Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^20 - 4*x^18 + x^16 + 5*x^14 + 7*x^12 - 7*x^10 - 5*x^8 + 7*x^6 - 3*x^4 - x^2 + 1)
gp: K = bnfinit(y^22 - y^20 - 4*y^18 + y^16 + 5*y^14 + 7*y^12 - 7*y^10 - 5*y^8 + 7*y^6 - 3*y^4 - y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - x^20 - 4*x^18 + x^16 + 5*x^14 + 7*x^12 - 7*x^10 - 5*x^8 + 7*x^6 - 3*x^4 - x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - x^20 - 4*x^18 + x^16 + 5*x^14 + 7*x^12 - 7*x^10 - 5*x^8 + 7*x^6 - 3*x^4 - x^2 + 1)
\( x^{22} - x^{20} - 4x^{18} + x^{16} + 5x^{14} + 7x^{12} - 7x^{10} - 5x^{8} + 7x^{6} - 3x^{4} - x^{2} + 1 \)
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: | $[8, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-1562900555060447098970765787136\)
\(\medspace = -\,2^{22}\cdot 610429790897^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(23.57\) | 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: |
\(2\), \(610429790897\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{97}a^{20}+\frac{38}{97}a^{18}+\frac{23}{97}a^{16}+\frac{25}{97}a^{14}+\frac{10}{97}a^{12}+\frac{9}{97}a^{10}-\frac{44}{97}a^{8}+\frac{25}{97}a^{6}+\frac{12}{97}a^{4}-\frac{20}{97}a^{2}-\frac{5}{97}$, $\frac{1}{97}a^{21}+\frac{38}{97}a^{19}+\frac{23}{97}a^{17}+\frac{25}{97}a^{15}+\frac{10}{97}a^{13}+\frac{9}{97}a^{11}-\frac{44}{97}a^{9}+\frac{25}{97}a^{7}+\frac{12}{97}a^{5}-\frac{20}{97}a^{3}-\frac{5}{97}a$
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: | $14$ | 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: |
$a^{21}-a^{19}-4a^{17}+a^{15}+5a^{13}+7a^{11}-7a^{9}-5a^{7}+7a^{5}-3a^{3}-a$, $\frac{196}{97}a^{20}-\frac{21}{97}a^{18}-\frac{827}{97}a^{16}-\frac{532}{97}a^{14}+\frac{602}{97}a^{12}+\frac{1958}{97}a^{10}+\frac{300}{97}a^{8}-\frac{920}{97}a^{6}+\frac{509}{97}a^{4}-\frac{137}{97}a^{2}-\frac{301}{97}$, $\frac{261}{97}a^{21}-\frac{73}{97}a^{19}-\frac{1078}{97}a^{17}-\frac{556}{97}a^{15}+\frac{864}{97}a^{13}+\frac{2543}{97}a^{11}+\frac{59}{97}a^{9}-\frac{1235}{97}a^{7}+\frac{610}{97}a^{5}-\frac{273}{97}a^{3}-\frac{238}{97}a$, $\frac{89}{97}a^{20}-\frac{13}{97}a^{18}-\frac{378}{97}a^{16}-\frac{200}{97}a^{14}+\frac{308}{97}a^{12}+\frac{801}{97}a^{10}-\frac{36}{97}a^{8}-\frac{491}{97}a^{6}+\frac{486}{97}a^{4}+\frac{63}{97}a^{2}-\frac{154}{97}$, $\frac{301}{97}a^{21}-\frac{105}{97}a^{19}-\frac{1225}{97}a^{17}-\frac{526}{97}a^{15}+\frac{973}{97}a^{13}+\frac{2709}{97}a^{11}-\frac{149}{97}a^{9}-\frac{1205}{97}a^{7}+\frac{1187}{97}a^{5}-\frac{394}{97}a^{3}-\frac{341}{97}a$, $\frac{42}{97}a^{20}+\frac{44}{97}a^{18}-\frac{198}{97}a^{16}-\frac{308}{97}a^{14}+\frac{32}{97}a^{12}+\frac{572}{97}a^{10}+\frac{577}{97}a^{8}-\frac{211}{97}a^{6}-\frac{78}{97}a^{4}+\frac{33}{97}a^{2}-\frac{113}{97}$, $\frac{183}{97}a^{20}-\frac{127}{97}a^{18}-\frac{738}{97}a^{16}-\frac{81}{97}a^{14}+\frac{763}{97}a^{12}+\frac{1550}{97}a^{10}-\frac{680}{97}a^{8}-\frac{857}{97}a^{6}+\frac{838}{97}a^{4}-\frac{362}{97}a^{2}-\frac{42}{97}$, $\frac{315}{97}a^{21}-\frac{155}{97}a^{19}-\frac{1291}{97}a^{17}-\frac{370}{97}a^{15}+\frac{1210}{97}a^{13}+\frac{2738}{97}a^{11}-\frac{668}{97}a^{9}-\frac{1437}{97}a^{7}+\frac{1452}{97}a^{5}-\frac{286}{97}a^{3}-\frac{508}{97}a$, $a^{21}-a^{19}-4a^{17}+a^{15}+5a^{13}+7a^{11}-7a^{9}-5a^{7}+7a^{5}-3a^{3}-a-1$, $\frac{210}{97}a^{21}+\frac{34}{97}a^{20}-\frac{71}{97}a^{19}+\frac{31}{97}a^{18}-\frac{893}{97}a^{17}-\frac{188}{97}a^{16}-\frac{376}{97}a^{15}-\frac{217}{97}a^{14}+\frac{839}{97}a^{13}+\frac{146}{97}a^{12}+\frac{1987}{97}a^{11}+\frac{500}{97}a^{10}-\frac{219}{97}a^{9}+\frac{347}{97}a^{8}-\frac{1152}{97}a^{7}-\frac{411}{97}a^{6}+\frac{774}{97}a^{5}-\frac{77}{97}a^{4}-\frac{29}{97}a^{3}+\frac{96}{97}a^{2}-\frac{371}{97}a-\frac{73}{97}$, $\frac{371}{97}a^{21}-\frac{196}{97}a^{20}-\frac{161}{97}a^{19}+\frac{21}{97}a^{18}-\frac{1555}{97}a^{17}+\frac{827}{97}a^{16}-\frac{522}{97}a^{15}+\frac{532}{97}a^{14}+\frac{1479}{97}a^{13}-\frac{602}{97}a^{12}+\frac{3436}{97}a^{11}-\frac{1958}{97}a^{10}-\frac{610}{97}a^{9}-\frac{300}{97}a^{8}-\frac{2074}{97}a^{7}+\frac{920}{97}a^{6}+\frac{1445}{97}a^{5}-\frac{509}{97}a^{4}-\frac{339}{97}a^{3}+\frac{137}{97}a^{2}-\frac{400}{97}a+\frac{301}{97}$, $\frac{113}{97}a^{21}-\frac{118}{97}a^{20}-\frac{71}{97}a^{19}-\frac{22}{97}a^{18}-\frac{408}{97}a^{17}+\frac{487}{97}a^{16}-\frac{85}{97}a^{15}+\frac{445}{97}a^{14}+\frac{257}{97}a^{13}-\frac{210}{97}a^{12}+\frac{823}{97}a^{11}-\frac{1159}{97}a^{10}-\frac{219}{97}a^{9}-\frac{531}{97}a^{8}+\frac{12}{97}a^{7}+\frac{348}{97}a^{6}+\frac{580}{97}a^{5}-\frac{252}{97}a^{4}-\frac{417}{97}a^{3}+\frac{32}{97}a^{2}+\frac{17}{97}a+\frac{202}{97}$, $\frac{76}{97}a^{21}-\frac{8}{97}a^{20}-\frac{22}{97}a^{19}-\frac{13}{97}a^{18}-\frac{289}{97}a^{17}+\frac{10}{97}a^{16}-\frac{137}{97}a^{15}+\frac{91}{97}a^{14}+\frac{178}{97}a^{13}+\frac{114}{97}a^{12}+\frac{587}{97}a^{11}-\frac{72}{97}a^{10}-\frac{46}{97}a^{9}-\frac{230}{97}a^{8}-\frac{137}{97}a^{7}-\frac{200}{97}a^{6}+\frac{427}{97}a^{5}+\frac{98}{97}a^{4}-\frac{65}{97}a^{3}+\frac{63}{97}a^{2}-\frac{186}{97}a-\frac{57}{97}$, $\frac{400}{97}a^{21}-\frac{99}{97}a^{20}-\frac{126}{97}a^{19}+\frac{21}{97}a^{18}-\frac{1664}{97}a^{17}+\frac{439}{97}a^{16}-\frac{767}{97}a^{15}+\frac{241}{97}a^{14}+\frac{1381}{97}a^{13}-\frac{408}{97}a^{12}+\frac{3794}{97}a^{11}-\frac{1085}{97}a^{10}-\frac{43}{97}a^{9}-\frac{106}{97}a^{8}-\frac{1931}{97}a^{7}+\frac{726}{97}a^{6}+\frac{1211}{97}a^{5}-\frac{121}{97}a^{4}-\frac{434}{97}a^{3}+\frac{40}{97}a^{2}-\frac{448}{97}a+\frac{204}{97}$
| 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: | \( 4235240.70716 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{7}\cdot 4235240.70716 \cdot 1}{2\cdot\sqrt{1562900555060447098970765787136}}\cr\approx \mathstrut & 0.167641475126 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - x^20 - 4*x^18 + x^16 + 5*x^14 + 7*x^12 - 7*x^10 - 5*x^8 + 7*x^6 - 3*x^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 - x^20 - 4*x^18 + x^16 + 5*x^14 + 7*x^12 - 7*x^10 - 5*x^8 + 7*x^6 - 3*x^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 - x^20 - 4*x^18 + x^16 + 5*x^14 + 7*x^12 - 7*x^10 - 5*x^8 + 7*x^6 - 3*x^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 - x^20 - 4*x^18 + x^16 + 5*x^14 + 7*x^12 - 7*x^10 - 5*x^8 + 7*x^6 - 3*x^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
$C_2^{10}.(C_2\times S_{11})$ (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
| 11.7.610429790897.1 |
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 | ${\href{/padicField/3.8.0.1}{8} }^{2}{,}\,{\href{/padicField/3.6.0.1}{6} }$ | $18{,}\,{\href{/padicField/5.4.0.1}{4} }$ | $16{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | $22$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.6.0.1}{6} }^{2}$ | $22$ | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.10.0.1}{10} }$ | $16{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.11.0.1}{11} }^{2}$ | ${\href{/padicField/41.7.0.1}{7} }^{2}{,}\,{\href{/padicField/41.4.0.1}{4} }^{2}$ | $20{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.11.0.1}{11} }^{2}$ | ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $22$ | $2$ | $11$ | $22$ | |||
|
\(610429790897\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ |