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Properties

Label 20.0.413...024.1
Degree $20$
Signature $[0, 10]$
Discriminant $4.140\times 10^{31}$
Root discriminant \(38.09\)
Ramified primes $2,3,401$
Class number $20$ (GRH)
Class group [2, 10] (GRH)
Galois group $C_2\times D_{10}$ (as 20T8)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 11*x^18 + 82*x^16 - 333*x^14 + 976*x^12 - 1499*x^10 + 1630*x^8 - 738*x^6 + 241*x^4 - 17*x^2 + 1)
 
gp: K = bnfinit(y^20 - 11*y^18 + 82*y^16 - 333*y^14 + 976*y^12 - 1499*y^10 + 1630*y^8 - 738*y^6 + 241*y^4 - 17*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 11*x^18 + 82*x^16 - 333*x^14 + 976*x^12 - 1499*x^10 + 1630*x^8 - 738*x^6 + 241*x^4 - 17*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 11*x^18 + 82*x^16 - 333*x^14 + 976*x^12 - 1499*x^10 + 1630*x^8 - 738*x^6 + 241*x^4 - 17*x^2 + 1)
 

\( x^{20} - 11 x^{18} + 82 x^{16} - 333 x^{14} + 976 x^{12} - 1499 x^{10} + 1630 x^{8} - 738 x^{6} + \cdots + 1 \) Copy content Toggle raw display

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:   \(41396863889937021633336022401024\) \(\medspace = 2^{20}\cdot 3^{10}\cdot 401^{8}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(38.09\)
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:  $2\cdot 3^{1/2}401^{1/2}\approx 69.3685807840985$
Ramified primes:   \(2\), \(3\), \(401\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $4$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{512}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3}a^{16}+\frac{1}{3}a^{8}+\frac{1}{3}$, $\frac{1}{3}a^{17}+\frac{1}{3}a^{9}+\frac{1}{3}a$, $\frac{1}{57050462421}a^{18}-\frac{9436997524}{57050462421}a^{16}-\frac{7171092781}{19016820807}a^{14}+\frac{1454930264}{19016820807}a^{12}-\frac{5860430873}{57050462421}a^{10}+\frac{8099492123}{57050462421}a^{8}+\frac{1846229572}{19016820807}a^{6}+\frac{1750305463}{19016820807}a^{4}-\frac{16428464837}{57050462421}a^{2}+\frac{23579076635}{57050462421}$, $\frac{1}{57050462421}a^{19}-\frac{9436997524}{57050462421}a^{17}-\frac{7171092781}{19016820807}a^{15}+\frac{1454930264}{19016820807}a^{13}-\frac{5860430873}{57050462421}a^{11}+\frac{8099492123}{57050462421}a^{9}+\frac{1846229572}{19016820807}a^{7}+\frac{1750305463}{19016820807}a^{5}-\frac{16428464837}{57050462421}a^{3}+\frac{23579076635}{57050462421}a$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $3$

Class group and class number

$C_{2}\times C_{10}$, which has order $20$ (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:   \( \frac{5611082278}{57050462421} a^{19} - \frac{59792284762}{57050462421} a^{17} + \frac{146262927500}{19016820807} a^{15} - \frac{569821635307}{19016820807} a^{13} + \frac{4828899803998}{57050462421} a^{11} - \frac{6516656977006}{57050462421} a^{9} + \frac{2081149570882}{19016820807} a^{7} - \frac{359338760282}{19016820807} a^{5} + \frac{76290742984}{57050462421} a^{3} + \frac{207837387275}{57050462421} a \)  (order $12$) Copy content Toggle raw display
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:   $\frac{5611082278}{57050462421}a^{19}-\frac{59792284762}{57050462421}a^{17}+\frac{146262927500}{19016820807}a^{15}-\frac{569821635307}{19016820807}a^{13}+\frac{4828899803998}{57050462421}a^{11}-\frac{6516656977006}{57050462421}a^{9}+\frac{2081149570882}{19016820807}a^{7}-\frac{359338760282}{19016820807}a^{5}+\frac{76290742984}{57050462421}a^{3}+\frac{207837387275}{57050462421}a+1$, $\frac{1072505552}{57050462421}a^{18}-\frac{11472655691}{57050462421}a^{16}+\frac{28064226250}{19016820807}a^{14}-\frac{109570415276}{19016820807}a^{12}+\frac{926546042489}{57050462421}a^{10}-\frac{1250384762033}{57050462421}a^{8}+\frac{390011230571}{19016820807}a^{6}-\frac{68948191051}{19016820807}a^{4}+\frac{14638300412}{57050462421}a^{2}+\frac{14764588510}{57050462421}$, $\frac{6278683217}{57050462421}a^{19}-\frac{66770695964}{57050462421}a^{17}+\frac{163333405000}{19016820807}a^{15}-\frac{635546302595}{19016820807}a^{13}+\frac{5392485032756}{57050462421}a^{11}-\frac{7277221859732}{57050462421}a^{9}+\frac{2357771266172}{19016820807}a^{7}-\frac{401277509404}{19016820807}a^{5}+\frac{85194704048}{57050462421}a^{3}+\frac{285658484323}{57050462421}a$, $\frac{2860360256}{57050462421}a^{18}-\frac{32122443596}{57050462421}a^{16}+\frac{80252077165}{19016820807}a^{14}-\frac{331826752115}{19016820807}a^{12}+\frac{2930150585846}{57050462421}a^{10}-\frac{4616411783249}{57050462421}a^{8}+\frac{1586617993541}{19016820807}a^{6}-\frac{661249798153}{19016820807}a^{4}-\frac{33294992380}{57050462421}a^{2}-\frac{8724260303}{57050462421}$, $\frac{2433713162}{57050462421}a^{18}-\frac{25910222807}{57050462421}a^{16}+\frac{63381171250}{19016820807}a^{14}-\frac{246761812049}{19016820807}a^{12}+\frac{2092542045053}{57050462421}a^{10}-\frac{2823910056341}{57050462421}a^{8}+\frac{909499800254}{19016820807}a^{6}-\frac{155714861527}{19016820807}a^{4}+\frac{33059618924}{57050462421}a^{2}+\frac{111729101701}{57050462421}$, $\frac{1591559533}{19016820807}a^{19}+\frac{1451404433}{57050462421}a^{18}-\frac{6211387702}{6338940269}a^{17}-\frac{15486155468}{57050462421}a^{16}+\frac{47592196098}{6338940269}a^{15}+\frac{37881985000}{19016820807}a^{14}-\frac{206959935680}{6338940269}a^{13}-\frac{147674101067}{19016820807}a^{12}+\frac{1916875038385}{19016820807}a^{11}+\frac{1250681311172}{57050462421}a^{10}-\frac{1144697765421}{6338940269}a^{9}-\frac{1687809112484}{57050462421}a^{8}+\frac{1374794514372}{6338940269}a^{7}+\frac{537155586749}{19016820807}a^{6}-\frac{910904667403}{6338940269}a^{5}-\frac{93068460748}{19016820807}a^{4}+\frac{901961003539}{19016820807}a^{3}+\frac{19759243376}{57050462421}a^{2}-\frac{38567122571}{6338940269}a+\frac{67436223298}{57050462421}$, $\frac{14478375778}{57050462421}a^{19}-\frac{2153152447}{19016820807}a^{18}-\frac{159821297356}{57050462421}a^{17}+\frac{8097248830}{6338940269}a^{16}+\frac{397204398416}{19016820807}a^{15}-\frac{60935486938}{6338940269}a^{14}-\frac{1616117721256}{19016820807}a^{13}+\frac{254097506932}{6338940269}a^{12}+\frac{14179150596562}{57050462421}a^{11}-\frac{2270740462768}{19016820807}a^{10}-\frac{21712278272866}{57050462421}a^{9}+\frac{1230911067484}{6338940269}a^{8}+\frac{7639945781308}{19016820807}a^{7}-\frac{1351325612321}{6338940269}a^{6}-\frac{3182594079386}{19016820807}a^{5}+\frac{688556796392}{6338940269}a^{4}+\frac{1842232853743}{57050462421}a^{3}-\frac{412100081518}{19016820807}a^{2}-\frac{41985849358}{57050462421}a+\frac{6340435993}{6338940269}$, $\frac{20287454084}{57050462421}a^{19}+\frac{6674068952}{19016820807}a^{18}-\frac{215798243360}{57050462421}a^{17}-\frac{72814902901}{19016820807}a^{16}+\frac{527882200000}{19016820807}a^{15}+\frac{180128252348}{6338940269}a^{14}-\frac{2054313008852}{19016820807}a^{13}-\frac{723424325596}{6338940269}a^{12}+\frac{17428136409440}{57050462421}a^{11}+\frac{6292579442690}{19016820807}a^{10}-\frac{23519474691680}{57050462421}a^{9}-\frac{9339157760848}{19016820807}a^{8}+\frac{7610469385265}{19016820807}a^{7}+\frac{3256762728310}{6338940269}a^{6}-\frac{1296900988960}{19016820807}a^{5}-\frac{1242858190667}{6338940269}a^{4}+\frac{275343355520}{57050462421}a^{3}+\frac{1066175914514}{19016820807}a^{2}+\frac{1095563063530}{57050462421}a+\frac{42131003942}{19016820807}$, $\frac{13239006328}{57050462421}a^{19}+\frac{12514349731}{57050462421}a^{18}-\frac{135602616844}{57050462421}a^{17}-\frac{131461286647}{57050462421}a^{16}+\frac{325866203891}{19016820807}a^{15}+\frac{319345595030}{19016820807}a^{14}-\frac{1204188400111}{19016820807}a^{13}-\frac{1220102783047}{19016820807}a^{12}+\frac{9780358272760}{57050462421}a^{11}+\frac{10162770138328}{57050462421}a^{10}-\frac{10896312246214}{57050462421}a^{9}-\frac{12802446222379}{57050462421}a^{8}+\frac{3020242373860}{19016820807}a^{7}+\frac{3825226334944}{19016820807}a^{6}+\frac{798696113236}{19016820807}a^{5}-\frac{64363220267}{19016820807}a^{4}+\frac{289762569577}{57050462421}a^{3}-\frac{176949188690}{57050462421}a^{2}-\frac{68283228670}{57050462421}a+\frac{26257894019}{57050462421}$ Copy content Toggle raw display (assuming GRH)
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:  \( 15405657.6163 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 15405657.6163 \cdot 20}{12\cdot\sqrt{41396863889937021633336022401024}}\cr\approx \mathstrut & 0.382687215677 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 - 11*x^18 + 82*x^16 - 333*x^14 + 976*x^12 - 1499*x^10 + 1630*x^8 - 738*x^6 + 241*x^4 - 17*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^20 - 11*x^18 + 82*x^16 - 333*x^14 + 976*x^12 - 1499*x^10 + 1630*x^8 - 738*x^6 + 241*x^4 - 17*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^20 - 11*x^18 + 82*x^16 - 333*x^14 + 976*x^12 - 1499*x^10 + 1630*x^8 - 738*x^6 + 241*x^4 - 17*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^20 - 11*x^18 + 82*x^16 - 333*x^14 + 976*x^12 - 1499*x^10 + 1630*x^8 - 738*x^6 + 241*x^4 - 17*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

$C_2\times D_{10}$ (as 20T8):

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 40
The 16 conjugacy class representatives for $C_2\times D_{10}$
Character table for $C_2\times D_{10}$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{12})\), 5.5.160801.1, 10.0.26477528679424.1, 10.10.6434039469100032.1, 10.0.6283241669043.1

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

Galois closure: deg 40
Degree 20 siblings: deg 20, deg 20, deg 20
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 ${\href{/padicField/5.10.0.1}{10} }^{2}$ ${\href{/padicField/7.10.0.1}{10} }^{2}$ ${\href{/padicField/11.10.0.1}{10} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{8}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ ${\href{/padicField/17.2.0.1}{2} }^{10}$ ${\href{/padicField/19.2.0.1}{2} }^{10}$ ${\href{/padicField/23.2.0.1}{2} }^{10}$ ${\href{/padicField/29.10.0.1}{10} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{10}$ ${\href{/padicField/37.2.0.1}{2} }^{8}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ ${\href{/padicField/41.10.0.1}{10} }^{2}$ ${\href{/padicField/43.10.0.1}{10} }^{2}$ ${\href{/padicField/47.10.0.1}{10} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{10}$ ${\href{/padicField/59.2.0.1}{2} }^{10}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display Deg $20$$2$$10$$20$
\(3\) Copy content Toggle raw display 3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(401\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$