Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 14*x^20 + 84*x^18 + 285*x^16 + 607*x^14 + 843*x^12 + 745*x^10 + 363*x^8 + 45*x^6 - 29*x^4 - 6*x^2 + 1)
gp: K = bnfinit(y^22 + 14*y^20 + 84*y^18 + 285*y^16 + 607*y^14 + 843*y^12 + 745*y^10 + 363*y^8 + 45*y^6 - 29*y^4 - 6*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + 14*x^20 + 84*x^18 + 285*x^16 + 607*x^14 + 843*x^12 + 745*x^10 + 363*x^8 + 45*x^6 - 29*x^4 - 6*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 14*x^20 + 84*x^18 + 285*x^16 + 607*x^14 + 843*x^12 + 745*x^10 + 363*x^8 + 45*x^6 - 29*x^4 - 6*x^2 + 1)
\( x^{22} + 14 x^{20} + 84 x^{18} + 285 x^{16} + 607 x^{14} + 843 x^{12} + 745 x^{10} + 363 x^{8} + \cdots + 1 \)
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: | $[4, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-316932538277153190033818958954496\)
\(\medspace = -\,2^{22}\cdot 151^{2}\cdot 2311^{2}\cdot 24910163^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(30.01\) | 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: |
\(2\), \(151\), \(2311\), \(24910163\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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: | $12$ | 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: |
$a^{2}+1$, $a^{17}+12a^{15}+59a^{13}+156a^{11}+245a^{9}+230a^{7}+106a^{5}-9a$, $a^{3}+2a$, $a$, $a^{17}+11a^{15}+50a^{13}+124a^{11}+186a^{9}+169a^{7}+77a^{5}+4a^{3}-5a$, $a^{20}+13a^{18}+71a^{16}+215a^{14}+402a^{12}+482a^{10}+355a^{8}+134a^{6}+14a^{4}-3a^{2}-1$, $a^{20}+13a^{18}+71a^{16}+214a^{14}+393a^{12}+450a^{10}+295a^{8}+68a^{6}-23a^{4}-5a^{2}+2$, $a^{21}+14a^{19}+83a^{17}+274a^{15}+557a^{13}+720a^{11}+566a^{9}+212a^{7}-9a^{5}-18a^{3}-a$, $2a^{20}+26a^{18}+142a^{16}+428a^{14}+785a^{12}+892a^{10}+566a^{8}+100a^{6}-76a^{4}-19a^{2}+4$, $11a^{21}+4a^{20}+156a^{19}+56a^{18}+952a^{17}+338a^{16}+3303a^{15}+1164a^{14}+7248a^{13}+2548a^{12}+10497a^{11}+3702a^{10}+9922a^{9}+3535a^{8}+5575a^{7}+2037a^{6}+1347a^{5}+532a^{4}-126a^{3}-34a^{2}-84a-32$, $6a^{21}-a^{20}+83a^{19}-14a^{18}+493a^{17}-84a^{16}+1663a^{15}-286a^{14}+3547a^{13}-619a^{12}+4984a^{11}-901a^{10}+4529a^{9}-888a^{8}+2379a^{7}-548a^{6}+480a^{5}-155a^{4}-83a^{3}+16a^{2}-35a+13$, $a^{21}+3a^{20}+14a^{19}+43a^{18}+84a^{17}+266a^{16}+285a^{15}+939a^{14}+606a^{13}+2105a^{12}+832a^{11}+3127a^{10}+697a^{9}+3049a^{8}+257a^{7}+1797a^{6}-80a^{5}+494a^{4}-104a^{3}-11a^{2}-24a-24$
| 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: | \( 31745730.4838 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{9}\cdot 31745730.4838 \cdot 1}{2\cdot\sqrt{316932538277153190033818958954496}}\cr\approx \mathstrut & 0.217726209913 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 + 14*x^20 + 84*x^18 + 285*x^16 + 607*x^14 + 843*x^12 + 745*x^10 + 363*x^8 + 45*x^6 - 29*x^4 - 6*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 + 14*x^20 + 84*x^18 + 285*x^16 + 607*x^14 + 843*x^12 + 745*x^10 + 363*x^8 + 45*x^6 - 29*x^4 - 6*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 + 14*x^20 + 84*x^18 + 285*x^16 + 607*x^14 + 843*x^12 + 745*x^10 + 363*x^8 + 45*x^6 - 29*x^4 - 6*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 + 14*x^20 + 84*x^18 + 285*x^16 + 607*x^14 + 843*x^12 + 745*x^10 + 363*x^8 + 45*x^6 - 29*x^4 - 6*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
$C_2^{10}.(C_2\times S_{11})$ (as 22T53):
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 81749606400 |
| The 752 conjugacy class representatives for $C_2^{10}.(C_2\times S_{11})$ are not computed |
| Character table for $C_2^{10}.(C_2\times S_{11})$ is not computed |
Intermediate fields
| 11.9.8692675390643.1 |
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 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $22$ | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.10.0.1}{10} }$ | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | $22$ | ${\href{/padicField/13.11.0.1}{11} }^{2}$ | ${\href{/padicField/17.9.0.1}{9} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | $16{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | $16{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.11.0.1}{11} }^{2}$ | $22$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.10.0.1}{10} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.10.10.12 | $x^{10} + 10 x^{9} + 34 x^{8} + 112 x^{7} + 328 x^{6} + 640 x^{5} + 1360 x^{4} + 2176 x^{3} + 1168 x^{2} + 1888 x - 1248$ | $2$ | $5$ | $10$ | $C_2 \times (C_2^4 : C_5)$ | $[2, 2, 2, 2, 2]^{5}$ |
| 2.12.12.2 | $x^{12} - 2 x^{11} - 8 x^{10} - 244 x^{9} + 500 x^{8} + 1696 x^{7} + 6656 x^{6} + 22336 x^{5} + 35952 x^{4} + 61344 x^{3} + 75648 x^{2} + 52288 x + 36544$ | $2$ | $6$ | $12$ | 12T105 | $[2, 2, 2, 2, 2]^{6}$ | |
|
\(151\)
| 151.2.0.1 | $x^{2} + 149 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 151.2.1.1 | $x^{2} + 453$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 151.2.1.1 | $x^{2} + 453$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 151.8.0.1 | $x^{8} + 9 x^{4} + 140 x^{3} + 122 x^{2} + 43 x + 6$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 151.8.0.1 | $x^{8} + 9 x^{4} + 140 x^{3} + 122 x^{2} + 43 x + 6$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
|
\(2311\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
|
\(24910163\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ |