Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 6*x^20 + 16*x^18 - 18*x^16 + 4*x^14 - 6*x^12 + 25*x^10 - 3*x^8 - 19*x^6 + 4*x^4 + 4*x^2 - 1)
gp: K = bnfinit(y^22 - 6*y^20 + 16*y^18 - 18*y^16 + 4*y^14 - 6*y^12 + 25*y^10 - 3*y^8 - 19*y^6 + 4*y^4 + 4*y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 6*x^20 + 16*x^18 - 18*x^16 + 4*x^14 - 6*x^12 + 25*x^10 - 3*x^8 - 19*x^6 + 4*x^4 + 4*x^2 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 6*x^20 + 16*x^18 - 18*x^16 + 4*x^14 - 6*x^12 + 25*x^10 - 3*x^8 - 19*x^6 + 4*x^4 + 4*x^2 - 1)
\( x^{22} - 6x^{20} + 16x^{18} - 18x^{16} + 4x^{14} - 6x^{12} + 25x^{10} - 3x^{8} - 19x^{6} + 4x^{4} + 4x^{2} - 1 \)
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: |
\(2613825179875044875466440704\)
\(\medspace = 2^{22}\cdot 971^{2}\cdot 25709231^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(17.63\) | 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: |
\(2\), \(971\), \(25709231\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{181121}a^{20}+\frac{52104}{181121}a^{18}-\frac{45455}{181121}a^{16}+\frac{40370}{181121}a^{14}-\frac{39711}{181121}a^{12}-\frac{32791}{181121}a^{10}-\frac{43471}{181121}a^{8}+\frac{6534}{181121}a^{6}-\frac{20759}{181121}a^{4}+\frac{84247}{181121}a^{2}-\frac{80745}{181121}$, $\frac{1}{181121}a^{21}+\frac{52104}{181121}a^{19}-\frac{45455}{181121}a^{17}+\frac{40370}{181121}a^{15}-\frac{39711}{181121}a^{13}-\frac{32791}{181121}a^{11}-\frac{43471}{181121}a^{9}+\frac{6534}{181121}a^{7}-\frac{20759}{181121}a^{5}+\frac{84247}{181121}a^{3}-\frac{80745}{181121}a$
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: | $13$ | 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: |
$\frac{25213}{181121}a^{20}-\frac{153582}{181121}a^{18}+\frac{439015}{181121}a^{16}-\frac{594573}{181121}a^{14}+\frac{365687}{181121}a^{12}-\frac{304360}{181121}a^{10}+\frac{653332}{181121}a^{8}-\frac{440610}{181121}a^{6}-\frac{138098}{181121}a^{4}+\frac{294765}{181121}a^{2}+\frac{157476}{181121}$, $\frac{251562}{181121}a^{20}-\frac{1427048}{181121}a^{18}+\frac{3583803}{181121}a^{16}-\frac{3518950}{181121}a^{14}+\frac{312415}{181121}a^{12}-\frac{1987049}{181121}a^{10}+\frac{5867908}{181121}a^{8}+\frac{938638}{181121}a^{6}-\frac{3717306}{181121}a^{4}-\frac{530001}{181121}a^{2}+\frac{527581}{181121}$, $\frac{42440}{181121}a^{20}-\frac{193650}{181121}a^{18}+\frac{371813}{181121}a^{16}-\frac{101860}{181121}a^{14}-\frac{185056}{181121}a^{12}-\frac{640760}{181121}a^{10}+\frac{713750}{181121}a^{8}+\frac{731193}{181121}a^{6}+\frac{322826}{181121}a^{4}-\frac{429223}{181121}a^{2}-\frac{189601}{181121}$, $a$, $\frac{426265}{181121}a^{21}-\frac{2386759}{181121}a^{19}+\frac{5882635}{181121}a^{17}-\frac{5421790}{181121}a^{15}-\frac{202997}{181121}a^{13}-\frac{2902618}{181121}a^{11}+\frac{9560866}{181121}a^{9}+\frac{2291345}{181121}a^{7}-\frac{6689036}{181121}a^{5}-\frac{762183}{181121}a^{3}+\frac{1105173}{181121}a$, $\frac{383825}{181121}a^{21}-\frac{2193109}{181121}a^{19}+\frac{5510822}{181121}a^{17}-\frac{5319930}{181121}a^{15}-\frac{17941}{181121}a^{13}-\frac{2261858}{181121}a^{11}+\frac{8847116}{181121}a^{9}+\frac{1560152}{181121}a^{7}-\frac{7011862}{181121}a^{5}-\frac{332960}{181121}a^{3}+\frac{1294774}{181121}a$, $\frac{25213}{181121}a^{20}-\frac{153582}{181121}a^{18}+\frac{439015}{181121}a^{16}-\frac{594573}{181121}a^{14}+\frac{365687}{181121}a^{12}-\frac{304360}{181121}a^{10}+\frac{653332}{181121}a^{8}-\frac{440610}{181121}a^{6}-\frac{138098}{181121}a^{4}+\frac{294765}{181121}a^{2}-a+\frac{157476}{181121}$, $\frac{2374}{2551}a^{21}+\frac{193767}{181121}a^{20}-\frac{13298}{2551}a^{19}-\frac{1097740}{181121}a^{18}+\frac{32844}{2551}a^{17}+\frac{2770939}{181121}a^{16}-\frac{30751}{2551}a^{15}-\frac{2777894}{181121}a^{14}+\frac{842}{2551}a^{13}+\frac{425469}{181121}a^{12}-\frac{19926}{2551}a^{11}-\frac{1719106}{181121}a^{10}+\frac{56673}{2551}a^{9}+\frac{4677115}{181121}a^{8}+\frac{11840}{2551}a^{7}+\frac{400030}{181121}a^{6}-\frac{34811}{2551}a^{5}-\frac{2609679}{181121}a^{4}-\frac{8777}{2551}a^{3}-\frac{147281}{181121}a^{2}+\frac{6265}{2551}a+\frac{421170}{181121}$, $a^{21}-\frac{251562}{181121}a^{20}-6a^{19}+\frac{1427048}{181121}a^{18}+16a^{17}-\frac{3583803}{181121}a^{16}-18a^{15}+\frac{3518950}{181121}a^{14}+4a^{13}-\frac{312415}{181121}a^{12}-6a^{11}+\frac{1987049}{181121}a^{10}+25a^{9}-\frac{5867908}{181121}a^{8}-3a^{7}-\frac{938638}{181121}a^{6}-19a^{5}+\frac{3717306}{181121}a^{4}+4a^{3}+\frac{530001}{181121}a^{2}+4a-\frac{527581}{181121}$, $\frac{295083}{181121}a^{21}+\frac{76590}{181121}a^{20}-\frac{1624905}{181121}a^{19}-\frac{355875}{181121}a^{18}+\frac{3902552}{181121}a^{17}+\frac{652775}{181121}a^{16}-\frac{3268759}{181121}a^{15}+\frac{21709}{181121}a^{14}-\frac{780160}{181121}a^{13}-\frac{987263}{181121}a^{12}-\frac{1669559}{181121}a^{11}-\frac{401146}{181121}a^{10}+\frac{6318725}{181121}a^{9}+\frac{1009058}{181121}a^{8}+\frac{2393850}{181121}a^{7}+\frac{1994068}{181121}a^{6}-\frac{5006044}{181121}a^{5}-\frac{776156}{181121}a^{4}-\frac{992080}{181121}a^{3}-\frac{863500}{181121}a^{2}+\frac{896320}{181121}a+\frac{116995}{181121}$, $\frac{332024}{181121}a^{21}-\frac{163470}{181121}a^{20}-\frac{1805029}{181121}a^{19}+\frac{860871}{181121}a^{18}+\frac{4284430}{181121}a^{17}-\frac{1952506}{181121}a^{16}-\frac{3492024}{181121}a^{15}+\frac{1308703}{181121}a^{14}-\frac{845232}{181121}a^{13}+\frac{905014}{181121}a^{12}-\frac{2208005}{181121}a^{11}+\frac{793259}{181121}a^{10}+\frac{6999784}{181121}a^{9}-\frac{2976001}{181121}a^{8}+\frac{2875414}{181121}a^{7}-\frac{2034774}{181121}a^{6}-\frac{4997949}{181121}a^{5}+\frac{2526368}{181121}a^{4}-\frac{1206917}{181121}a^{3}+\frac{583750}{181121}a^{2}+\frac{977024}{181121}a-\frac{532209}{181121}$, $\frac{23645}{181121}a^{21}+\frac{193767}{181121}a^{20}-\frac{167083}{181121}a^{19}-\frac{1097740}{181121}a^{18}+\frac{531902}{181121}a^{17}+\frac{2770939}{181121}a^{16}-\frac{864625}{181121}a^{15}-\frac{2777894}{181121}a^{14}+\frac{689153}{181121}a^{13}+\frac{425469}{181121}a^{12}-\frac{507557}{181121}a^{11}-\frac{1719106}{181121}a^{10}+\frac{895485}{181121}a^{9}+\frac{4677115}{181121}a^{8}-\frac{724267}{181121}a^{7}+\frac{400030}{181121}a^{6}-\frac{8645}{181121}a^{5}-\frac{2609679}{181121}a^{4}+\frac{232678}{181121}a^{3}-\frac{147281}{181121}a^{2}-\frac{381306}{181121}a+\frac{421170}{181121}$, $\frac{237106}{181121}a^{21}+\frac{25213}{181121}a^{20}-\frac{1179112}{181121}a^{19}-\frac{153582}{181121}a^{18}+\frac{2487569}{181121}a^{17}+\frac{439015}{181121}a^{16}-\frac{1181235}{181121}a^{15}-\frac{594573}{181121}a^{14}-\frac{1590149}{181121}a^{13}+\frac{365687}{181121}a^{12}-\frac{1772889}{181121}a^{11}-\frac{304360}{181121}a^{10}+\frac{3983604}{181121}a^{9}+\frac{653332}{181121}a^{8}+\frac{4107353}{181121}a^{7}-\frac{440610}{181121}a^{6}-\frac{2293731}{181121}a^{5}-\frac{138098}{181121}a^{4}-\frac{1814876}{181121}a^{3}+\frac{294765}{181121}a^{2}+\frac{271335}{181121}a+\frac{157476}{181121}$
| 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: | \( 107375.053215 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{8}\cdot 107375.053215 \cdot 1}{2\cdot\sqrt{2613825179875044875466440704}}\cr\approx \mathstrut & 0.163250322759 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 6*x^20 + 16*x^18 - 18*x^16 + 4*x^14 - 6*x^12 + 25*x^10 - 3*x^8 - 19*x^6 + 4*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 - 6*x^20 + 16*x^18 - 18*x^16 + 4*x^14 - 6*x^12 + 25*x^10 - 3*x^8 - 19*x^6 + 4*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 - 6*x^20 + 16*x^18 - 18*x^16 + 4*x^14 - 6*x^12 + 25*x^10 - 3*x^8 - 19*x^6 + 4*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 - 6*x^20 + 16*x^18 - 18*x^16 + 4*x^14 - 6*x^12 + 25*x^10 - 3*x^8 - 19*x^6 + 4*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
$C_2^{10}.S_{11}$ (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
| 11.3.24963663301.1 |
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: | 22.6.9841275385698713547061800803050974420965363933658706138702352603003020290397779358031956909097305281204793677693797269504.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.11.0.1}{11} }^{2}$ | ${\href{/padicField/5.11.0.1}{11} }^{2}$ | $16{,}\,{\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.11.0.1}{11} }^{2}$ | $16{,}\,{\href{/padicField/13.6.0.1}{6} }$ | $18{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.7.0.1}{7} }^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.9.0.1}{9} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | $18{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.5.0.1}{5} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.5.0.1}{5} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.7.0.1}{7} }^{2}{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.10.10.8 | $x^{10} + 4 x^{9} + 14 x^{8} + 240 x^{7} + 928 x^{6} + 4400 x^{5} + 6368 x^{4} + 13888 x^{3} - 336 x^{2} + 2432 x - 17632$ | $2$ | $5$ | $10$ | $C_2 \times (C_2^4 : C_5)$ | $[2, 2, 2, 2, 2]^{5}$ |
| 2.12.12.3 | $x^{12} - 4 x^{11} + 52 x^{10} - 128 x^{9} + 532 x^{8} - 1184 x^{7} + 6080 x^{6} - 2304 x^{5} + 36144 x^{4} - 10688 x^{3} + 86208 x^{2} - 38656 x + 107968$ | $2$ | $6$ | $12$ | 12T134 | $[2, 2, 2, 2, 2, 2]^{6}$ | |
|
\(971\)
| $\Q_{971}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{971}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 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 $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
|
\(25709231\)
| 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 $4$ | $2$ | $2$ | $2$ | ||||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |