Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 5*x^20 - 8*x^19 + 16*x^18 - 27*x^17 + 41*x^16 - 61*x^15 + 80*x^14 - 104*x^13 + 121*x^12 - 138*x^11 + 143*x^10 - 142*x^9 + 131*x^8 - 112*x^7 + 89*x^6 - 64*x^5 + 43*x^4 - 24*x^3 + 12*x^2 - 4*x + 1)
gp: K = bnfinit(y^22 - y^21 + 5*y^20 - 8*y^19 + 16*y^18 - 27*y^17 + 41*y^16 - 61*y^15 + 80*y^14 - 104*y^13 + 121*y^12 - 138*y^11 + 143*y^10 - 142*y^9 + 131*y^8 - 112*y^7 + 89*y^6 - 64*y^5 + 43*y^4 - 24*y^3 + 12*y^2 - 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - x^21 + 5*x^20 - 8*x^19 + 16*x^18 - 27*x^17 + 41*x^16 - 61*x^15 + 80*x^14 - 104*x^13 + 121*x^12 - 138*x^11 + 143*x^10 - 142*x^9 + 131*x^8 - 112*x^7 + 89*x^6 - 64*x^5 + 43*x^4 - 24*x^3 + 12*x^2 - 4*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - x^21 + 5*x^20 - 8*x^19 + 16*x^18 - 27*x^17 + 41*x^16 - 61*x^15 + 80*x^14 - 104*x^13 + 121*x^12 - 138*x^11 + 143*x^10 - 142*x^9 + 131*x^8 - 112*x^7 + 89*x^6 - 64*x^5 + 43*x^4 - 24*x^3 + 12*x^2 - 4*x + 1)
\( x^{22} - x^{21} + 5 x^{20} - 8 x^{19} + 16 x^{18} - 27 x^{17} + 41 x^{16} - 61 x^{15} + 80 x^{14} + \cdots + 1 \)
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: | $[2, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(635524829577904360423933313\)
\(\medspace = 60821\cdot 1487459\cdot 7024799976319967\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(16.53\) | 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: | $60821^{1/2}1487459^{1/2}7024799976319967^{1/2}\approx 25209617799123.895$ | ||
| Ramified primes: |
\(60821\), \(1487459\), \(7024799976319967\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{63552\!\cdots\!33313}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$a^{21}-a^{20}+5a^{19}-8a^{18}+16a^{17}-27a^{16}+41a^{15}-61a^{14}+80a^{13}-104a^{12}+121a^{11}-138a^{10}+143a^{9}-142a^{8}+131a^{7}-112a^{6}+89a^{5}-64a^{4}+43a^{3}-24a^{2}+12a-4$, $2a^{21}-a^{20}+8a^{19}-11a^{18}+20a^{17}-35a^{16}+47a^{15}-70a^{14}+85a^{13}-108a^{12}+119a^{11}-130a^{10}+130a^{9}-121a^{8}+110a^{7}-87a^{6}+67a^{5}-45a^{4}+30a^{3}-15a^{2}+6a-2$, $2a^{21}-3a^{20}+9a^{19}-19a^{18}+31a^{17}-55a^{16}+82a^{15}-117a^{14}+154a^{13}-192a^{12}+223a^{11}-244a^{10}+251a^{9}-238a^{8}+216a^{7}-177a^{6}+134a^{5}-92a^{4}+58a^{3}-30a^{2}+11a-3$, $3a^{21}-a^{20}+12a^{19}-14a^{18}+28a^{17}-46a^{16}+61a^{15}-90a^{14}+104a^{13}-132a^{12}+136a^{11}-148a^{10}+135a^{9}-122a^{8}+101a^{7}-71a^{6}+48a^{5}-25a^{4}+15a^{3}-a+2$, $2a^{20}-a^{19}+9a^{18}-11a^{17}+24a^{16}-38a^{15}+55a^{14}-81a^{13}+99a^{12}-128a^{11}+138a^{10}-155a^{9}+148a^{8}-141a^{7}+121a^{6}-94a^{5}+68a^{4}-42a^{3}+25a^{2}-9a+4$, $3a^{21}-2a^{20}+14a^{19}-19a^{18}+40a^{17}-65a^{16}+96a^{15}-142a^{14}+179a^{13}-232a^{12}+259a^{11}-293a^{10}+292a^{9}-283a^{8}+253a^{7}-207a^{6}+157a^{5}-107a^{4}+68a^{3}-33a^{2}+14a-3$, $3a^{21}-2a^{20}+14a^{19}-19a^{18}+40a^{17}-65a^{16}+96a^{15}-142a^{14}+179a^{13}-232a^{12}+259a^{11}-293a^{10}+291a^{9}-283a^{8}+251a^{7}-205a^{6}+156a^{5}-103a^{4}+66a^{3}-30a^{2}+13a-1$, $3a^{21}+13a^{19}-11a^{18}+29a^{17}-45a^{16}+61a^{15}-95a^{14}+109a^{13}-147a^{12}+152a^{11}-175a^{10}+164a^{9}-158a^{8}+137a^{7}-105a^{6}+80a^{5}-50a^{4}+32a^{3}-11a^{2}+6a$, $a^{21}+a^{20}+4a^{19}+a^{18}+5a^{17}-3a^{16}+3a^{15}-6a^{14}-a^{13}-5a^{12}-8a^{11}+a^{10}-15a^{9}+11a^{8}-16a^{7}+18a^{6}-15a^{5}+15a^{4}-9a^{3}+10a^{2}-3a+4$, $2a^{21}-a^{20}+9a^{19}-11a^{18}+23a^{17}-38a^{16}+52a^{15}-79a^{14}+95a^{13}-123a^{12}+134a^{11}-150a^{10}+148a^{9}-141a^{8}+127a^{7}-101a^{6}+79a^{5}-53a^{4}+35a^{3}-17a^{2}+9a-2$, $2a^{20}+9a^{18}-7a^{17}+20a^{16}-30a^{15}+40a^{14}-64a^{13}+71a^{12}-97a^{11}+99a^{10}-114a^{9}+106a^{8}-101a^{7}+88a^{6}-65a^{5}+50a^{4}-30a^{3}+19a^{2}-6a+3$
| 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: | \( 66805.8364906 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 66805.8364906 \cdot 1}{2\cdot\sqrt{635524829577904360423933313}}\cr\approx \mathstrut & 0.508249340861 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 5*x^20 - 8*x^19 + 16*x^18 - 27*x^17 + 41*x^16 - 61*x^15 + 80*x^14 - 104*x^13 + 121*x^12 - 138*x^11 + 143*x^10 - 142*x^9 + 131*x^8 - 112*x^7 + 89*x^6 - 64*x^5 + 43*x^4 - 24*x^3 + 12*x^2 - 4*x + 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 - x^21 + 5*x^20 - 8*x^19 + 16*x^18 - 27*x^17 + 41*x^16 - 61*x^15 + 80*x^14 - 104*x^13 + 121*x^12 - 138*x^11 + 143*x^10 - 142*x^9 + 131*x^8 - 112*x^7 + 89*x^6 - 64*x^5 + 43*x^4 - 24*x^3 + 12*x^2 - 4*x + 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 - x^21 + 5*x^20 - 8*x^19 + 16*x^18 - 27*x^17 + 41*x^16 - 61*x^15 + 80*x^14 - 104*x^13 + 121*x^12 - 138*x^11 + 143*x^10 - 142*x^9 + 131*x^8 - 112*x^7 + 89*x^6 - 64*x^5 + 43*x^4 - 24*x^3 + 12*x^2 - 4*x + 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 - x^21 + 5*x^20 - 8*x^19 + 16*x^18 - 27*x^17 + 41*x^16 - 61*x^15 + 80*x^14 - 104*x^13 + 121*x^12 - 138*x^11 + 143*x^10 - 142*x^9 + 131*x^8 - 112*x^7 + 89*x^6 - 64*x^5 + 43*x^4 - 24*x^3 + 12*x^2 - 4*x + 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
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 $S_{22}$ are not computed |
| Character table for $S_{22}$ is not computed |
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 | $21{,}\,{\href{/padicField/2.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | $22$ | ${\href{/padicField/11.11.0.1}{11} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | $22$ | ${\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.10.0.1}{10} }$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }$ | $22$ | $15{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $22$ | $15{,}\,{\href{/padicField/53.7.0.1}{7} }$ | $21{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(60821\)
| $\Q_{60821}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{60821}$ | $x$ | $1$ | $1$ | $0$ | 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 $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
|
\(1487459\)
| $\Q_{1487459}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{1487459}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
|
\(7024799976319967\)
| $\Q_{7024799976319967}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ |