Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 4*x^20 - 4*x^18 - 31*x^16 - 8*x^14 + 81*x^12 + 48*x^10 - 88*x^8 - 60*x^6 + 35*x^4 + 22*x^2 - 1)
gp: K = bnfinit(y^22 + 4*y^20 - 4*y^18 - 31*y^16 - 8*y^14 + 81*y^12 + 48*y^10 - 88*y^8 - 60*y^6 + 35*y^4 + 22*y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + 4*x^20 - 4*x^18 - 31*x^16 - 8*x^14 + 81*x^12 + 48*x^10 - 88*x^8 - 60*x^6 + 35*x^4 + 22*x^2 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 4*x^20 - 4*x^18 - 31*x^16 - 8*x^14 + 81*x^12 + 48*x^10 - 88*x^8 - 60*x^6 + 35*x^4 + 22*x^2 - 1)
\( x^{22} + 4 x^{20} - 4 x^{18} - 31 x^{16} - 8 x^{14} + 81 x^{12} + 48 x^{10} - 88 x^{8} - 60 x^{6} + \cdots - 1 \)
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: | $[10, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(12701184200335173359101913935642624\)
\(\medspace = 2^{22}\cdot 55029067682009^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(35.50\) | 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: |
\(2\), \(55029067682009\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$
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: | $15$ | 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: |
$a$, $2a^{20}+5a^{18}-15a^{16}-38a^{14}+38a^{12}+94a^{10}-41a^{8}-89a^{6}+16a^{4}+26a^{2}+1$, $a^{21}+3a^{19}-6a^{17}-22a^{15}+9a^{13}+53a^{11}-a^{9}-53a^{7}-4a^{5}+20a^{3}+a$, $a^{20}+3a^{18}-7a^{16}-24a^{14}+16a^{12}+65a^{10}-17a^{8}-71a^{6}+11a^{4}+25a^{2}-2$, $a^{21}+2a^{19}-9a^{17}-16a^{15}+30a^{13}+42a^{11}-47a^{9}-41a^{7}+34a^{5}+11a^{3}-9a$, $2a^{20}+5a^{18}-16a^{16}-40a^{14}+45a^{12}+107a^{10}-56a^{8}-113a^{6}+26a^{4}+38a^{2}+1$, $a^{4}-2$, $a^{2}+1$, $a^{18}+3a^{16}-5a^{14}-19a^{12}+3a^{10}+33a^{8}+9a^{6}-15a^{4}-9a^{2}$, $2a^{20}+5a^{18}-16a^{16}-40a^{14}+45a^{12}+107a^{10}-56a^{8}-113a^{6}+27a^{4}+38a^{2}-1$, $2a^{21}+2a^{20}+5a^{19}+5a^{18}-15a^{17}-15a^{16}-38a^{15}-38a^{14}+38a^{13}+38a^{12}+94a^{11}+94a^{10}-41a^{9}-41a^{8}-89a^{7}-89a^{6}+16a^{5}+16a^{4}+25a^{3}+25a^{2}$, $a^{21}+4a^{19}-3a^{17}-27a^{15}-11a^{13}+55a^{11}+39a^{9}-39a^{7}-32a^{5}+6a^{3}+a^{2}+6a+1$, $a^{19}-a^{18}+4a^{17}-4a^{16}-2a^{15}+2a^{14}-24a^{13}+23a^{12}-15a^{11}+12a^{10}+38a^{9}-34a^{8}+38a^{7}-22a^{6}-14a^{5}+15a^{4}-23a^{3}+5a^{2}-6a-2$, $3a^{21}-2a^{20}+11a^{19}-8a^{18}-16a^{17}+6a^{16}-91a^{15}+54a^{14}-a^{13}+21a^{12}+256a^{11}-112a^{10}+112a^{9}-73a^{8}-290a^{7}+88a^{6}-174a^{5}+61a^{4}+111a^{3}-21a^{2}+72a-15$, $a^{21}-a^{20}+3a^{19}-2a^{18}-7a^{17}+10a^{16}-25a^{15}+18a^{14}+14a^{13}-37a^{12}+73a^{11}-55a^{10}-2a^{9}+62a^{8}-91a^{7}+65a^{6}-22a^{5}-44a^{4}+40a^{3}-24a^{2}+17a+5$
| 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: | \( 513315796.771 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{6}\cdot 513315796.771 \cdot 1}{2\cdot\sqrt{12701184200335173359101913935642624}}\cr\approx \mathstrut & 0.143486715133 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 + 4*x^20 - 4*x^18 - 31*x^16 - 8*x^14 + 81*x^12 + 48*x^10 - 88*x^8 - 60*x^6 + 35*x^4 + 22*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 + 4*x^20 - 4*x^18 - 31*x^16 - 8*x^14 + 81*x^12 + 48*x^10 - 88*x^8 - 60*x^6 + 35*x^4 + 22*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 + 4*x^20 - 4*x^18 - 31*x^16 - 8*x^14 + 81*x^12 + 48*x^10 - 88*x^8 - 60*x^6 + 35*x^4 + 22*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 + 4*x^20 - 4*x^18 - 31*x^16 - 8*x^14 + 81*x^12 + 48*x^10 - 88*x^8 - 60*x^6 + 35*x^4 + 22*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
| 11.11.55029067682009.1 |
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: | 22.10.58772424788590906152372716857826825553922490852986504756287773011687467420528898741718186038588040141551514870575537045678129943405925512713655442416794075136.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.14.0.1}{14} }{,}\,{\href{/padicField/3.8.0.1}{8} }$ | ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.11.0.1}{11} }^{2}$ | ${\href{/padicField/11.11.0.1}{11} }^{2}$ | ${\href{/padicField/13.10.0.1}{10} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/17.2.0.1}{2} }$ | $16{,}\,{\href{/padicField/19.6.0.1}{6} }$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}{,}\,{\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.11.0.1}{11} }^{2}$ | $20{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{5}$ |
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 $22$ | $2$ | $11$ | $22$ | |||
|
\(55029067682009\)
| $\Q_{55029067682009}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{55029067682009}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ |