Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 10*x^20 + 44*x^18 + 112*x^16 + 182*x^14 + 199*x^12 + 151*x^10 + 67*x^8 - 13*x^6 - 36*x^4 - 13*x^2 - 1)
gp: K = bnfinit(y^22 + 10*y^20 + 44*y^18 + 112*y^16 + 182*y^14 + 199*y^12 + 151*y^10 + 67*y^8 - 13*y^6 - 36*y^4 - 13*y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + 10*x^20 + 44*x^18 + 112*x^16 + 182*x^14 + 199*x^12 + 151*x^10 + 67*x^8 - 13*x^6 - 36*x^4 - 13*x^2 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 10*x^20 + 44*x^18 + 112*x^16 + 182*x^14 + 199*x^12 + 151*x^10 + 67*x^8 - 13*x^6 - 36*x^4 - 13*x^2 - 1)
\( x^{22} + 10 x^{20} + 44 x^{18} + 112 x^{16} + 182 x^{14} + 199 x^{12} + 151 x^{10} + 67 x^{8} - 13 x^{6} + \cdots - 1 \)
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: | $[2, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(10994531599582075769676214829056\)
\(\medspace = 2^{22}\cdot 1619043113033^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(25.76\) | 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: |
\(2\), \(1619043113033\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{13}a^{20}+\frac{2}{13}a^{18}+\frac{2}{13}a^{16}+\frac{5}{13}a^{14}-\frac{1}{13}a^{12}-\frac{1}{13}a^{10}+\frac{3}{13}a^{8}+\frac{4}{13}a^{6}-\frac{6}{13}a^{4}-\frac{1}{13}a^{2}-\frac{5}{13}$, $\frac{1}{13}a^{21}+\frac{2}{13}a^{19}+\frac{2}{13}a^{17}+\frac{5}{13}a^{15}-\frac{1}{13}a^{13}-\frac{1}{13}a^{11}+\frac{3}{13}a^{9}+\frac{4}{13}a^{7}-\frac{6}{13}a^{5}-\frac{1}{13}a^{3}-\frac{5}{13}a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| 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: |
$a^{2}+1$, $\frac{20}{13}a^{21}+\frac{144}{13}a^{19}+\frac{430}{13}a^{17}+\frac{672}{13}a^{15}+\frac{539}{13}a^{13}+\frac{175}{13}a^{11}-\frac{109}{13}a^{9}-\frac{362}{13}a^{7}-\frac{198}{13}a^{5}+\frac{136}{13}a^{3}+\frac{43}{13}a$, $a^{21}+10a^{19}+44a^{17}+112a^{15}+182a^{13}+199a^{11}+151a^{9}+67a^{7}-13a^{5}-36a^{3}-13a$, $a^{21}+9a^{19}+35a^{17}+77a^{15}+105a^{13}+94a^{11}+57a^{9}+10a^{7}-23a^{5}-13a^{3}+a$, $\frac{88}{13}a^{20}+\frac{722}{13}a^{18}+\frac{2568}{13}a^{16}+\frac{5185}{13}a^{14}+\frac{6503}{13}a^{12}+\frac{5450}{13}a^{10}+\frac{3059}{13}a^{8}+\frac{79}{13}a^{6}-\frac{1412}{13}a^{4}-\frac{543}{13}a^{2}-\frac{37}{13}$, $\frac{68}{13}a^{20}+\frac{578}{13}a^{18}+\frac{2138}{13}a^{16}+\frac{4513}{13}a^{14}+\frac{5964}{13}a^{12}+\frac{5275}{13}a^{10}+\frac{3168}{13}a^{8}+\frac{441}{13}a^{6}-\frac{1214}{13}a^{4}-\frac{666}{13}a^{2}-\frac{67}{13}$, $\frac{47}{13}a^{20}+\frac{393}{13}a^{18}+\frac{1420}{13}a^{16}+\frac{2900}{13}a^{14}+\frac{3658}{13}a^{12}+\frac{3047}{13}a^{10}+\frac{1688}{13}a^{8}+\frac{19}{13}a^{6}-\frac{867}{13}a^{4}-\frac{320}{13}a^{2}-\frac{14}{13}$, $\frac{33}{13}a^{21}+\frac{261}{13}a^{19}+\frac{898}{13}a^{17}+\frac{1764}{13}a^{15}+\frac{2177}{13}a^{13}+\frac{1852}{13}a^{11}+\frac{1087}{13}a^{9}+\frac{80}{13}a^{7}-\frac{380}{13}a^{5}-\frac{137}{13}a^{3}-\frac{48}{13}a$, $\frac{49}{13}a^{21}+\frac{14}{13}a^{20}+\frac{410}{13}a^{19}+\frac{119}{13}a^{18}+\frac{1489}{13}a^{17}+\frac{444}{13}a^{16}+\frac{3079}{13}a^{15}+\frac{954}{13}a^{14}+\frac{3981}{13}a^{13}+\frac{1299}{13}a^{12}+\frac{3461}{13}a^{11}+\frac{1208}{13}a^{10}+\frac{2045}{13}a^{9}+\frac{796}{13}a^{8}+\frac{196}{13}a^{7}+\frac{225}{13}a^{6}-\frac{840}{13}a^{5}-\frac{188}{13}a^{4}-\frac{374}{13}a^{3}-\frac{170}{13}a^{2}-\frac{37}{13}a-\frac{31}{13}$, $4a^{21}+\frac{68}{13}a^{20}+32a^{19}+\frac{565}{13}a^{18}+111a^{17}+\frac{2034}{13}a^{16}+219a^{15}+\frac{4162}{13}a^{14}+269a^{13}+\frac{5314}{13}a^{12}+223a^{11}+\frac{4534}{13}a^{10}+126a^{9}+\frac{2570}{13}a^{8}+3a^{7}+\frac{129}{13}a^{6}-54a^{5}-\frac{1175}{13}a^{4}-16a^{3}-\frac{549}{13}a^{2}-2a-\frac{41}{13}$, $\frac{33}{13}a^{21}+\frac{19}{13}a^{20}+\frac{313}{13}a^{19}+\frac{181}{13}a^{18}+\frac{1327}{13}a^{17}+\frac{779}{13}a^{16}+\frac{3337}{13}a^{15}+\frac{2006}{13}a^{14}+\frac{5570}{13}a^{13}+\frac{3452}{13}a^{12}+\frac{6662}{13}a^{11}+\frac{4271}{13}a^{10}+\frac{6027}{13}a^{9}+\frac{3996}{13}a^{8}+\frac{3928}{13}a^{7}+\frac{2754}{13}a^{6}+\frac{1505}{13}a^{5}+\frac{1212}{13}a^{4}+\frac{227}{13}a^{3}+\frac{254}{13}a^{2}-\frac{22}{13}a+\frac{9}{13}$
| 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: | \( 3635928.32614 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 3635928.32614 \cdot 1}{2\cdot\sqrt{10994531599582075769676214829056}}\cr\approx \mathstrut & 0.210307919391 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 + 10*x^20 + 44*x^18 + 112*x^16 + 182*x^14 + 199*x^12 + 151*x^10 + 67*x^8 - 13*x^6 - 36*x^4 - 13*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 + 10*x^20 + 44*x^18 + 112*x^16 + 182*x^14 + 199*x^12 + 151*x^10 + 67*x^8 - 13*x^6 - 36*x^4 - 13*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 + 10*x^20 + 44*x^18 + 112*x^16 + 182*x^14 + 199*x^12 + 151*x^10 + 67*x^8 - 13*x^6 - 36*x^4 - 13*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 + 10*x^20 + 44*x^18 + 112*x^16 + 182*x^14 + 199*x^12 + 151*x^10 + 67*x^8 - 13*x^6 - 36*x^4 - 13*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
$C_2^{10}.S_{11}$ (as 22T51):
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 40874803200 |
| The 376 conjugacy class representatives for $C_2^{10}.S_{11}$ |
| Character table for $C_2^{10}.S_{11}$ |
Intermediate fields
| 11.7.1619043113033.1 |
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 sibling: | data not computed |
| Minimal sibling: | 22.2.840431756599608455702367012311719571812153210508326264722041853019088432659346485389647494483321252259172032729969587402093091559398721978368.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.10.0.1}{10} }$ | $16{,}\,{\href{/padicField/5.6.0.1}{6} }$ | ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.7.0.1}{7} }^{2}{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.11.0.1}{11} }^{2}$ | ${\href{/padicField/31.11.0.1}{11} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $22$ | $2$ | $11$ | $22$ | |||
|
\(1619043113033\)
| 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |