Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 8*x + 6)
gp: K = bnfinit(y^22 - 8*y + 6, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 8*x + 6);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 8*x + 6)
\( x^{22} - 8x + 6 \)
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: | $[2, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(423616943333188060347578772565944553233123901440\)
\(\medspace = 2^{43}\cdot 3^{21}\cdot 5\cdot 67\cdot 43223\cdot 317963629399156207\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(146.17\) | 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: |
\(2\), \(3\), \(5\), \(67\), \(43223\), \(317963629399156207\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{27624\!\cdots\!73610}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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
$C_{2}$, which has order $2$ (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: | $11$ | 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: |
$a-1$, $a^{15}-2a^{8}+1$, $10a^{21}+7a^{20}+9a^{19}-13a^{18}+15a^{17}-27a^{16}+14a^{15}-23a^{14}-a^{13}+14a^{12}-35a^{11}+50a^{10}-58a^{9}+62a^{8}-11a^{7}+17a^{6}+57a^{5}-78a^{4}+110a^{3}-105a^{2}+112a-149$, $20a^{21}+11a^{20}+15a^{19}+16a^{18}-12a^{17}-23a^{16}-9a^{15}-11a^{14}-21a^{13}+15a^{12}+46a^{11}+13a^{10}+11a^{9}+31a^{8}-12a^{7}-74a^{6}-30a^{5}+7a^{4}-52a^{3}+8a^{2}+113a-97$, $18a^{21}+9a^{20}-17a^{19}-2a^{18}+9a^{17}+7a^{16}+2a^{15}-28a^{14}-3a^{13}+50a^{12}-21a^{11}-58a^{10}+46a^{9}+36a^{8}-36a^{7}-10a^{6}-14a^{5}+46a^{4}+73a^{3}-128a^{2}-46a+67$, $23a^{21}+26a^{20}+21a^{19}+9a^{18}+2a^{17}-4a^{16}+3a^{15}+24a^{14}+39a^{13}+51a^{12}+39a^{11}+5a^{10}-33a^{9}-88a^{8}-116a^{7}-111a^{6}-88a^{5}-23a^{4}+29a^{3}+81a^{2}+129a-85$, $1724a^{21}+3727a^{20}-1796a^{19}-4821a^{18}+1725a^{17}+6137a^{16}-1525a^{15}-7777a^{14}+1069a^{13}+9734a^{12}-311a^{11}-12066a^{10}-833a^{9}+14848a^{8}+2503a^{7}-18165a^{6}-4951a^{5}+22013a^{4}+8428a^{3}-26368a^{2}-13168a+17467$, $43a^{21}+62a^{20}+4a^{19}+87a^{18}+18a^{17}+82a^{16}+33a^{15}+39a^{14}+124a^{13}-8a^{12}+139a^{11}-44a^{10}+169a^{9}+35a^{8}+17a^{7}+166a^{6}-65a^{5}+339a^{4}-182a^{3}+259a^{2}+20a-233$, $2360a^{21}+2364a^{20}+2440a^{19}+2685a^{18}+2887a^{17}+2845a^{16}+2778a^{15}+2966a^{14}+3271a^{13}+3473a^{12}+3509a^{11}+3396a^{10}+3409a^{9}+3915a^{8}+4457a^{7}+4276a^{6}+3839a^{5}+4163a^{4}+4906a^{3}+5215a^{2}+5199a-13855$, $1670a^{21}-7187a^{20}+9892a^{19}-7425a^{18}-145a^{17}+9681a^{16}-16042a^{15}+14517a^{14}-3945a^{13}-11890a^{12}+24907a^{11}-26574a^{10}+12853a^{9}+12320a^{8}-36929a^{7}+46183a^{6}-30381a^{5}-7914a^{4}+51813a^{3}-76750a^{2}+62589a-20525$, $4032a^{21}+2049a^{20}-9875a^{19}+19514a^{18}-30848a^{17}+44080a^{16}-59416a^{15}+76511a^{14}-94868a^{13}+114536a^{12}-135391a^{11}+156274a^{10}-175901a^{9}+193940a^{8}-209729a^{7}+220951a^{6}-225212a^{5}+221636a^{4}-208912a^{3}+183580a^{2}-142421a+52135$
| 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: | \( 2419229210220000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 2419229210220000 \cdot 2}{2\cdot\sqrt{423616943333188060347578772565944553233123901440}}\cr\approx \mathstrut & 1.42576855238952 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 8*x + 6)
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 - 8*x + 6, 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 - 8*x + 6);
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 - 8*x + 6);
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
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 1124000727777607680000 |
| The 1002 conjugacy class representatives for $S_{22}$ |
| Character table for $S_{22}$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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 44 sibling: | 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 | R | R | ${\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.11.0.1}{11} }{,}\,{\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.11.0.1}{11} }{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | $17{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.11.0.1}{11} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.13.0.1}{13} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 $22$ | $22$ | $1$ | $43$ | |||
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 3.3.3.1 | $x^{3} + 6 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $[3/2]_{2}$ | |
| Deg $18$ | $3$ | $6$ | $18$ | ||||
|
\(5\)
| 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 5.11.0.1 | $x^{11} + 3 x + 3$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | |
|
\(67\)
| 67.2.1.2 | $x^{2} + 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 67.20.0.1 | $x^{20} - 2 x + 7$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ | |
|
\(43223\)
| $\Q_{43223}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
|
\(317963629399156207\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ |