Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 2*x^20 - x^16 - 7*x^14 - 14*x^12 - x^10 + 9*x^8 + 15*x^6 + 22*x^4 + 10*x^2 + 1)
gp: K = bnfinit(y^22 + 2*y^20 - y^16 - 7*y^14 - 14*y^12 - y^10 + 9*y^8 + 15*y^6 + 22*y^4 + 10*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + 2*x^20 - x^16 - 7*x^14 - 14*x^12 - x^10 + 9*x^8 + 15*x^6 + 22*x^4 + 10*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 2*x^20 - x^16 - 7*x^14 - 14*x^12 - x^10 + 9*x^8 + 15*x^6 + 22*x^4 + 10*x^2 + 1)
\( x^{22} + 2x^{20} - x^{16} - 7x^{14} - 14x^{12} - x^{10} + 9x^{8} + 15x^{6} + 22x^{4} + 10x^{2} + 1 \)
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: | $[0, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-117045760309296676701139369984\)
\(\medspace = -\,2^{22}\cdot 2089^{2}\cdot 79966751^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(20.96\) | 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: |
\(2\), \(2089\), \(79966751\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5}a^{18}+\frac{2}{5}a^{16}-\frac{2}{5}a^{14}+\frac{2}{5}a^{10}+\frac{1}{5}a^{8}+\frac{2}{5}a^{4}-\frac{2}{5}$, $\frac{1}{5}a^{19}+\frac{2}{5}a^{17}-\frac{2}{5}a^{15}+\frac{2}{5}a^{11}+\frac{1}{5}a^{9}+\frac{2}{5}a^{5}-\frac{2}{5}a$, $\frac{1}{1315}a^{20}-\frac{12}{1315}a^{18}-\frac{19}{263}a^{16}-\frac{512}{1315}a^{14}-\frac{203}{1315}a^{12}+\frac{198}{1315}a^{10}+\frac{646}{1315}a^{8}-\frac{93}{1315}a^{6}+\frac{2}{1315}a^{4}-\frac{532}{1315}a^{2}-\frac{432}{1315}$, $\frac{1}{1315}a^{21}-\frac{12}{1315}a^{19}-\frac{19}{263}a^{17}-\frac{512}{1315}a^{15}-\frac{203}{1315}a^{13}+\frac{198}{1315}a^{11}+\frac{646}{1315}a^{9}-\frac{93}{1315}a^{7}+\frac{2}{1315}a^{5}-\frac{532}{1315}a^{3}-\frac{432}{1315}a$
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: | $10$ | 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: |
$\frac{597}{1315}a^{20}+\frac{463}{1315}a^{18}-\frac{696}{1315}a^{16}-\frac{58}{1315}a^{14}-\frac{4156}{1315}a^{12}-\frac{660}{263}a^{10}+\frac{4049}{1315}a^{8}+\frac{2339}{1315}a^{6}+\frac{5928}{1315}a^{4}+\frac{4571}{1315}a^{2}-\frac{953}{1315}$, $a^{21}+2a^{19}-a^{15}-7a^{13}-14a^{11}-a^{9}+9a^{7}+15a^{5}+22a^{3}+10a$, $\frac{16}{263}a^{21}+\frac{71}{263}a^{19}+\frac{58}{263}a^{17}-\frac{39}{263}a^{15}-\frac{92}{263}a^{13}-\frac{514}{263}a^{11}-\frac{447}{263}a^{9}+\frac{353}{263}a^{7}+\frac{295}{263}a^{5}+\frac{956}{263}a^{3}+\frac{978}{263}a$, $\frac{1971}{1315}a^{21}+\frac{2911}{1315}a^{19}-\frac{1304}{1315}a^{17}-\frac{1073}{1315}a^{15}-\frac{13503}{1315}a^{13}-\frac{20811}{1315}a^{11}+\frac{7184}{1315}a^{9}+\frac{12632}{1315}a^{7}+\frac{24193}{1315}a^{5}+\frac{32358}{1315}a^{3}+\frac{6697}{1315}a$, $\frac{1374}{1315}a^{21}+\frac{2448}{1315}a^{19}-\frac{608}{1315}a^{17}-\frac{203}{263}a^{15}-\frac{9347}{1315}a^{13}-\frac{17511}{1315}a^{11}+\frac{627}{263}a^{9}+\frac{10293}{1315}a^{7}+\frac{3653}{263}a^{5}+\frac{27787}{1315}a^{3}+\frac{1530}{263}a$, $\frac{836}{1315}a^{20}+\frac{1014}{1315}a^{18}-\frac{783}{1315}a^{16}-\frac{394}{1315}a^{14}-\frac{5333}{1315}a^{12}-\frac{1400}{263}a^{10}+\frac{4062}{1315}a^{8}+\frac{5097}{1315}a^{6}+\frac{7984}{1315}a^{4}+\frac{10238}{1315}a^{2}+\frac{2051}{1315}$, $\frac{141}{1315}a^{20}+\frac{412}{1315}a^{18}+\frac{18}{1315}a^{16}-\frac{26}{263}a^{14}-\frac{1008}{1315}a^{12}-\frac{3379}{1315}a^{10}-\frac{35}{263}a^{8}+\frac{1352}{1315}a^{6}+\frac{635}{263}a^{4}+\frac{6518}{1315}a^{2}+\frac{389}{263}$, $\frac{977}{1315}a^{21}+\frac{146}{263}a^{20}+\frac{1426}{1315}a^{19}+\frac{1497}{1315}a^{18}-\frac{153}{263}a^{17}-\frac{181}{1315}a^{16}-\frac{524}{1315}a^{15}-\frac{1089}{1315}a^{14}-\frac{6341}{1315}a^{13}-\frac{971}{263}a^{12}-\frac{10379}{1315}a^{11}-\frac{9841}{1315}a^{10}+\frac{3887}{1315}a^{9}+\frac{547}{1315}a^{8}+\frac{6449}{1315}a^{7}+\frac{1676}{263}a^{6}+\frac{9844}{1315}a^{5}+\frac{10139}{1315}a^{4}+\frac{16756}{1315}a^{3}+\frac{2543}{263}a^{2}+\frac{5311}{1315}a+\frac{3396}{1315}$, $\frac{474}{1315}a^{21}-\frac{159}{1315}a^{20}+\frac{887}{1315}a^{19}-\frac{196}{1315}a^{18}-\frac{64}{263}a^{17}+\frac{377}{1315}a^{16}-\frac{728}{1315}a^{15}+\frac{141}{1315}a^{14}-\frac{2857}{1315}a^{13}+\frac{717}{1315}a^{12}-\frac{6088}{1315}a^{11}+\frac{226}{263}a^{10}+\frac{1124}{1315}a^{9}-\frac{2248}{1315}a^{8}+\frac{5888}{1315}a^{7}-\frac{993}{1315}a^{6}+\frac{4893}{1315}a^{5}+\frac{734}{1315}a^{4}+\frac{8202}{1315}a^{3}-\frac{887}{1315}a^{2}+\frac{3002}{1315}a+\frac{571}{1315}$, $\frac{191}{1315}a^{21}+\frac{234}{1315}a^{20}+\frac{15}{263}a^{19}+\frac{17}{263}a^{18}-\frac{261}{1315}a^{17}-\frac{664}{1315}a^{16}+\frac{44}{1315}a^{15}-\frac{669}{1315}a^{14}-\frac{1953}{1315}a^{13}-\frac{1477}{1315}a^{12}-\frac{2158}{1315}a^{11}-\frac{482}{1315}a^{10}+\frac{828}{1315}a^{9}+\frac{1517}{1315}a^{8}+\frac{1962}{1315}a^{7}+\frac{1908}{1315}a^{6}+\frac{3801}{1315}a^{5}+\frac{3624}{1315}a^{4}+\frac{3588}{1315}a^{3}+\frac{1752}{1315}a^{2}+\frac{2174}{1315}a+\frac{956}{1315}$
| 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: | \( 229900.23166 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 229900.23166 \cdot 1}{2\cdot\sqrt{117045760309296676701139369984}}\cr\approx \mathstrut & 0.20244642950 \end{aligned}\] (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^16 - 7*x^14 - 14*x^12 - x^10 + 9*x^8 + 15*x^6 + 22*x^4 + 10*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^16 - 7*x^14 - 14*x^12 - x^10 + 9*x^8 + 15*x^6 + 22*x^4 + 10*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^16 - 7*x^14 - 14*x^12 - x^10 + 9*x^8 + 15*x^6 + 22*x^4 + 10*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^16 - 7*x^14 - 14*x^12 - x^10 + 9*x^8 + 15*x^6 + 22*x^4 + 10*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
| 11.5.167050542839.1 |
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 | $22$ | ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | $22$ | $16{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | $16{,}\,{\href{/padicField/13.6.0.1}{6} }$ | ${\href{/padicField/17.11.0.1}{11} }^{2}$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.14.0.1}{14} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | $22$ | $18{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{5}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.7.0.1}{7} }^{2}{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.5.0.1}{5} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $22$ | $2$ | $11$ | $22$ | |||
|
\(2089\)
| $\Q_{2089}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{2089}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
|
\(79966751\)
| 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 $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |