Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 12*x^18 + 162*x^16 - 1080*x^14 + 7452*x^12 - 38880*x^10 + 256608*x^8 - 769824*x^6 + 3464208*x^4 - 13856832*x^2 + 20785248)
gp: K = bnfinit(y^20 - 12*y^18 + 162*y^16 - 1080*y^14 + 7452*y^12 - 38880*y^10 + 256608*y^8 - 769824*y^6 + 3464208*y^4 - 13856832*y^2 + 20785248, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 12*x^18 + 162*x^16 - 1080*x^14 + 7452*x^12 - 38880*x^10 + 256608*x^8 - 769824*x^6 + 3464208*x^4 - 13856832*x^2 + 20785248);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 12*x^18 + 162*x^16 - 1080*x^14 + 7452*x^12 - 38880*x^10 + 256608*x^8 - 769824*x^6 + 3464208*x^4 - 13856832*x^2 + 20785248)
\( x^{20} - 12 x^{18} + 162 x^{16} - 1080 x^{14} + 7452 x^{12} - 38880 x^{10} + 256608 x^{8} + \cdots + 20785248 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(5016449852600330836716407488512\)
\(\medspace = 2^{55}\cdot 3^{10}\cdot 11^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(34.28\) | 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: | $2^{11/4}3^{1/2}11^{9/10}\approx 100.84317994603103$ | ||
| Ramified primes: |
\(2\), \(3\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{22}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $10$ | 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 \(a\))
$1$, $a$, $\frac{1}{3}a^{2}$, $\frac{1}{3}a^{3}$, $\frac{1}{18}a^{4}$, $\frac{1}{18}a^{5}$, $\frac{1}{54}a^{6}$, $\frac{1}{54}a^{7}$, $\frac{1}{324}a^{8}$, $\frac{1}{324}a^{9}$, $\frac{1}{972}a^{10}$, $\frac{1}{972}a^{11}$, $\frac{1}{5832}a^{12}$, $\frac{1}{5832}a^{13}$, $\frac{1}{17496}a^{14}$, $\frac{1}{17496}a^{15}$, $\frac{1}{104976}a^{16}$, $\frac{1}{104976}a^{17}$, $\frac{1}{6337718462448}a^{18}+\frac{837739}{528143205204}a^{16}-\frac{1251545}{44011933767}a^{14}+\frac{284801}{19560859452}a^{12}+\frac{203587}{9780429726}a^{10}-\frac{544400}{543357207}a^{8}-\frac{7047079}{1086714414}a^{6}+\frac{5489405}{362238138}a^{4}+\frac{970885}{60373023}a^{2}-\frac{4788209}{20124341}$, $\frac{1}{6337718462448}a^{19}+\frac{837739}{528143205204}a^{17}-\frac{1251545}{44011933767}a^{15}+\frac{284801}{19560859452}a^{13}+\frac{203587}{9780429726}a^{11}-\frac{544400}{543357207}a^{9}-\frac{7047079}{1086714414}a^{7}+\frac{5489405}{362238138}a^{5}+\frac{970885}{60373023}a^{3}-\frac{4788209}{20124341}a$
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
$C_{2}\times C_{2}$, which has order $4$ (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: | $9$ | 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: |
$\frac{1170421}{6337718462448}a^{18}-\frac{1772507}{1056286410408}a^{16}+\frac{1516919}{58682578356}a^{14}-\frac{837701}{6520286484}a^{12}+\frac{381581}{362238138}a^{10}-\frac{23816237}{6520286484}a^{8}+\frac{1209431}{40248682}a^{6}-\frac{390416}{60373023}a^{4}+\frac{7759340}{20124341}a^{2}-\frac{14008650}{20124341}$, $\frac{2897}{352095470136}a^{18}-\frac{701327}{2112572820816}a^{16}+\frac{57431}{14670644589}a^{14}-\frac{546}{20124341}a^{12}+\frac{1315649}{19560859452}a^{10}+\frac{1363912}{1630071621}a^{8}-\frac{3257437}{543357207}a^{6}+\frac{943373}{181119069}a^{4}-\frac{5072746}{60373023}a^{2}+\frac{16876614}{20124341}$, $\frac{2651761}{6337718462448}a^{18}-\frac{2145929}{528143205204}a^{16}+\frac{3258913}{58682578356}a^{14}-\frac{32631263}{117365156712}a^{12}+\frac{37114423}{19560859452}a^{10}-\frac{5490467}{724476276}a^{8}+\frac{12071455}{181119069}a^{6}-\frac{9993563}{120746046}a^{4}+\frac{48034355}{60373023}a^{2}-\frac{54921555}{20124341}$, $\frac{292961}{1584429615612}a^{18}-\frac{1534481}{704190940272}a^{16}+\frac{4773331}{176047735068}a^{14}-\frac{18492941}{117365156712}a^{12}+\frac{16408087}{19560859452}a^{10}-\frac{22545095}{6520286484}a^{8}+\frac{13291519}{543357207}a^{6}-\frac{9643915}{181119069}a^{4}+\frac{16267846}{60373023}a^{2}-\frac{25602799}{20124341}$, $\frac{11924231}{3168859231224}a^{19}+\frac{13714555}{3168859231224}a^{18}-\frac{80358221}{2112572820816}a^{17}-\frac{10177847}{264071602602}a^{16}+\frac{15444581}{29341289178}a^{15}+\frac{32882039}{58682578356}a^{14}-\frac{341412655}{117365156712}a^{13}-\frac{83347087}{29341289178}a^{12}+\frac{99659131}{4890214863}a^{11}+\frac{437521531}{19560859452}a^{10}-\frac{200569853}{2173428828}a^{9}-\frac{177769567}{1630071621}a^{8}+\frac{381527420}{543357207}a^{7}+\frac{17054234}{20124341}a^{6}-\frac{222740321}{181119069}a^{5}-\frac{403171289}{362238138}a^{4}+\frac{516351163}{60373023}a^{3}+\frac{225122995}{20124341}a^{2}-\frac{556475852}{20124341}a-\frac{877378472}{20124341}$, $\frac{9035}{7710119784}a^{19}+\frac{4655}{2891294919}a^{18}-\frac{49369}{3855059892}a^{17}-\frac{14383}{963764973}a^{16}+\frac{240379}{1285019964}a^{15}+\frac{136001}{642509982}a^{14}-\frac{263279}{214169994}a^{13}-\frac{115612}{107084997}a^{12}+\frac{336983}{35694999}a^{11}+\frac{285257}{35694999}a^{10}-\frac{569933}{11898333}a^{9}-\frac{405416}{11898333}a^{8}+\frac{412085}{1322037}a^{7}+\frac{1111264}{3966111}a^{6}-\frac{325238}{440679}a^{5}-\frac{257752}{1322037}a^{4}+\frac{1970452}{440679}a^{3}+\frac{1379048}{440679}a^{2}-\frac{2121812}{146893}a-\frac{2653245}{146893}$, $\frac{10555555}{2112572820816}a^{19}+\frac{73029883}{6337718462448}a^{18}-\frac{29268631}{528143205204}a^{17}-\frac{15316421}{78243437808}a^{16}+\frac{54499831}{88023867534}a^{15}+\frac{894765527}{352095470136}a^{14}-\frac{22842413}{9780429726}a^{13}-\frac{2348495621}{117365156712}a^{12}+\frac{12168541}{6520286484}a^{11}+\frac{128708215}{1086714414}a^{10}+\frac{541179497}{6520286484}a^{9}-\frac{182015515}{362238138}a^{8}-\frac{403265861}{1086714414}a^{7}+\frac{2358955697}{1086714414}a^{6}+\frac{689055479}{181119069}a^{5}-\frac{911972989}{181119069}a^{4}-\frac{895768796}{60373023}a^{3}-\frac{23400675}{20124341}a^{2}+\frac{377781902}{20124341}a+\frac{319222258}{20124341}$, $\frac{2501321}{1584429615612}a^{19}-\frac{5096053}{2112572820816}a^{18}-\frac{26524883}{2112572820816}a^{17}+\frac{5818037}{528143205204}a^{16}+\frac{69391181}{352095470136}a^{15}-\frac{76215623}{352095470136}a^{14}-\frac{87922853}{117365156712}a^{13}+\frac{16062701}{117365156712}a^{12}+\frac{20193131}{3260143242}a^{11}-\frac{99818887}{19560859452}a^{10}-\frac{29525137}{2173428828}a^{9}+\frac{9286303}{1630071621}a^{8}+\frac{65989351}{362238138}a^{7}-\frac{172440098}{543357207}a^{6}+\frac{73647161}{181119069}a^{5}-\frac{12708156}{20124341}a^{4}+\frac{161999050}{60373023}a^{3}-\frac{152430737}{20124341}a^{2}+\frac{51806696}{20124341}a+\frac{127145547}{20124341}$, $\frac{15972497}{6337718462448}a^{19}+\frac{1384807}{352095470136}a^{18}-\frac{3804221}{176047735068}a^{17}-\frac{175709}{4890214863}a^{16}+\frac{14131075}{44011933767}a^{15}+\frac{172375715}{352095470136}a^{14}-\frac{90754085}{58682578356}a^{13}-\frac{2416511}{1086714414}a^{12}+\frac{60926150}{4890214863}a^{11}+\frac{101840021}{6520286484}a^{10}-\frac{63344653}{1086714414}a^{9}-\frac{42454625}{724476276}a^{8}+\frac{86500577}{181119069}a^{7}+\frac{330301006}{543357207}a^{6}-\frac{92219099}{120746046}a^{5}-\frac{26422856}{60373023}a^{4}+\frac{148516295}{20124341}a^{3}+\frac{485293711}{60373023}a^{2}-\frac{456682320}{20124341}a-\frac{658756728}{20124341}$
| 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: | \( 3611756.8811914893 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 3611756.8811914893 \cdot 4}{2\cdot\sqrt{5016449852600330836716407488512}}\cr\approx \mathstrut & 0.309277944673370 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 12*x^18 + 162*x^16 - 1080*x^14 + 7452*x^12 - 38880*x^10 + 256608*x^8 - 769824*x^6 + 3464208*x^4 - 13856832*x^2 + 20785248)
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^20 - 12*x^18 + 162*x^16 - 1080*x^14 + 7452*x^12 - 38880*x^10 + 256608*x^8 - 769824*x^6 + 3464208*x^4 - 13856832*x^2 + 20785248, 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^20 - 12*x^18 + 162*x^16 - 1080*x^14 + 7452*x^12 - 38880*x^10 + 256608*x^8 - 769824*x^6 + 3464208*x^4 - 13856832*x^2 + 20785248);
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^20 - 12*x^18 + 162*x^16 - 1080*x^14 + 7452*x^12 - 38880*x^10 + 256608*x^8 - 769824*x^6 + 3464208*x^4 - 13856832*x^2 + 20785248);
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
$C_{10}\wr C_2$ (as 20T53):
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 solvable group of order 200 |
| The 65 conjugacy class representatives for $C_{10}\wr C_2$ |
| Character table for $C_{10}\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{-2}) \), 4.0.202752.6, 10.0.479756288.1 |
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 20 siblings: | data not computed |
| Degree 40 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 | R | $20$ | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{10}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | $20$ | $20$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }^{2}$ | $20$ | $20$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{5}$ |
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 $20$ | $4$ | $5$ | $55$ | |||
|
\(3\)
| 3.10.5.1 | $x^{10} + 162 x^{2} - 243$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
| 3.10.5.1 | $x^{10} + 162 x^{2} - 243$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
|
\(11\)
| 11.10.9.4 | $x^{10} + 22$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |
| 11.10.0.1 | $x^{10} + 7 x^{5} + 8 x^{4} + 10 x^{3} + 6 x^{2} + 6 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |