Properties

Label 20.0.709...856.1
Degree $20$
Signature $[0, 10]$
Discriminant $7.090\times 10^{20}$
Root discriminant \(11.03\)
Ramified primes $2,83,983$
Class number $1$
Class group trivial
Galois group $C_3^5.D_6$ (as 20T669)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 15*x^18 - 37*x^17 + 78*x^16 - 139*x^15 + 219*x^14 - 307*x^13 + 386*x^12 - 442*x^11 + 463*x^10 - 442*x^9 + 386*x^8 - 307*x^7 + 219*x^6 - 139*x^5 + 78*x^4 - 37*x^3 + 15*x^2 - 5*x + 1)
 
gp: K = bnfinit(y^20 - 5*y^19 + 15*y^18 - 37*y^17 + 78*y^16 - 139*y^15 + 219*y^14 - 307*y^13 + 386*y^12 - 442*y^11 + 463*y^10 - 442*y^9 + 386*y^8 - 307*y^7 + 219*y^6 - 139*y^5 + 78*y^4 - 37*y^3 + 15*y^2 - 5*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 5*x^19 + 15*x^18 - 37*x^17 + 78*x^16 - 139*x^15 + 219*x^14 - 307*x^13 + 386*x^12 - 442*x^11 + 463*x^10 - 442*x^9 + 386*x^8 - 307*x^7 + 219*x^6 - 139*x^5 + 78*x^4 - 37*x^3 + 15*x^2 - 5*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 5*x^19 + 15*x^18 - 37*x^17 + 78*x^16 - 139*x^15 + 219*x^14 - 307*x^13 + 386*x^12 - 442*x^11 + 463*x^10 - 442*x^9 + 386*x^8 - 307*x^7 + 219*x^6 - 139*x^5 + 78*x^4 - 37*x^3 + 15*x^2 - 5*x + 1)
 

\( x^{20} - 5 x^{19} + 15 x^{18} - 37 x^{17} + 78 x^{16} - 139 x^{15} + 219 x^{14} - 307 x^{13} + 386 x^{12} + \cdots + 1 \) 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:   \(709000307415298179856\) \(\medspace = 2^{4}\cdot 83^{4}\cdot 983^{4}\) 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:  \(11.03\)
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:  $2\cdot 83^{1/2}983^{1/2}\approx 571.2757652832825$
Ramified primes:   \(2\), \(83\), \(983\) 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 not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{62}a^{18}-\frac{9}{31}a^{17}-\frac{19}{62}a^{15}+\frac{15}{62}a^{14}-\frac{5}{62}a^{13}+\frac{21}{62}a^{12}-\frac{17}{62}a^{11}+\frac{14}{31}a^{10}+\frac{17}{62}a^{9}+\frac{14}{31}a^{8}-\frac{17}{62}a^{7}+\frac{21}{62}a^{6}-\frac{5}{62}a^{5}+\frac{15}{62}a^{4}-\frac{19}{62}a^{3}-\frac{9}{31}a+\frac{1}{62}$, $\frac{1}{3782}a^{19}-\frac{3}{1891}a^{18}-\frac{325}{1891}a^{17}+\frac{1345}{3782}a^{16}-\frac{1267}{3782}a^{15}-\frac{1251}{3782}a^{14}-\frac{1031}{3782}a^{13}+\frac{297}{3782}a^{12}+\frac{563}{1891}a^{11}-\frac{1507}{3782}a^{10}-\frac{845}{1891}a^{9}+\frac{1187}{3782}a^{8}-\frac{679}{3782}a^{7}+\frac{433}{3782}a^{6}+\frac{823}{3782}a^{5}-\frac{1389}{3782}a^{4}-\frac{517}{1891}a^{3}-\frac{691}{1891}a^{2}+\frac{1397}{3782}a-\frac{335}{1891}$ Copy content Toggle raw display

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 $1$

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 $2$) 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:   $\frac{2634}{1891}a^{19}-\frac{12571}{1891}a^{18}+\frac{37502}{1891}a^{17}-\frac{93663}{1891}a^{16}+\frac{197977}{1891}a^{15}-\frac{353409}{1891}a^{14}+\frac{562303}{1891}a^{13}-\frac{793088}{1891}a^{12}+\frac{1001013}{1891}a^{11}-\frac{1152092}{1891}a^{10}+\frac{1206534}{1891}a^{9}-\frac{1149237}{1891}a^{8}+\frac{1002508}{1891}a^{7}-\frac{792263}{1891}a^{6}+\frac{17923}{61}a^{5}-\frac{348156}{1891}a^{4}+\frac{187669}{1891}a^{3}-\frac{81326}{1891}a^{2}+\frac{30495}{1891}a-\frac{8590}{1891}$, $\frac{599}{1891}a^{19}-\frac{2008}{1891}a^{18}+\frac{3795}{1891}a^{17}-\frac{7475}{1891}a^{16}+\frac{12717}{1891}a^{15}-\frac{14543}{1891}a^{14}+\frac{15550}{1891}a^{13}-\frac{15711}{1891}a^{12}+\frac{12136}{1891}a^{11}-\frac{13008}{1891}a^{10}+\frac{14991}{1891}a^{9}-\frac{14216}{1891}a^{8}+\frac{16375}{1891}a^{7}-\frac{21233}{1891}a^{6}+\frac{17970}{1891}a^{5}-\frac{17783}{1891}a^{4}+\frac{16132}{1891}a^{3}-\frac{9015}{1891}a^{2}+\frac{4580}{1891}a-\frac{2634}{1891}$, $\frac{5240}{1891}a^{19}-\frac{24730}{1891}a^{18}+\frac{71793}{1891}a^{17}-\frac{173929}{1891}a^{16}+\frac{358718}{1891}a^{15}-\frac{622746}{1891}a^{14}+\frac{959372}{1891}a^{13}-\frac{1307609}{1891}a^{12}+\frac{1595714}{1891}a^{11}-\frac{1774842}{1891}a^{10}+\frac{1797013}{1891}a^{9}-\frac{1653573}{1891}a^{8}+\frac{1388286}{1891}a^{7}-\frac{1050700}{1891}a^{6}+\frac{703089}{1891}a^{5}-\frac{413603}{1891}a^{4}+\frac{208663}{1891}a^{3}-\frac{84245}{1891}a^{2}+\frac{30719}{1891}a-\frac{7631}{1891}$, $\frac{1183}{3782}a^{19}-\frac{2523}{3782}a^{18}+\frac{1716}{1891}a^{17}-\frac{4869}{3782}a^{16}-\frac{564}{1891}a^{15}+\frac{12926}{1891}a^{14}-\frac{31282}{1891}a^{13}+\frac{64869}{1891}a^{12}-\frac{213133}{3782}a^{11}+\frac{289267}{3782}a^{10}-\frac{355747}{3782}a^{9}+\frac{390157}{3782}a^{8}-\frac{187122}{1891}a^{7}+\frac{168005}{1891}a^{6}-\frac{133535}{1891}a^{5}+\frac{90143}{1891}a^{4}-\frac{107471}{3782}a^{3}+\frac{27824}{1891}a^{2}-\frac{18139}{3782}a+\frac{6185}{3782}$, $\frac{3301}{3782}a^{19}-\frac{18037}{3782}a^{18}+\frac{26944}{1891}a^{17}-\frac{132593}{3782}a^{16}+\frac{140410}{1891}a^{15}-\frac{249385}{1891}a^{14}+\frac{389140}{1891}a^{13}-\frac{544514}{1891}a^{12}+\frac{1349577}{3782}a^{11}-\frac{1528839}{3782}a^{10}+\frac{1584235}{3782}a^{9}-\frac{1485825}{3782}a^{8}+\frac{638032}{1891}a^{7}-\frac{497800}{1891}a^{6}+\frac{342254}{1891}a^{5}-\frac{208632}{1891}a^{4}+\frac{225477}{3782}a^{3}-\frac{47720}{1891}a^{2}+\frac{33691}{3782}a-\frac{12559}{3782}$, $\frac{936}{1891}a^{19}-\frac{7145}{3782}a^{18}+\frac{10994}{1891}a^{17}-\frac{26960}{1891}a^{16}+\frac{107155}{3782}a^{15}-\frac{185339}{3782}a^{14}+\frac{288477}{3782}a^{13}-\frac{375547}{3782}a^{12}+\frac{453735}{3782}a^{11}-\frac{245208}{1891}a^{10}+\frac{472225}{3782}a^{9}-\frac{210289}{1891}a^{8}+\frac{334859}{3782}a^{7}-\frac{226851}{3782}a^{6}+\frac{136009}{3782}a^{5}-\frac{69257}{3782}a^{4}+\frac{17633}{3782}a^{3}-\frac{108}{1891}a^{2}+\frac{57}{1891}a+\frac{5471}{3782}$, $\frac{197}{62}a^{19}-\frac{461}{31}a^{18}+\frac{1308}{31}a^{17}-\frac{6285}{62}a^{16}+\frac{12929}{62}a^{15}-\frac{22261}{62}a^{14}+\frac{34107}{62}a^{13}-\frac{46515}{62}a^{12}+\frac{28349}{31}a^{11}-\frac{63237}{62}a^{10}+\frac{32223}{31}a^{9}-\frac{59581}{62}a^{8}+\frac{50483}{62}a^{7}-\frac{38757}{62}a^{6}+\frac{26415}{62}a^{5}-\frac{15905}{62}a^{4}+\frac{4243}{31}a^{3}-\frac{1835}{31}a^{2}+\frac{1387}{62}a-\frac{207}{31}$, $\frac{4413}{1891}a^{19}-\frac{45575}{3782}a^{18}+\frac{66138}{1891}a^{17}-\frac{159208}{1891}a^{16}+\frac{21367}{122}a^{15}-\frac{1154141}{3782}a^{14}+\frac{1766995}{3782}a^{13}-\frac{2423927}{3782}a^{12}+\frac{2955833}{3782}a^{11}-\frac{1645585}{1891}a^{10}+\frac{3351791}{3782}a^{9}-\frac{1545457}{1891}a^{8}+\frac{2595397}{3782}a^{7}-\frac{1983773}{3782}a^{6}+\frac{1334537}{3782}a^{5}-\frac{787471}{3782}a^{4}+\frac{408033}{3782}a^{3}-\frac{85386}{1891}a^{2}+\frac{28422}{1891}a-\frac{21247}{3782}$, $\frac{7059}{3782}a^{19}-\frac{13979}{1891}a^{18}+\frac{36452}{1891}a^{17}-\frac{168655}{3782}a^{16}+\frac{332273}{3782}a^{15}-\frac{17429}{122}a^{14}+\frac{800401}{3782}a^{13}-\frac{1046563}{3782}a^{12}+\frac{618241}{1891}a^{11}-\frac{1343343}{3782}a^{10}+\frac{668224}{1891}a^{9}-\frac{1198577}{3782}a^{8}+\frac{994497}{3782}a^{7}-\frac{744607}{3782}a^{6}+\frac{491943}{3782}a^{5}-\frac{292855}{3782}a^{4}+\frac{77048}{1891}a^{3}-\frac{31136}{1891}a^{2}+\frac{26271}{3782}a-\frac{1381}{1891}$ Copy content Toggle raw display
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:  \( 103.364702597 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 103.364702597 \cdot 1}{2\cdot\sqrt{709000307415298179856}}\cr\approx \mathstrut & 0.186130582115 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 15*x^18 - 37*x^17 + 78*x^16 - 139*x^15 + 219*x^14 - 307*x^13 + 386*x^12 - 442*x^11 + 463*x^10 - 442*x^9 + 386*x^8 - 307*x^7 + 219*x^6 - 139*x^5 + 78*x^4 - 37*x^3 + 15*x^2 - 5*x + 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 - 5*x^19 + 15*x^18 - 37*x^17 + 78*x^16 - 139*x^15 + 219*x^14 - 307*x^13 + 386*x^12 - 442*x^11 + 463*x^10 - 442*x^9 + 386*x^8 - 307*x^7 + 219*x^6 - 139*x^5 + 78*x^4 - 37*x^3 + 15*x^2 - 5*x + 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 - 5*x^19 + 15*x^18 - 37*x^17 + 78*x^16 - 139*x^15 + 219*x^14 - 307*x^13 + 386*x^12 - 442*x^11 + 463*x^10 - 442*x^9 + 386*x^8 - 307*x^7 + 219*x^6 - 139*x^5 + 78*x^4 - 37*x^3 + 15*x^2 - 5*x + 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 - 5*x^19 + 15*x^18 - 37*x^17 + 78*x^16 - 139*x^15 + 219*x^14 - 307*x^13 + 386*x^12 - 442*x^11 + 463*x^10 - 442*x^9 + 386*x^8 - 307*x^7 + 219*x^6 - 139*x^5 + 78*x^4 - 37*x^3 + 15*x^2 - 5*x + 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

$C_3^5.D_6$ (as 20T669):

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 61440
The 126 conjugacy class representatives for $C_3^5.D_6$
Character table for $C_3^5.D_6$

Intermediate fields

5.5.81589.1, 10.2.6656764921.1, 10.4.26627059684.1, 10.4.26627059684.2

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: 20.0.209139509314799821952907668928868954259822604297616.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.10.0.1}{10} }^{2}$ ${\href{/padicField/5.5.0.1}{5} }^{4}$ ${\href{/padicField/7.10.0.1}{10} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}$ ${\href{/padicField/23.10.0.1}{10} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{6}$ ${\href{/padicField/37.5.0.1}{5} }^{4}$ ${\href{/padicField/41.5.0.1}{5} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{6}$ ${\href{/padicField/59.10.0.1}{10} }^{2}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x^{2} + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x^{2} + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
\(83\) Copy content Toggle raw display 83.2.1.1$x^{2} + 166$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.1$x^{2} + 166$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.1$x^{2} + 166$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.1$x^{2} + 166$$2$$1$$1$$C_2$$[\ ]_{2}$
83.6.0.1$x^{6} + x^{4} + 76 x^{3} + 32 x^{2} + 17 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
83.6.0.1$x^{6} + x^{4} + 76 x^{3} + 32 x^{2} + 17 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
\(983\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
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$