Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 5*x + 6)
gp: K = bnfinit(y^22 - 5*y + 6, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 5*x + 6);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 5*x + 6)
\( x^{22} - 5x + 6 \)
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: | $[0, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-7475956680035860099059007110342551356066477779\)
\(\medspace = -\,3^{21}\cdot 31\cdot 61\cdot 67\cdot 181\cdot 143137\cdot 217732675899704637585877\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(121.67\) | 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: |
\(3\), \(31\), \(61\), \(67\), \(181\), \(143137\), \(217732675899704637585877\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-21440\!\cdots\!70179}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: | $10$ | 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: |
$29a^{21}+39a^{20}+37a^{19}+43a^{18}+55a^{17}+48a^{16}+27a^{15}+23a^{14}+44a^{13}+55a^{12}+42a^{11}+49a^{10}+79a^{9}+77a^{8}+32a^{7}+22a^{5}+32a^{4}-13a^{3}-18a^{2}+48a-61$, $369a^{21}+349a^{20}+285a^{19}+347a^{18}+242a^{17}+226a^{16}+236a^{15}+82a^{14}+179a^{13}+63a^{12}-40a^{11}+66a^{10}-190a^{9}-69a^{8}-104a^{7}-373a^{6}-97a^{5}-421a^{4}-359a^{3}-145a^{2}-690a-2057$, $484a^{21}-489a^{20}+483a^{19}-426a^{18}+383a^{17}-275a^{16}+150a^{15}-a^{14}-235a^{13}+457a^{12}-766a^{11}+1115a^{10}-1478a^{9}+1942a^{8}-2365a^{7}+2863a^{6}-3376a^{5}+3818a^{4}-4325a^{3}+4702a^{2}-5053a+2897$, $256a^{21}+40a^{20}-388a^{19}+388a^{18}+5a^{17}-506a^{16}+566a^{15}-64a^{14}-629a^{13}+771a^{12}-122a^{11}-836a^{10}+1103a^{9}-249a^{8}-1110a^{7}+1617a^{6}-551a^{5}-1327a^{4}+2192a^{3}-903a^{2}-1633a+1681$, $55a^{21}-23a^{19}-a^{18}+55a^{17}+105a^{16}+98a^{15}+23a^{14}-72a^{13}-105a^{12}-24a^{11}+108a^{10}+142a^{9}+35a^{8}-95a^{7}-160a^{6}-157a^{5}-39a^{4}+169a^{3}+222a^{2}-26a-589$, $347a^{21}+342a^{20}-50a^{19}-493a^{18}-511a^{17}+34a^{16}+779a^{15}+1115a^{14}+670a^{13}-298a^{12}-1035a^{11}-888a^{10}+49a^{9}+1000a^{8}+974a^{7}-166a^{6}-1603a^{5}-2028a^{4}-881a^{3}+1240a^{2}+2525a+127$, $321a^{21}-1079a^{20}+286a^{19}+1237a^{18}-1026a^{17}-1007a^{16}+1714a^{15}+211a^{14}-2153a^{13}+1087a^{12}+2125a^{11}-2560a^{10}-1312a^{9}+3704a^{8}-489a^{7}-4091a^{6}+3126a^{5}+3330a^{4}-5851a^{3}-1070a^{2}+7565a-4433$, $239a^{21}-256a^{20}+275a^{19}-277a^{18}+240a^{17}-190a^{16}+146a^{15}-117a^{14}+39a^{13}+109a^{12}-287a^{11}+407a^{10}-523a^{9}+738a^{8}-1061a^{7}+1327a^{6}-1485a^{5}+1668a^{4}-2008a^{3}+2375a^{2}-2577a+1429$, $1727a^{21}+752a^{20}+703a^{19}+766a^{18}-938a^{17}-963a^{16}-754a^{15}-2569a^{14}-1758a^{13}-706a^{12}-2327a^{11}-250a^{10}+1465a^{9}-473a^{8}+2277a^{7}+3797a^{6}+253a^{5}+3080a^{4}+3604a^{3}-2215a^{2}+1101a-7579$, $205a^{21}+95a^{20}-60a^{19}-164a^{18}-268a^{17}-378a^{16}-393a^{15}-342a^{14}-266a^{13}-129a^{12}+52a^{11}+257a^{10}+420a^{9}+503a^{8}+615a^{7}+627a^{6}+408a^{5}+287a^{4}+175a^{3}-259a^{2}-499a-1483$
| 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: | \( 2404518202930000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 2404518202930000 \cdot 1}{2\cdot\sqrt{7475956680035860099059007110342551356066477779}}\cr\approx \mathstrut & 8.37805558410868 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 5*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 - 5*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 - 5*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 - 5*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 | ${\href{/padicField/5.5.0.1}{5} }^{4}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.11.0.1}{11} }{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.11.0.1}{11} }{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | $20{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 3.3.3.1 | $x^{3} + 6 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $[3/2]_{2}$ | |
| Deg $18$ | $3$ | $6$ | $18$ | ||||
|
\(31\)
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 31.2.1.1 | $x^{2} + 93$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.9.0.1 | $x^{9} + 4 x^{3} + 20 x^{2} + 29 x + 28$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
| 31.10.0.1 | $x^{10} + 30 x^{5} + 26 x^{4} + 13 x^{3} + 13 x^{2} + 13 x + 3$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
|
\(61\)
| 61.2.1.1 | $x^{2} + 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 61.3.0.1 | $x^{3} + 7 x + 59$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 61.4.0.1 | $x^{4} + 3 x^{2} + 40 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 61.13.0.1 | $x^{13} + 3 x + 59$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | |
|
\(67\)
| 67.2.1.1 | $x^{2} + 134$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 67.20.0.1 | $x^{20} - 2 x + 7$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ | |
|
\(181\)
| $\Q_{181}$ | $x + 179$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{181}$ | $x + 179$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 181.2.1.1 | $x^{2} + 181$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 181.4.0.1 | $x^{4} + 6 x^{2} + 105 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 181.14.0.1 | $x^{14} - x + 2$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
|
\(143137\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
|
\(217\!\cdots\!877\)
| $\Q_{21\!\cdots\!77}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ |