Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 15*x^20 + 91*x^18 + 278*x^16 + 410*x^14 + 123*x^12 - 404*x^10 - 417*x^8 - 19*x^6 + 77*x^4 - 3*x^2 - 1)
gp: K = bnfinit(y^22 + 15*y^20 + 91*y^18 + 278*y^16 + 410*y^14 + 123*y^12 - 404*y^10 - 417*y^8 - 19*y^6 + 77*y^4 - 3*y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + 15*x^20 + 91*x^18 + 278*x^16 + 410*x^14 + 123*x^12 - 404*x^10 - 417*x^8 - 19*x^6 + 77*x^4 - 3*x^2 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 15*x^20 + 91*x^18 + 278*x^16 + 410*x^14 + 123*x^12 - 404*x^10 - 417*x^8 - 19*x^6 + 77*x^4 - 3*x^2 - 1)
\( x^{22} + 15 x^{20} + 91 x^{18} + 278 x^{16} + 410 x^{14} + 123 x^{12} - 404 x^{10} - 417 x^{8} + \cdots - 1 \)
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: | $[6, 8]$ | 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
$C_{2}$, which has order $2$ (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: | $13$ | 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: |
$a^{21}+15a^{19}+91a^{17}+278a^{15}+410a^{13}+123a^{11}-404a^{9}-417a^{7}-19a^{5}+77a^{3}-3a$, $a^{3}+2a$, $a^{19}+13a^{17}+65a^{15}+149a^{13}+122a^{11}-85a^{9}-184a^{7}-50a^{5}+22a^{3}+2a$, $a^{21}+14a^{19}+79a^{17}+224a^{15}+306a^{13}+80a^{11}-284a^{9}-294a^{7}-41a^{5}+39a^{3}+2a$, $2a^{20}+25a^{18}+120a^{16}+262a^{14}+192a^{12}-182a^{10}-331a^{8}-63a^{6}+65a^{4}+3a^{2}-3$, $2a^{21}+28a^{19}+157a^{17}+436a^{15}+558a^{13}+54a^{11}-626a^{9}-503a^{7}+25a^{5}+89a^{3}-10a$, $a^{16}+10a^{14}+35a^{12}+43a^{10}-15a^{8}-60a^{6}-14a^{4}+13a^{2}$, $a^{4}+2a^{2}-1$, $2a^{20}+25a^{18}+119a^{16}+252a^{14}+157a^{12}-225a^{10}-316a^{8}-3a^{6}+79a^{4}-8a^{2}-1$, $2a^{20}+25a^{18}+119a^{16}+252a^{14}+157a^{12}-225a^{10}-316a^{8}-3a^{6}+79a^{4}-9a^{2}-2$, $3a^{21}-a^{20}+45a^{19}-15a^{18}+274a^{17}-91a^{16}+847a^{15}-279a^{14}+1295a^{13}-420a^{12}+518a^{11}-158a^{10}-1090a^{9}+361a^{8}-1337a^{7}+432a^{6}-246a^{5}+80a^{4}+176a^{3}-59a^{2}+16a-7$, $7a^{21}+2a^{20}+104a^{19}+31a^{18}+627a^{17}+196a^{16}+1918a^{15}+634a^{14}+2897a^{13}+1033a^{12}+1124a^{11}+509a^{10}-2444a^{9}-791a^{8}-2957a^{7}-1113a^{6}-548a^{5}-251a^{4}+383a^{3}+143a^{2}+37a+16$, $a^{21}+2a^{20}+16a^{19}+28a^{18}+105a^{17}+158a^{16}+358a^{15}+451a^{14}+647a^{13}+646a^{12}+497a^{11}+305a^{10}-144a^{9}-289a^{8}-458a^{7}-381a^{6}-182a^{5}-103a^{4}+12a^{3}+15a^{2}+3$
| 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: | \( 132523335.017 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{8}\cdot 132523335.017 \cdot 2}{2\cdot\sqrt{12701184200335173359101913935642624}}\cr\approx \mathstrut & 0.182805463022 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 + 15*x^20 + 91*x^18 + 278*x^16 + 410*x^14 + 123*x^12 - 404*x^10 - 417*x^8 - 19*x^6 + 77*x^4 - 3*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 + 15*x^20 + 91*x^18 + 278*x^16 + 410*x^14 + 123*x^12 - 404*x^10 - 417*x^8 - 19*x^6 + 77*x^4 - 3*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 + 15*x^20 + 91*x^18 + 278*x^16 + 410*x^14 + 123*x^12 - 404*x^10 - 417*x^8 - 19*x^6 + 77*x^4 - 3*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 + 15*x^20 + 91*x^18 + 278*x^16 + 410*x^14 + 123*x^12 - 404*x^10 - 417*x^8 - 19*x^6 + 77*x^4 - 3*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: | 22.6.58772424788590906152372716857826825553922490852986504756287773011687467420528898741718186038588040141551514870575537045678129943405925512713655442416794075136.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.14.0.1}{14} }{,}\,{\href{/padicField/3.8.0.1}{8} }$ | ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.11.0.1}{11} }^{2}$ | ${\href{/padicField/11.11.0.1}{11} }^{2}$ | $20{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.10.0.1}{10} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/19.6.0.1}{6} }$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.11.0.1}{11} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |