Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 7*x^20 + 11*x^18 - 20*x^16 - 54*x^14 + 17*x^12 + 80*x^10 - 11*x^8 - 43*x^6 + 9*x^4 + 5*x^2 - 1)
gp: K = bnfinit(y^22 + 7*y^20 + 11*y^18 - 20*y^16 - 54*y^14 + 17*y^12 + 80*y^10 - 11*y^8 - 43*y^6 + 9*y^4 + 5*y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + 7*x^20 + 11*x^18 - 20*x^16 - 54*x^14 + 17*x^12 + 80*x^10 - 11*x^8 - 43*x^6 + 9*x^4 + 5*x^2 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 7*x^20 + 11*x^18 - 20*x^16 - 54*x^14 + 17*x^12 + 80*x^10 - 11*x^8 - 43*x^6 + 9*x^4 + 5*x^2 - 1)
\( x^{22} + 7 x^{20} + 11 x^{18} - 20 x^{16} - 54 x^{14} + 17 x^{12} + 80 x^{10} - 11 x^{8} - 43 x^{6} + \cdots - 1 \)
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: |
$a^{21}+7a^{19}+11a^{17}-20a^{15}-54a^{13}+17a^{11}+80a^{9}-11a^{7}-43a^{5}+9a^{3}+5a$, $a^{16}+6a^{14}+7a^{12}-15a^{10}-25a^{8}+12a^{6}+18a^{4}-4a^{2}-1$, $2a^{21}+15a^{19}+30a^{17}-22a^{15}-116a^{13}-34a^{11}+129a^{9}+55a^{7}-45a^{5}-12a^{3}+2a$, $a^{21}+7a^{19}+11a^{17}-20a^{15}-54a^{13}+17a^{11}+80a^{9}-11a^{7}-43a^{5}+8a^{3}+4a$, $a^{5}+a^{3}-2a$, $a^{20}+7a^{18}+12a^{16}-14a^{14}-47a^{12}+a^{10}+51a^{8}+a^{6}-15a^{4}+5a^{2}-1$, $a^{15}+5a^{13}+3a^{11}-14a^{9}-11a^{7}+13a^{5}+5a^{3}-3a$, $a^{20}+7a^{18}+12a^{16}-15a^{14}-53a^{12}-6a^{10}+65a^{8}+23a^{6}-25a^{4}-6a^{2}+1$, $a^{20}+7a^{18}+12a^{16}-14a^{14}-47a^{12}+a^{10}+51a^{8}+a^{6}-14a^{4}+7a^{2}-2$, $2a^{20}+15a^{18}+29a^{16}-28a^{14}-123a^{12}-19a^{10}+154a^{8}+43a^{6}-63a^{4}-8a^{2}+3$, $a+1$, $5a^{21}-a^{20}+37a^{19}-7a^{18}+69a^{17}-11a^{16}-76a^{15}+20a^{14}-299a^{13}+54a^{12}-15a^{11}-17a^{10}+396a^{9}-80a^{8}+66a^{7}+11a^{6}-188a^{5}+43a^{4}-6a^{3}-9a^{2}+19a-5$, $2a^{21}+a^{20}+16a^{19}+8a^{18}+38a^{17}+19a^{16}-3a^{15}-a^{14}-116a^{13}-55a^{12}-84a^{11}-38a^{10}+95a^{9}+42a^{8}+88a^{7}+31a^{6}-21a^{5}-12a^{4}-18a^{3}-3a^{2}+2$, $13a^{21}-6a^{20}+93a^{19}-44a^{18}+157a^{17}-79a^{16}-238a^{15}+102a^{14}-742a^{13}+360a^{12}+113a^{11}-18a^{10}+1075a^{9}-505a^{8}+22a^{7}-42a^{6}-577a^{5}+263a^{4}+22a^{3}-5a^{2}+75a-34$, $10a^{21}-7a^{20}+76a^{19}-53a^{18}+155a^{17}-107a^{16}-110a^{15}+81a^{14}-606a^{13}+429a^{12}-176a^{11}+126a^{10}+709a^{9}-498a^{8}+290a^{7}-210a^{6}-278a^{5}+191a^{4}-64a^{3}+46a^{2}+20a-13$
| 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: | \( 644486533.056 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{6}\cdot 644486533.056 \cdot 1}{2\cdot\sqrt{12701184200335173359101913935642624}}\cr\approx \mathstrut & 0.180152756173 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 + 7*x^20 + 11*x^18 - 20*x^16 - 54*x^14 + 17*x^12 + 80*x^10 - 11*x^8 - 43*x^6 + 9*x^4 + 5*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 + 7*x^20 + 11*x^18 - 20*x^16 - 54*x^14 + 17*x^12 + 80*x^10 - 11*x^8 - 43*x^6 + 9*x^4 + 5*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 + 7*x^20 + 11*x^18 - 20*x^16 - 54*x^14 + 17*x^12 + 80*x^10 - 11*x^8 - 43*x^6 + 9*x^4 + 5*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 + 7*x^20 + 11*x^18 - 20*x^16 - 54*x^14 + 17*x^12 + 80*x^10 - 11*x^8 - 43*x^6 + 9*x^4 + 5*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 $C_2^{10}.S_{11}$ are not computed |
| Character table for $C_2^{10}.S_{11}$ is not computed |
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: | 22.10.58772424788590906152372716857826825553922490852986504756287773011687467420528898741718186038588040141551514870575537045678129943405925512713655442416794075136.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.7.0.1}{7} }^{2}{,}\,{\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.11.0.1}{11} }^{2}$ | ${\href{/padicField/11.11.0.1}{11} }^{2}$ | ${\href{/padicField/13.10.0.1}{10} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.11.0.1}{11} }^{2}$ | $20{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 $22$ | $2$ | $11$ | $22$ | |||
|
\(55029067682009\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ |