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: |
$\frac{2651761}{6337718462448}a^{18}+\frac{2145929}{528143205204}a^{16}+\frac{3258913}{58682578356}a^{14}+\frac{32631263}{117365156712}a^{12}+\frac{37114423}{19560859452}a^{10}+\frac{5490467}{724476276}a^{8}+\frac{12071455}{181119069}a^{6}+\frac{9993563}{120746046}a^{4}+\frac{48034355}{60373023}a^{2}+\frac{54921555}{20124341}$, $\frac{988555}{6337718462448}a^{18}+\frac{3090407}{2112572820816}a^{16}+\frac{2072857}{117365156712}a^{14}+\frac{8592551}{117365156712}a^{12}+\frac{6949229}{19560859452}a^{10}+\frac{5165527}{3260143242}a^{8}+\frac{17943125}{1086714414}a^{6}+\frac{564661}{20124341}a^{4}+\frac{3150203}{60373023}a^{2}+\frac{22074408}{20124341}$, $\frac{292961}{1584429615612}a^{18}+\frac{1534481}{704190940272}a^{16}+\frac{4773331}{176047735068}a^{14}+\frac{18492941}{117365156712}a^{12}+\frac{16408087}{19560859452}a^{10}+\frac{22545095}{6520286484}a^{8}+\frac{13291519}{543357207}a^{6}+\frac{9643915}{181119069}a^{4}+\frac{16267846}{60373023}a^{2}+\frac{25602799}{20124341}$, $\frac{1465}{6337718462448}a^{18}+\frac{148583}{264071602602}a^{16}+\frac{2537189}{352095470136}a^{14}+\frac{12141835}{117365156712}a^{12}+\frac{12904021}{19560859452}a^{10}+\frac{5269061}{1086714414}a^{8}+\frac{19940539}{1086714414}a^{6}+\frac{48006757}{362238138}a^{4}+\frac{13642655}{60373023}a^{2}+\frac{31579858}{20124341}$, $\frac{3769169}{3168859231224}a^{19}-\frac{4204687}{792214807806}a^{18}+\frac{5825987}{704190940272}a^{17}-\frac{7115461}{132035801301}a^{16}+\frac{48679253}{352095470136}a^{15}-\frac{11243846}{14670644589}a^{14}+\frac{17864671}{39121718904}a^{13}-\frac{257990465}{58682578356}a^{12}+\frac{5055343}{1086714414}a^{11}-\frac{633155731}{19560859452}a^{10}+\frac{79003595}{6520286484}a^{9}-\frac{245690986}{1630071621}a^{8}+\frac{187107799}{1086714414}a^{7}-\frac{196229638}{181119069}a^{6}-\frac{36317774}{181119069}a^{5}-\frac{750250505}{362238138}a^{4}+\frac{179824637}{60373023}a^{3}-\frac{301028789}{20124341}a^{2}-\frac{41155686}{20124341}a-\frac{975050222}{20124341}$, $\frac{11936233}{704190940272}a^{19}-\frac{55142665}{6337718462448}a^{18}+\frac{324257501}{2112572820816}a^{17}-\frac{229826279}{2112572820816}a^{16}+\frac{400495117}{176047735068}a^{15}-\frac{485955611}{352095470136}a^{14}+\frac{335956213}{29341289178}a^{13}-\frac{1111397701}{117365156712}a^{12}+\frac{65335249}{724476276}a^{11}-\frac{1159809811}{19560859452}a^{10}+\frac{1248180335}{3260143242}a^{9}-\frac{2147653895}{6520286484}a^{8}+\frac{567573557}{181119069}a^{7}-\frac{1091347280}{543357207}a^{6}+\frac{1269220547}{362238138}a^{5}-\frac{265539349}{40248682}a^{4}+\frac{2741244203}{60373023}a^{3}-\frac{395598508}{20124341}a^{2}+\frac{2058477553}{20124341}a-\frac{2848430165}{20124341}$, $\frac{185099}{117365156712}a^{19}+\frac{208639}{234730313424}a^{18}+\frac{42665597}{2112572820816}a^{17}+\frac{2478215}{528143205204}a^{16}+\frac{77482957}{352095470136}a^{15}-\frac{8600177}{352095470136}a^{14}+\frac{125341759}{117365156712}a^{13}-\frac{90620717}{58682578356}a^{12}+\frac{13052917}{6520286484}a^{11}-\frac{190109807}{9780429726}a^{10}-\frac{38170111}{2173428828}a^{9}-\frac{1010545507}{6520286484}a^{8}-\frac{51366320}{543357207}a^{7}-\frac{442490228}{543357207}a^{6}-\frac{117306431}{120746046}a^{5}-\frac{1392513007}{362238138}a^{4}-\frac{110026975}{20124341}a^{3}-\frac{836785669}{60373023}a^{2}-\frac{143582978}{20124341}a-\frac{447621132}{20124341}$, $\frac{971227}{2112572820816}a^{19}+\frac{2309}{6520286484}a^{18}-\frac{690757}{132035801301}a^{17}+\frac{2134013}{58682578356}a^{16}-\frac{3891817}{39121718904}a^{15}+\frac{54125509}{117365156712}a^{14}-\frac{8077507}{4890214863}a^{13}+\frac{604439135}{117365156712}a^{12}-\frac{118247615}{9780429726}a^{11}+\frac{192477877}{6520286484}a^{10}-\frac{217343753}{3260143242}a^{9}+\frac{465978881}{3260143242}a^{8}-\frac{14232476}{60373023}a^{7}+\frac{293950388}{543357207}a^{6}-\frac{635537419}{362238138}a^{5}+\frac{778062929}{181119069}a^{4}-\frac{431923505}{60373023}a^{3}+\frac{644516435}{60373023}a^{2}-\frac{268173400}{20124341}a+\frac{129944778}{20124341}$, $\frac{1318589}{704190940272}a^{19}+\frac{3559423}{1584429615612}a^{18}+\frac{41413681}{2112572820816}a^{17}+\frac{42862585}{2112572820816}a^{16}+\frac{15747967}{58682578356}a^{15}+\frac{11711359}{39121718904}a^{14}+\frac{45507421}{29341289178}a^{13}+\frac{95262827}{58682578356}a^{12}+\frac{101071637}{9780429726}a^{11}+\frac{260706971}{19560859452}a^{10}+\frac{35173219}{724476276}a^{9}+\frac{107398199}{1630071621}a^{8}+\frac{117582791}{362238138}a^{7}+\frac{571471331}{1086714414}a^{6}+\frac{111837791}{181119069}a^{5}+\frac{264369371}{362238138}a^{4}+\frac{216946406}{60373023}a^{3}+\frac{516489478}{60373023}a^{2}+\frac{289822742}{20124341}a+\frac{330510553}{20124341}$
| 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: | \( 2295848.9348199023 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 2295848.9348199023 \cdot 4}{2\cdot\sqrt{5016449852600330836716407488512}}\cr\approx \mathstrut & 0.196595580267132 \end{aligned}\] (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
| \(\Q(\sqrt{-2}) \), 4.0.202752.5, 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 | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{5}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | $20$ | $20$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }^{2}$ | $20$ | $20$ | ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{10}$ |
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 $20$ | $4$ | $5$ | $55$ | |||
|
\(3\)
| 3.10.5.2 | $x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
| 3.10.5.2 | $x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
|
\(11\)
| 11.5.0.1 | $x^{5} + 10 x^{2} + 9$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
| 11.5.0.1 | $x^{5} + 10 x^{2} + 9$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 11.10.9.10 | $x^{10} + 99$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |