Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 + 44*x^20 - 55*x^19 - 112*x^18 + 381*x^17 - 87*x^16 - 834*x^15 + 674*x^14 + 1020*x^13 - 1213*x^12 - 834*x^11 + 1220*x^10 + 511*x^9 - 776*x^8 - 247*x^7 + 323*x^6 + 85*x^5 - 89*x^4 - 18*x^3 + 16*x^2 + x - 1)
gp: K = bnfinit(y^22 - 11*y^21 + 44*y^20 - 55*y^19 - 112*y^18 + 381*y^17 - 87*y^16 - 834*y^15 + 674*y^14 + 1020*y^13 - 1213*y^12 - 834*y^11 + 1220*y^10 + 511*y^9 - 776*y^8 - 247*y^7 + 323*y^6 + 85*y^5 - 89*y^4 - 18*y^3 + 16*y^2 + y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 11*x^21 + 44*x^20 - 55*x^19 - 112*x^18 + 381*x^17 - 87*x^16 - 834*x^15 + 674*x^14 + 1020*x^13 - 1213*x^12 - 834*x^11 + 1220*x^10 + 511*x^9 - 776*x^8 - 247*x^7 + 323*x^6 + 85*x^5 - 89*x^4 - 18*x^3 + 16*x^2 + x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 11*x^21 + 44*x^20 - 55*x^19 - 112*x^18 + 381*x^17 - 87*x^16 - 834*x^15 + 674*x^14 + 1020*x^13 - 1213*x^12 - 834*x^11 + 1220*x^10 + 511*x^9 - 776*x^8 - 247*x^7 + 323*x^6 + 85*x^5 - 89*x^4 - 18*x^3 + 16*x^2 + x - 1)
\( x^{22} - 11 x^{21} + 44 x^{20} - 55 x^{19} - 112 x^{18} + 381 x^{17} - 87 x^{16} - 834 x^{15} + 674 x^{14} + \cdots - 1 \)
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: | $[8, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-14799420881562079578038490499\)
\(\medspace = -\,97\cdot 2381^{2}\cdot 2467\cdot 104446171^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(19.08\) | 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: | $97^{1/2}2381^{1/2}2467^{1/2}104446171^{1/2}\approx 243947516.5618645$ | ||
| Ramified primes: |
\(97\), \(2381\), \(2467\), \(104446171\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-239299}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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: | $14$ | 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: |
$a^{2}-a-1$, $a^{20}-10a^{19}+35a^{18}-30a^{17}-106a^{16}+236a^{15}+69a^{14}-533a^{13}+99a^{12}+719a^{11}-212a^{10}-684a^{9}+152a^{8}+465a^{7}-29a^{6}-206a^{5}-18a^{4}+49a^{3}+9a^{2}-6a-1$, $a^{16}-8a^{15}+19a^{14}+7a^{13}-85a^{12}+55a^{11}+153a^{10}-149a^{9}-168a^{8}+169a^{7}+131a^{6}-95a^{5}-70a^{4}+23a^{3}+20a^{2}-3a-3$, $a^{18}-9a^{17}+26a^{16}-4a^{15}-112a^{14}+140a^{13}+169a^{12}-364a^{11}-122a^{10}+467a^{9}+59a^{8}-360a^{7}-41a^{6}+168a^{5}+28a^{4}-46a^{3}-8a^{2}+8a+1$, $a^{20}-10a^{19}+34a^{18}-21a^{17}-133a^{16}+248a^{15}+160a^{14}-666a^{13}-12a^{12}+1008a^{11}-133a^{10}-1035a^{9}+92a^{8}+738a^{7}+32a^{6}-337a^{5}-60a^{4}+85a^{3}+21a^{2}-12a-3$, $a$, $a^{16}-8a^{15}+20a^{14}-72a^{12}+68a^{11}+93a^{10}-135a^{9}-71a^{8}+126a^{7}+44a^{6}-68a^{5}-19a^{4}+21a^{3}+5a^{2}-5a$, $a-1$, $3a^{21}-31a^{20}+111a^{19}-87a^{18}-412a^{17}+899a^{16}+360a^{15}-2409a^{14}+523a^{13}+3644a^{12}-1536a^{11}-3734a^{10}+1612a^{9}+2748a^{8}-867a^{7}-1418a^{6}+231a^{5}+472a^{4}-29a^{3}-97a^{2}+2a+9$, $2a^{21}-21a^{20}+77a^{19}-66a^{18}-279a^{17}+651a^{16}+199a^{15}-1736a^{14}+521a^{13}+2629a^{12}-1352a^{11}-2723a^{10}+1454a^{9}+2055a^{8}-851a^{7}-1106a^{6}+267a^{5}+386a^{4}-42a^{3}-79a^{2}+4a+7$, $a^{20}-11a^{19}+44a^{18}-55a^{17}-111a^{16}+374a^{15}-76a^{14}-807a^{13}+589a^{12}+1003a^{11}-993a^{10}-884a^{9}+908a^{8}+600a^{7}-491a^{6}-298a^{5}+157a^{4}+89a^{3}-32a^{2}-14a+4$, $a^{21}-10a^{20}+34a^{19}-20a^{18}-141a^{17}+266a^{16}+176a^{15}-778a^{14}+56a^{13}+1245a^{12}-404a^{11}-1292a^{10}+489a^{9}+919a^{8}-282a^{7}-438a^{6}+83a^{5}+128a^{4}-18a^{3}-22a^{2}+4a+1$, $2a^{21}-21a^{20}+78a^{19}-76a^{18}-244a^{17}+620a^{16}+101a^{15}-1519a^{14}+584a^{13}+2175a^{12}-1300a^{11}-2141a^{10}+1350a^{9}+1532a^{8}-803a^{7}-782a^{6}+284a^{5}+259a^{4}-62a^{3}-54a^{2}+9a+4$, $a^{21}-10a^{20}+34a^{19}-21a^{18}-133a^{17}+247a^{16}+169a^{15}-693a^{14}+a^{13}+1093a^{12}-260a^{11}-1120a^{10}+334a^{9}+771a^{8}-207a^{7}-346a^{6}+81a^{5}+95a^{4}-26a^{3}-18a^{2}+5a+1$
| 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: | \( 439864.358869 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{7}\cdot 439864.358869 \cdot 1}{2\cdot\sqrt{14799420881562079578038490499}}\cr\approx \mathstrut & 0.178922707378 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 + 44*x^20 - 55*x^19 - 112*x^18 + 381*x^17 - 87*x^16 - 834*x^15 + 674*x^14 + 1020*x^13 - 1213*x^12 - 834*x^11 + 1220*x^10 + 511*x^9 - 776*x^8 - 247*x^7 + 323*x^6 + 85*x^5 - 89*x^4 - 18*x^3 + 16*x^2 + x - 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 - 11*x^21 + 44*x^20 - 55*x^19 - 112*x^18 + 381*x^17 - 87*x^16 - 834*x^15 + 674*x^14 + 1020*x^13 - 1213*x^12 - 834*x^11 + 1220*x^10 + 511*x^9 - 776*x^8 - 247*x^7 + 323*x^6 + 85*x^5 - 89*x^4 - 18*x^3 + 16*x^2 + x - 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 - 11*x^21 + 44*x^20 - 55*x^19 - 112*x^18 + 381*x^17 - 87*x^16 - 834*x^15 + 674*x^14 + 1020*x^13 - 1213*x^12 - 834*x^11 + 1220*x^10 + 511*x^9 - 776*x^8 - 247*x^7 + 323*x^6 + 85*x^5 - 89*x^4 - 18*x^3 + 16*x^2 + x - 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 - 11*x^21 + 44*x^20 - 55*x^19 - 112*x^18 + 381*x^17 - 87*x^16 - 834*x^15 + 674*x^14 + 1020*x^13 - 1213*x^12 - 834*x^11 + 1220*x^10 + 511*x^9 - 776*x^8 - 247*x^7 + 323*x^6 + 85*x^5 - 89*x^4 - 18*x^3 + 16*x^2 + x - 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 $C_2^{10}.(C_2\times S_{11})$ |
| Character table for $C_2^{10}.(C_2\times S_{11})$ |
Intermediate fields
| 11.5.248686333151.1 |
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 | $22$ | $22$ | ${\href{/padicField/5.11.0.1}{11} }^{2}$ | $16{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | $18{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.7.0.1}{7} }^{2}{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.9.0.1}{9} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | $16{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.11.0.1}{11} }^{2}$ | ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.5.0.1}{5} }^{2}$ | $20{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.11.0.1}{11} }^{2}$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(97\)
| 97.2.1.2 | $x^{2} + 485$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 97.10.0.1 | $x^{10} + 22 x^{5} + 66 x^{4} + 34 x^{3} + 34 x^{2} + 20 x + 5$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
| 97.10.0.1 | $x^{10} + 22 x^{5} + 66 x^{4} + 34 x^{3} + 34 x^{2} + 20 x + 5$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
|
\(2381\)
| Deg $4$ | $2$ | $2$ | $2$ | |||
| Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
|
\(2467\)
| 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
|
\(104446171\)
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ |