Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 9*x + 6)
gp: K = bnfinit(y^22 - 9*y + 6, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 9*x + 6);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 9*x + 6)
\( x^{22} - 9x + 6 \)
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: |
\(5746119802689231329092338026699843340583720730997\)
\(\medspace = 3^{21}\cdot 67\cdot 54631\cdot 398887\cdot 3137422219\cdot 119919973547835079\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(164.56\) | 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: | $3^{21/22}67^{1/2}54631^{1/2}398887^{1/2}3137422219^{1/2}119919973547835079^{1/2}\approx 6.688798124490173e+19$ | ||
| Ramified primes: |
\(3\), \(67\), \(54631\), \(398887\), \(3137422219\), \(119919973547835079\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{16479\!\cdots\!04197}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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
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: | $11$ | 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: |
$a^{21}-a^{20}+a^{19}-a^{18}+a^{17}-a^{16}+a^{15}-a^{14}+a^{13}-a^{12}+a^{11}-a^{10}+a^{9}-a^{8}+a^{7}-a^{6}+a^{5}-a^{4}+4a^{2}-10a+5$, $33a^{21}+23a^{20}-9a^{19}-40a^{18}-48a^{17}-21a^{16}+22a^{15}+54a^{14}+42a^{13}+4a^{12}-42a^{11}-43a^{10}-7a^{9}+44a^{8}+51a^{7}+3a^{6}-80a^{5}-106a^{4}-49a^{3}+89a^{2}+198a-127$, $3a^{21}-2a^{20}-12a^{19}+29a^{18}-48a^{17}+60a^{16}-67a^{15}+57a^{14}-44a^{13}+39a^{12}-59a^{11}+87a^{10}-123a^{9}+151a^{8}-161a^{7}+120a^{6}-51a^{5}-18a^{4}+51a^{3}-60a^{2}+24a+1$, $163a^{21}-230a^{20}+304a^{19}-389a^{18}+483a^{17}-581a^{16}+684a^{15}-788a^{14}+884a^{13}-975a^{12}+1050a^{11}-1096a^{10}+1116a^{9}-1088a^{8}+1004a^{7}-868a^{6}+645a^{5}-331a^{4}-83a^{3}+634a^{2}-1304a+635$, $5a^{21}-47a^{20}-142a^{19}-198a^{18}-226a^{17}-297a^{16}-281a^{15}-225a^{14}-216a^{13}-100a^{12}+62a^{11}+110a^{10}+290a^{9}+462a^{8}+444a^{7}+555a^{6}+623a^{5}+391a^{4}+361a^{3}+263a^{2}-206a-343$, $54a^{21}-21a^{20}-69a^{19}-43a^{18}+52a^{17}+127a^{16}+117a^{15}+55a^{14}-43a^{13}-133a^{12}-88a^{11}+106a^{10}+271a^{9}+235a^{8}+10a^{7}-213a^{6}-214a^{5}-20a^{4}+194a^{3}+426a^{2}+447a-511$, $1034a^{21}+581a^{20}-19a^{19}-696a^{18}-1379a^{17}-2021a^{16}-2535a^{15}-2807a^{14}-2798a^{13}-2500a^{12}-1851a^{11}-872a^{10}+309a^{9}+1601a^{8}+2956a^{7}+4143a^{6}+4974a^{5}+5404a^{4}+5348a^{3}+4571a^{2}+3175a-7973$, $10846a^{21}+7222a^{20}+4820a^{19}+3215a^{18}+2135a^{17}+1435a^{16}+956a^{15}+623a^{14}+435a^{13}+289a^{12}+166a^{11}+134a^{10}+93a^{9}+33a^{8}+51a^{7}+30a^{6}-9a^{5}+52a^{4}+14a^{3}-56a^{2}+50a-97591$, $165a^{21}-48a^{20}+126a^{19}-20a^{18}-31a^{17}+189a^{16}-346a^{15}+542a^{14}-728a^{13}+875a^{12}-971a^{11}+965a^{10}-881a^{9}+698a^{8}-468a^{7}+235a^{6}-49a^{5}-a^{4}-81a^{3}+362a^{2}-725a-331$, $1140a^{21}+54a^{20}-1341a^{19}-2764a^{18}-3155a^{17}-1827a^{16}+434a^{15}+2365a^{14}+3719a^{13}+4451a^{12}+3460a^{11}+11a^{10}-4145a^{9}-6247a^{8}-5803a^{7}-4040a^{6}-755a^{5}+4971a^{4}+10465a^{3}+10766a^{2}+5313a-11359$, $321808a^{21}+335198a^{20}-145416a^{19}-522242a^{18}-211496a^{17}+485432a^{16}+623035a^{15}-131954a^{14}-864256a^{13}-482815a^{12}+700217a^{11}+1115523a^{10}-24773a^{9}-1386811a^{8}-1009775a^{7}+944110a^{6}+1949451a^{5}+305793a^{4}-2168842a^{3}-2009097a^{2}+1160942a+454955$
| 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: | \( 26906527131600000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 26906527131600000 \cdot 1}{2\cdot\sqrt{5746119802689231329092338026699843340583720730997}}\cr\approx \mathstrut & 2.15277671479775 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 9*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 - 9*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 - 9*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 - 9*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 | ${\href{/padicField/2.6.0.1}{6} }^{2}{,}\,{\href{/padicField/2.3.0.1}{3} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | R | $22$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | $18{,}\,{\href{/padicField/13.4.0.1}{4} }$ | $21{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.11.0.1}{11} }{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $22$ | ${\href{/padicField/37.9.0.1}{9} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.8.0.1}{8} }$ | $19{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.22.21.1 | $x^{22} + 6$ | $22$ | $1$ | $21$ | 22T5 | $[\ ]_{22}^{5}$ |
|
\(67\)
| 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 | $[\ ]^{20}$ | |
|
\(54631\)
| $\Q_{54631}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 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 $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
|
\(398887\)
| $\Q_{398887}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 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 $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
|
\(3137422219\)
| $\Q_{3137422219}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ | ||
|
\(119919973547835079\)
| 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 $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ |