12}-56a^{11}+30a^{10}-30a^{9}+14a^{8}+91a^{7}-54a^{6}+51a^{5}-22a^{4}-31a^{3}+19a^{2}-12a+5$, $5a^{21}+9a^{20}-14a^{19}-21a^{18}+5a^{17}+a^{16}+18a^{15}+29a^{14}-18a^{13}-19a^{12}-14a^{11}-29a^{10}+7a^{9}-5a^{8}+39a^{7}+64a^{6}-14a^{5}+3a^{4}-20a^{3}-26a^{2}+7a+4$, $5a^{21}+a^{20}-5a^{19}-14a^{17}-5a^{16}+15a^{15}+3a^{14}+10a^{13}+5a^{12}-27a^{11}-7a^{10}-25a^{9}-9a^{8}+27a^{7}+4a^{6}+47a^{5}+17a^{4}-4a^{3}-14a-6$, $8a^{21}+6a^{20}-17a^{19}-14a^{18}-3a^{17}+a^{16}+26a^{15}+19a^{14}-12a^{13}-13a^{12}-29a^{11}-19a^{10}-9a^{9}-3a^{8}+55a^{7}+41a^{6}+15a^{5}-21a^{3}-16a^{2}-a+3$
| 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: | \( 6351542.86721 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{7}\cdot 6351542.86721 \cdot 1}{2\cdot\sqrt{2741869623221317890608793124864}}\cr\approx \mathstrut & 0.189812531631 \end{aligned}\] (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 + 4*x^16 - x^14 - 5*x^12 - x^10 + 8*x^8 + 3*x^6 - 5*x^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 + 4*x^16 - x^14 - 5*x^12 - x^10 + 8*x^8 + 3*x^6 - 5*x^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 + 4*x^16 - x^14 - 5*x^12 - x^10 + 8*x^8 + 3*x^6 - 5*x^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 + 4*x^16 - x^14 - 5*x^12 - x^10 + 8*x^8 + 3*x^6 - 5*x^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 $C_2^{10}.(C_2\times S_{11})$ are not computed |
| Character table for $C_2^{10}.(C_2\times S_{11})$ is not computed |
Intermediate fields
| 11.7.808524990121.1 |
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 | $22$ | ${\href{/padicField/5.11.0.1}{11} }^{2}$ | $22$ | ${\href{/padicField/11.9.0.1}{9} }^{2}{,}\,{\href{/padicField/11.4.0.1}{4} }$ | ${\href{/padicField/13.7.0.1}{7} }^{2}{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.11.0.1}{11} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.11.0.1}{11} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 $22$ | $2$ | $11$ | $22$ | |||
|
\(43\)
| 43.2.1.2 | $x^{2} + 43$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 43.2.1.2 | $x^{2} + 43$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 43.14.0.1 | $x^{14} + 38 x^{7} + 22 x^{6} + 24 x^{5} + 37 x^{4} + 18 x^{3} + 4 x^{2} + 19 x + 3$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
|
\(547\)
| $\Q_{547}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{547}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
|
\(34374601\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |