Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^20 - 2*x^19 + 5*x^18 + 6*x^17 - 5*x^16 - 12*x^15 + 2*x^14 + 18*x^13 + 4*x^12 - 18*x^11 - 11*x^10 + 14*x^9 + 14*x^8 - 8*x^7 - 11*x^6 + 2*x^5 + 8*x^4 - 4*x^2 + 1)
gp: K = bnfinit(y^22 - 2*y^20 - 2*y^19 + 5*y^18 + 6*y^17 - 5*y^16 - 12*y^15 + 2*y^14 + 18*y^13 + 4*y^12 - 18*y^11 - 11*y^10 + 14*y^9 + 14*y^8 - 8*y^7 - 11*y^6 + 2*y^5 + 8*y^4 - 4*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 2*x^20 - 2*x^19 + 5*x^18 + 6*x^17 - 5*x^16 - 12*x^15 + 2*x^14 + 18*x^13 + 4*x^12 - 18*x^11 - 11*x^10 + 14*x^9 + 14*x^8 - 8*x^7 - 11*x^6 + 2*x^5 + 8*x^4 - 4*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 2*x^20 - 2*x^19 + 5*x^18 + 6*x^17 - 5*x^16 - 12*x^15 + 2*x^14 + 18*x^13 + 4*x^12 - 18*x^11 - 11*x^10 + 14*x^9 + 14*x^8 - 8*x^7 - 11*x^6 + 2*x^5 + 8*x^4 - 4*x^2 + 1)
\( x^{22} - 2 x^{20} - 2 x^{19} + 5 x^{18} + 6 x^{17} - 5 x^{16} - 12 x^{15} + 2 x^{14} + 18 x^{13} + \cdots + 1 \)
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: | $[0, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-577860666762298833296687104\)
\(\medspace = -\,2^{30}\cdot 73^{2}\cdot 577\cdot 53149\cdot 3293113\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(16.46\) | 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: |
\(2\), \(73\), \(577\), \(53149\), \(3293113\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-100989807456949}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
\( 2 a^{21} - 3 a^{19} - 4 a^{18} + 8 a^{17} + 10 a^{16} - 5 a^{15} - 18 a^{14} - a^{13} + 24 a^{12} + 10 a^{11} - 18 a^{10} - 18 a^{9} + 11 a^{8} + 17 a^{7} - 3 a^{6} - 9 a^{5} - 2 a^{4} + 7 a^{3} + a^{2} - 2 a \)
(order $4$)
| 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: |
$a^{21}-2a^{19}-2a^{18}+5a^{17}+6a^{16}-5a^{15}-12a^{14}+2a^{13}+18a^{12}+4a^{11}-18a^{10}-11a^{9}+14a^{8}+14a^{7}-8a^{6}-11a^{5}+2a^{4}+8a^{3}-4a$, $3a^{21}+a^{20}-4a^{19}-7a^{18}+10a^{17}+17a^{16}-3a^{15}-27a^{14}-6a^{13}+34a^{12}+20a^{11}-26a^{10}-30a^{9}+14a^{8}+28a^{7}-5a^{6}-18a^{5}-4a^{4}+12a^{3}+3a^{2}-4a-1$, $5a^{21}+2a^{20}-8a^{19}-14a^{18}+17a^{17}+35a^{16}-4a^{15}-57a^{14}-21a^{13}+68a^{12}+53a^{11}-49a^{10}-75a^{9}+18a^{8}+68a^{7}+4a^{6}-41a^{5}-17a^{4}+23a^{3}+12a^{2}-8a-4$, $2a^{21}+a^{20}-4a^{19}-6a^{18}+7a^{17}+16a^{16}-3a^{15}-26a^{14}-10a^{13}+31a^{12}+24a^{11}-24a^{10}-35a^{9}+8a^{8}+33a^{7}+2a^{6}-21a^{5}-9a^{4}+11a^{3}+8a^{2}-5a-3$, $a^{20}-2a^{18}-2a^{17}+4a^{16}+5a^{15}-4a^{14}-9a^{13}+11a^{11}+2a^{10}-10a^{9}-6a^{8}+5a^{7}+5a^{6}-4a^{5}-2a^{4}+2a^{2}-1$, $a^{19}-a^{17}-2a^{16}+4a^{15}+4a^{14}-a^{13}-8a^{12}+a^{11}+10a^{10}+5a^{9}-8a^{8}-7a^{7}+6a^{6}+8a^{5}-a^{4}-5a^{3}-a^{2}+3a+1$, $a^{4}-a+1$, $2a^{21}-2a^{19}-3a^{18}+7a^{17}+7a^{16}-4a^{15}-11a^{14}+3a^{13}+18a^{12}+a^{11}-14a^{10}-6a^{9}+14a^{8}+8a^{7}-10a^{6}-4a^{5}+3a^{4}+7a^{3}-3a^{2}-2a+2$, $2a^{21}+2a^{20}-2a^{19}-7a^{18}+2a^{17}+16a^{16}+10a^{15}-18a^{14}-24a^{13}+13a^{12}+37a^{11}+5a^{10}-36a^{9}-20a^{8}+23a^{7}+23a^{6}-8a^{5}-17a^{4}+2a^{3}+9a^{2}-3$, $a^{21}+3a^{20}-a^{19}-7a^{18}-3a^{17}+17a^{16}+15a^{15}-17a^{14}-32a^{13}+9a^{12}+46a^{11}+8a^{10}-44a^{9}-27a^{8}+30a^{7}+29a^{6}-14a^{5}-21a^{4}+2a^{3}+14a^{2}-5$
| 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: | \( 65028.759848 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 65028.759848 \cdot 1}{4\cdot\sqrt{577860666762298833296687104}}\cr\approx \mathstrut & 0.40748578691 \end{aligned}\] (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 - 2*x^19 + 5*x^18 + 6*x^17 - 5*x^16 - 12*x^15 + 2*x^14 + 18*x^13 + 4*x^12 - 18*x^11 - 11*x^10 + 14*x^9 + 14*x^8 - 8*x^7 - 11*x^6 + 2*x^5 + 8*x^4 - 4*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 - 2*x^19 + 5*x^18 + 6*x^17 - 5*x^16 - 12*x^15 + 2*x^14 + 18*x^13 + 4*x^12 - 18*x^11 - 11*x^10 + 14*x^9 + 14*x^8 - 8*x^7 - 11*x^6 + 2*x^5 + 8*x^4 - 4*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 - 2*x^19 + 5*x^18 + 6*x^17 - 5*x^16 - 12*x^15 + 2*x^14 + 18*x^13 + 4*x^12 - 18*x^11 - 11*x^10 + 14*x^9 + 14*x^8 - 8*x^7 - 11*x^6 + 2*x^5 + 8*x^4 - 4*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 - 2*x^19 + 5*x^18 + 6*x^17 - 5*x^16 - 12*x^15 + 2*x^14 + 18*x^13 + 4*x^12 - 18*x^11 - 11*x^10 + 14*x^9 + 14*x^8 - 8*x^7 - 11*x^6 + 2*x^5 + 8*x^4 - 4*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
$S_{11}\wr C_2$ (as 22T57):
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 3186701844480000 |
| The 1652 conjugacy class representatives for $S_{11}\wr C_2$ |
| Character table for $S_{11}\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \) |
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 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 | $22$ | ${\href{/padicField/5.8.0.1}{8} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.8.0.1}{8} }^{2}{,}\,{\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | $16{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | $16{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.16.65 | $x^{8} + 2 x^{6} + 4 x^{3} + 4 x + 2$ | $8$ | $1$ | $16$ | $QD_{16}$ | $[2, 2, 5/2]^{2}$ |
| 2.14.14.38 | $x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1948 x^{10} + 8392 x^{9} + 30520 x^{8} + 84992 x^{7} + 178608 x^{6} + 284064 x^{5} + 325984 x^{4} + 242688 x^{3} + 97600 x^{2} + 11648 x - 5504$ | $2$ | $7$ | $14$ | $C_{14}$ | $[2]^{7}$ | |
|
\(73\)
| $\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 73.2.0.1 | $x^{2} + 70 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 73.2.1.1 | $x^{2} + 73$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 73.2.0.1 | $x^{2} + 70 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 73.2.1.1 | $x^{2} + 73$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 73.4.0.1 | $x^{4} + 16 x^{2} + 56 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 73.9.0.1 | $x^{9} + 72 x^{2} + 15 x + 68$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
|
\(577\)
| $\Q_{577}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{577}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{577}$ | $x$ | $1$ | $1$ | $0$ | 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 $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
|
\(53149\)
| $\Q_{53149}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{53149}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| 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 $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
|
\(3293113\)
| $\Q_{3293113}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |