Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 11*x^20 + 43*x^18 + 62*x^16 - 16*x^14 - 116*x^12 - 53*x^10 + 56*x^8 + 32*x^6 - 10*x^4 - 4*x^2 + 1)
gp: K = bnfinit(y^22 + 11*y^20 + 43*y^18 + 62*y^16 - 16*y^14 - 116*y^12 - 53*y^10 + 56*y^8 + 32*y^6 - 10*y^4 - 4*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + 11*x^20 + 43*x^18 + 62*x^16 - 16*x^14 - 116*x^12 - 53*x^10 + 56*x^8 + 32*x^6 - 10*x^4 - 4*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 11*x^20 + 43*x^18 + 62*x^16 - 16*x^14 - 116*x^12 - 53*x^10 + 56*x^8 + 32*x^6 - 10*x^4 - 4*x^2 + 1)
\( x^{22} + 11 x^{20} + 43 x^{18} + 62 x^{16} - 16 x^{14} - 116 x^{12} - 53 x^{10} + 56 x^{8} + 32 x^{6} + \cdots + 1 \)
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: | $[8, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-13844017509319141075535476685799424\)
\(\medspace = -\,2^{22}\cdot 233^{2}\cdot 246572816873^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(35.63\) | 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: |
\(2\), \(233\), \(246572816873\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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: | $14$ | 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: |
$a$, $a^{2}+1$, $3a^{21}+29a^{19}+87a^{17}+34a^{15}-230a^{13}-227a^{11}+207a^{9}+221a^{7}-88a^{5}-52a^{3}+14a$, $2a^{20}+23a^{18}+97a^{16}+167a^{14}+30a^{12}-248a^{10}-222a^{8}+58a^{6}+115a^{4}+9a^{2}-11$, $14a^{21}+157a^{19}+637a^{17}+1018a^{15}+39a^{13}-1572a^{11}-1126a^{9}+450a^{7}+542a^{5}+17a^{3}-45a$, $2a^{18}+21a^{16}+76a^{14}+91a^{12}-61a^{10}-187a^{8}-35a^{6}+93a^{4}+23a^{2}-12$, $10a^{20}+109a^{18}+421a^{16}+598a^{14}-146a^{12}-1053a^{10}-475a^{8}+427a^{6}+233a^{4}-38a^{2}-13$, $2a^{20}+23a^{18}+97a^{16}+167a^{14}+30a^{12}-248a^{10}-222a^{8}+58a^{6}+115a^{4}+8a^{2}-12$, $13a^{21}+147a^{19}+604a^{17}+989a^{15}+84a^{13}-1501a^{11}-1144a^{9}+413a^{7}+552a^{5}+21a^{3}-50a$, $8a^{20}+92a^{18}+388a^{16}+669a^{14}+130a^{12}-958a^{10}-851a^{8}+208a^{6}+405a^{4}+34a^{2}-37$, $20a^{21}+10a^{20}+225a^{19}+114a^{18}+918a^{17}+475a^{16}+1488a^{15}+803a^{14}+120a^{13}+132a^{12}-2204a^{11}-1153a^{10}-1654a^{9}-985a^{8}+544a^{7}+244a^{6}+734a^{5}+451a^{4}+57a^{3}+47a^{2}-47a-33$, $66a^{21}+26a^{20}+749a^{19}+292a^{18}+3095a^{17}+1187a^{16}+5128a^{15}+1903a^{14}+574a^{13}+86a^{12}-7656a^{11}-2916a^{10}-6068a^{9}-2088a^{8}+1961a^{7}+827a^{6}+2897a^{5}+983a^{4}+172a^{3}+34a^{2}-255a-76$, $50a^{21}-27a^{20}+567a^{19}-307a^{18}+2341a^{17}-1274a^{16}+3878a^{15}-2138a^{14}+459a^{13}-330a^{12}-5693a^{11}+3043a^{10}-4480a^{9}+2508a^{8}+1456a^{7}-702a^{6}+2074a^{5}-1128a^{4}+77a^{3}-56a^{2}-195a+107$, $132a^{21}+40a^{20}+1489a^{19}+451a^{18}+6092a^{17}+1844a^{16}+9877a^{15}+2986a^{14}+602a^{13}+173a^{12}-15216a^{11}-4611a^{10}-11237a^{9}-3385a^{8}+4364a^{7}+1359a^{6}+5482a^{5}+1678a^{4}+161a^{3}+32a^{2}-497a-161$
| 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: | \( 525272724.88 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{7}\cdot 525272724.88 \cdot 1}{2\cdot\sqrt{13844017509319141075535476685799424}}\cr\approx \mathstrut & 0.22091379107 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 + 11*x^20 + 43*x^18 + 62*x^16 - 16*x^14 - 116*x^12 - 53*x^10 + 56*x^8 + 32*x^6 - 10*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 + 11*x^20 + 43*x^18 + 62*x^16 - 16*x^14 - 116*x^12 - 53*x^10 + 56*x^8 + 32*x^6 - 10*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 + 11*x^20 + 43*x^18 + 62*x^16 - 16*x^14 - 116*x^12 - 53*x^10 + 56*x^8 + 32*x^6 - 10*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 + 11*x^20 + 43*x^18 + 62*x^16 - 16*x^14 - 116*x^12 - 53*x^10 + 56*x^8 + 32*x^6 - 10*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
$C_2^{10}.(C_2\times S_{11})$ (as 22T53):
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 81749606400 |
| The 752 conjugacy class representatives for $C_2^{10}.(C_2\times S_{11})$ |
| Character table for $C_2^{10}.(C_2\times S_{11})$ |
Intermediate fields
| 11.11.57451466331409.1 |
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 22 sibling: | data not computed |
| 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 | $22$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.14.0.1}{14} }{,}\,{\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.9.0.1}{9} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}$ | $22$ | ${\href{/padicField/29.8.0.1}{8} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | $16{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | $22$ | ${\href{/padicField/47.7.0.1}{7} }^{2}{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $22$ | $2$ | $11$ | $22$ | |||
|
\(233\)
| Deg $4$ | $2$ | $2$ | $2$ | |||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
|
\(246572816873\)
| Deg $4$ | $2$ | $2$ | $2$ | |||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |