Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 12*x^20 + 58*x^18 - 144*x^16 + 193*x^14 - 130*x^12 + 21*x^10 + 40*x^8 - 45*x^6 + 18*x^4 - 1)
gp: K = bnfinit(y^22 - 12*y^20 + 58*y^18 - 144*y^16 + 193*y^14 - 130*y^12 + 21*y^10 + 40*y^8 - 45*y^6 + 18*y^4 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 12*x^20 + 58*x^18 - 144*x^16 + 193*x^14 - 130*x^12 + 21*x^10 + 40*x^8 - 45*x^6 + 18*x^4 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 12*x^20 + 58*x^18 - 144*x^16 + 193*x^14 - 130*x^12 + 21*x^10 + 40*x^8 - 45*x^6 + 18*x^4 - 1)
\( x^{22} - 12x^{20} + 58x^{18} - 144x^{16} + 193x^{14} - 130x^{12} + 21x^{10} + 40x^{8} - 45x^{6} + 18x^{4} - 1 \)
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: | $[14, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(315595142827230969392399757869056\)
\(\medspace = 2^{22}\cdot 8674315276967^{2}\)
| 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: |
\(2\), \(8674315276967\)
| 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: | $17$ | 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: |
$a$, $a^{20}-12a^{18}+57a^{16}-134a^{14}+156a^{12}-69a^{10}-20a^{8}+44a^{6}-35a^{4}+5a^{2}+3$, $a^{21}-11a^{19}+47a^{17}-97a^{15}+96a^{13}-34a^{11}-13a^{9}+27a^{7}-18a^{5}$, $a^{20}-11a^{18}+48a^{16}-106a^{14}+125a^{12}-73a^{10}+5a^{8}+29a^{6}-27a^{4}+9a^{2}$, $4a^{21}-46a^{19}+209a^{17}-472a^{15}+540a^{13}-261a^{11}-35a^{9}+140a^{7}-113a^{5}+21a^{3}+7a$, $4a^{20}-46a^{18}+209a^{16}-472a^{14}+540a^{12}-261a^{10}-35a^{8}+140a^{6}-112a^{4}+18a^{2}+8$, $4a^{20}-45a^{18}+199a^{16}-434a^{14}+472a^{12}-204a^{10}-51a^{8}+129a^{6}-95a^{4}+11a^{2}+8$, $4a^{21}-45a^{19}+199a^{17}-434a^{15}+472a^{13}-204a^{11}-51a^{9}+129a^{7}-95a^{5}+11a^{3}+9a$, $6a^{21}-69a^{19}+313a^{17}-703a^{15}+792a^{13}-365a^{11}-63a^{9}+205a^{7}-165a^{5}+22a^{3}+12a$, $a-1$, $4a^{21}-45a^{19}-a^{18}+199a^{17}+10a^{16}-434a^{15}-37a^{14}+472a^{13}+60a^{12}-204a^{11}-35a^{10}-50a^{9}-7a^{8}+124a^{7}+16a^{6}-89a^{5}-13a^{4}+12a^{3}+3a^{2}+7a+1$, $a^{2}+a-1$, $a^{21}+a^{20}-11a^{19}-12a^{18}+47a^{17}+57a^{16}-97a^{15}-134a^{14}+96a^{13}+156a^{12}-34a^{11}-69a^{10}-13a^{9}-20a^{8}+27a^{7}+44a^{6}-18a^{5}-34a^{4}+2a^{2}+4$, $a^{21}+a^{20}-11a^{19}-12a^{18}+47a^{17}+57a^{16}-97a^{15}-134a^{14}+96a^{13}+156a^{12}-34a^{11}-70a^{10}-13a^{9}-15a^{8}+28a^{7}+38a^{6}-22a^{5}-35a^{4}+2a^{3}+5a^{2}+3a+3$, $2a^{21}+a^{20}-23a^{19}-11a^{18}+105a^{17}+47a^{16}-240a^{15}-97a^{14}+281a^{13}+96a^{12}-142a^{11}-34a^{10}-14a^{9}-13a^{8}+68a^{7}+28a^{6}-56a^{5}-21a^{4}+14a^{3}+3a+2$, $a^{21}+4a^{20}-11a^{19}-45a^{18}+47a^{17}+199a^{16}-97a^{15}-434a^{14}+96a^{13}+472a^{12}-34a^{11}-204a^{10}-13a^{9}-51a^{8}+27a^{7}+129a^{6}-18a^{5}-95a^{4}+12a^{2}+7$, $4a^{21}+3a^{20}-46a^{19}-36a^{18}+209a^{17}+171a^{16}-472a^{15}-404a^{14}+540a^{13}+482a^{12}-261a^{11}-240a^{10}-35a^{9}-30a^{8}+140a^{7}+122a^{6}-113a^{5}-103a^{4}+20a^{3}+17a^{2}+8a+8$
| 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: | \( 262128409.062 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{4}\cdot 262128409.062 \cdot 1}{2\cdot\sqrt{315595142827230969392399757869056}}\cr\approx \mathstrut & 0.188390220060 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 12*x^20 + 58*x^18 - 144*x^16 + 193*x^14 - 130*x^12 + 21*x^10 + 40*x^8 - 45*x^6 + 18*x^4 - 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 - 12*x^20 + 58*x^18 - 144*x^16 + 193*x^14 - 130*x^12 + 21*x^10 + 40*x^8 - 45*x^6 + 18*x^4 - 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 - 12*x^20 + 58*x^18 - 144*x^16 + 193*x^14 - 130*x^12 + 21*x^10 + 40*x^8 - 45*x^6 + 18*x^4 - 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 - 12*x^20 + 58*x^18 - 144*x^16 + 193*x^14 - 130*x^12 + 21*x^10 + 40*x^8 - 45*x^6 + 18*x^4 - 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
| 11.9.8674315276967.1 |
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: | 22.4.87750002479670632534243431065508320574647330919687899613924773423575897287324522118580170655687148440732006564632195586415317917246456646341473861632.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.11.0.1}{11} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{3}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | $18{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.11.0.1}{11} }^{2}$ | $16{,}\,{\href{/padicField/13.6.0.1}{6} }$ | $18{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.11.0.1}{11} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.5.0.1}{5} }^{2}$ | ${\href{/padicField/47.7.0.1}{7} }^{2}{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 $22$ | $2$ | $11$ | $22$ | |||
|
\(8674315276967\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |