Properties

Label 20.0.294...449.1
Degree $20$
Signature $[0, 10]$
Discriminant $2.941\times 10^{21}$
Root discriminant \(11.84\)
Ramified primes $11,23$
Class number $1$
Class group trivial
Galois group $C_2^5:C_{10}$ (as 20T86)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 24*x^18 - 72*x^17 + 177*x^16 - 363*x^15 + 637*x^14 - 973*x^13 + 1307*x^12 - 1556*x^11 + 1649*x^10 - 1556*x^9 + 1307*x^8 - 973*x^7 + 637*x^6 - 363*x^5 + 177*x^4 - 72*x^3 + 24*x^2 - 6*x + 1)
 
gp: K = bnfinit(y^20 - 6*y^19 + 24*y^18 - 72*y^17 + 177*y^16 - 363*y^15 + 637*y^14 - 973*y^13 + 1307*y^12 - 1556*y^11 + 1649*y^10 - 1556*y^9 + 1307*y^8 - 973*y^7 + 637*y^6 - 363*y^5 + 177*y^4 - 72*y^3 + 24*y^2 - 6*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 6*x^19 + 24*x^18 - 72*x^17 + 177*x^16 - 363*x^15 + 637*x^14 - 973*x^13 + 1307*x^12 - 1556*x^11 + 1649*x^10 - 1556*x^9 + 1307*x^8 - 973*x^7 + 637*x^6 - 363*x^5 + 177*x^4 - 72*x^3 + 24*x^2 - 6*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 6*x^19 + 24*x^18 - 72*x^17 + 177*x^16 - 363*x^15 + 637*x^14 - 973*x^13 + 1307*x^12 - 1556*x^11 + 1649*x^10 - 1556*x^9 + 1307*x^8 - 973*x^7 + 637*x^6 - 363*x^5 + 177*x^4 - 72*x^3 + 24*x^2 - 6*x + 1)
 

\( x^{20} - 6 x^{19} + 24 x^{18} - 72 x^{17} + 177 x^{16} - 363 x^{15} + 637 x^{14} - 973 x^{13} + 1307 x^{12} - 1556 x^{11} + 1649 x^{10} - 1556 x^{9} + 1307 x^{8} - 973 x^{7} + 637 x^{6} + \cdots + 1 \) 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:   \(2941196258837390453449\) \(\medspace = 11^{18}\cdot 23^{2}\) 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:  \(11.84\)
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:  $11^{9/10}23^{1/2}\approx 41.50661671665305$
Ramified primes:   \(11\), \(23\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{109}a^{19}-\frac{7}{109}a^{18}+\frac{31}{109}a^{17}+\frac{6}{109}a^{16}-\frac{47}{109}a^{15}+\frac{11}{109}a^{14}-\frac{28}{109}a^{13}+\frac{36}{109}a^{12}-\frac{37}{109}a^{11}+\frac{7}{109}a^{10}+\frac{7}{109}a^{9}-\frac{37}{109}a^{8}+\frac{36}{109}a^{7}-\frac{28}{109}a^{6}+\frac{11}{109}a^{5}-\frac{47}{109}a^{4}+\frac{6}{109}a^{3}+\frac{31}{109}a^{2}-\frac{7}{109}a+\frac{1}{109}$ 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:   \( \frac{20}{109} a^{19} - \frac{140}{109} a^{18} + \frac{511}{109} a^{17} - \frac{1515}{109} a^{16} + \frac{3638}{109} a^{15} - \frac{7410}{109} a^{14} + \frac{12956}{109} a^{13} - \frac{20208}{109} a^{12} + \frac{28254}{109} a^{11} - \frac{34849}{109} a^{10} + \frac{38399}{109} a^{9} - \frac{37364}{109} a^{8} + \frac{31894}{109} a^{7} - \frac{23995}{109} a^{6} + \frac{15480}{109} a^{5} - \frac{8461}{109} a^{4} + \frac{4153}{109} a^{3} - \frac{1669}{109} a^{2} + \frac{623}{109} a - \frac{198}{109} \)  (order $22$) 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:   $\frac{285}{109}a^{19}-\frac{1341}{109}a^{18}+\frac{4693}{109}a^{17}-\frac{12024}{109}a^{16}+\frac{25191}{109}a^{15}-\frac{41882}{109}a^{14}+\frac{56548}{109}a^{13}-\frac{60481}{109}a^{12}+\frac{46135}{109}a^{11}-\frac{14028}{109}a^{10}-\frac{26999}{109}a^{9}+\frac{64120}{109}a^{8}-\frac{84025}{109}a^{7}+\frac{83580}{109}a^{6}-\frac{67061}{109}a^{5}+\frac{43830}{109}a^{4}-\frac{23469}{109}a^{3}+\frac{10034}{109}a^{2}-\frac{3085}{109}a+\frac{721}{109}$, $\frac{105}{109}a^{19}-\frac{626}{109}a^{18}+\frac{2492}{109}a^{17}-\frac{7654}{109}a^{16}+\frac{19263}{109}a^{15}-\frac{40919}{109}a^{14}+\frac{74777}{109}a^{13}-\frac{120044}{109}a^{12}+\frac{169425}{109}a^{11}-\frac{211488}{109}a^{10}+\frac{233668}{109}a^{9}-\frac{227771}{109}a^{8}+\frac{195293}{109}a^{7}-\frac{146275}{109}a^{6}+\frac{94132}{109}a^{5}-\frac{51369}{109}a^{4}+\frac{22975}{109}a^{3}-\frac{8081}{109}a^{2}+\frac{2317}{109}a-\frac{440}{109}$, $\frac{114}{109}a^{19}-\frac{798}{109}a^{18}+\frac{3207}{109}a^{17}-\frac{9671}{109}a^{16}+\frac{23309}{109}a^{15}-\frac{46597}{109}a^{14}+\frac{77904}{109}a^{13}-\frac{111545}{109}a^{12}+\frac{137918}{109}a^{11}-\frac{148532}{109}a^{10}+\frac{139337}{109}a^{9}-\frac{113436}{109}a^{8}+\frac{79314}{109}a^{7}-\frac{47010}{109}a^{6}+\frac{22727}{109}a^{5}-\frac{8410}{109}a^{4}+\frac{1992}{109}a^{3}-\frac{63}{109}a^{2}-\frac{144}{109}a+\frac{5}{109}$, $\frac{15}{109}a^{19}+\frac{4}{109}a^{18}-\frac{407}{109}a^{17}+\frac{2161}{109}a^{16}-\frac{7572}{109}a^{15}+\frac{20112}{109}a^{14}-\frac{43475}{109}a^{13}+\frac{77385}{109}a^{12}-\frac{116967}{109}a^{11}+\frac{152051}{109}a^{10}-\frac{171679}{109}a^{9}+\frac{169158}{109}a^{8}-\frac{145411}{109}a^{7}+\frac{108471}{109}a^{6}-\frac{69813}{109}a^{5}+\frac{38099}{109}a^{4}-\frac{17241}{109}a^{3}+\frac{6133}{109}a^{2}-\frac{1631}{109}a+\frac{233}{109}$, $\frac{23}{109}a^{19}-\frac{379}{109}a^{18}+\frac{2021}{109}a^{17}-\frac{7274}{109}a^{16}+\frac{19956}{109}a^{15}-\frac{44655}{109}a^{14}+\frac{82414}{109}a^{13}-\frac{127465}{109}a^{12}+\frac{167118}{109}a^{11}-\frac{187864}{109}a^{10}+\frac{181101}{109}a^{9}-\frac{149636}{109}a^{8}+\frac{104705}{109}a^{7}-\frac{60812}{109}a^{6}+\frac{28484}{109}a^{5}-\frac{10346}{109}a^{4}+\frac{2645}{109}a^{3}-\frac{595}{109}a^{2}+\frac{57}{109}a+\frac{23}{109}$, $\frac{20}{109}a^{19}-\frac{249}{109}a^{18}+\frac{1056}{109}a^{17}-\frac{3477}{109}a^{16}+\frac{8870}{109}a^{15}-\frac{18964}{109}a^{14}+\frac{33775}{109}a^{13}-\frac{52036}{109}a^{12}+\frac{70110}{109}a^{11}-\frac{82809}{109}a^{10}+\frac{86359}{109}a^{9}-\frac{79329}{109}a^{8}+\frac{63613}{109}a^{7}-\frac{44814}{109}a^{6}+\frac{27034}{109}a^{5}-\frac{13802}{109}a^{4}+\frac{6006}{109}a^{3}-\frac{2214}{109}a^{2}+\frac{732}{109}a-\frac{198}{109}$, $\frac{288}{109}a^{19}-\frac{1253}{109}a^{18}+\frac{4241}{109}a^{17}-\frac{10589}{109}a^{16}+\frac{22216}{109}a^{15}-\frac{37816}{109}a^{14}+\frac{55592}{109}a^{13}-\frac{72581}{109}a^{12}+\frac{85373}{109}a^{11}-\frac{92051}{109}a^{10}+\frac{90960}{109}a^{9}-\frac{81615}{109}a^{8}+\frac{66721}{109}a^{7}-\frac{48285}{109}a^{6}+\frac{30200}{109}a^{5}-\frac{15825}{109}a^{4}+\frac{6633}{109}a^{3}-\frac{2081}{109}a^{2}+\frac{600}{109}a-\frac{39}{109}$, $\frac{134}{109}a^{19}-\frac{720}{109}a^{18}+\frac{2955}{109}a^{17}-\frac{8897}{109}a^{16}+\frac{22260}{109}a^{15}-\frac{46159}{109}a^{14}+\frac{82358}{109}a^{13}-\frac{126739}{109}a^{12}+\frac{169769}{109}a^{11}-\frac{199077}{109}a^{10}+\frac{204986}{109}a^{9}-\frac{184590}{109}a^{8}+\frac{145325}{109}a^{7}-\frac{98473}{109}a^{6}+\frac{56737}{109}a^{5}-\frac{27008}{109}a^{4}+\frac{10069}{109}a^{3}-\frac{2604}{109}a^{2}+\frac{370}{109}a+\frac{134}{109}$, $\frac{28}{109}a^{19}+\frac{131}{109}a^{18}-\frac{767}{109}a^{17}+\frac{3111}{109}a^{16}-\frac{8619}{109}a^{15}+\frac{19710}{109}a^{14}-\frac{36100}{109}a^{13}+\frac{56053}{109}a^{12}-\frac{75810}{109}a^{11}+\frac{90339}{109}a^{10}-\frac{96814}{109}a^{9}+\frac{93576}{109}a^{8}-\frac{81505}{109}a^{7}+\frac{64398}{109}a^{6}-\frac{45472}{109}a^{5}+\frac{28223}{109}a^{4}-\frac{14983}{109}a^{3}+\frac{6427}{109}a^{2}-\frac{2049}{109}a+\frac{573}{109}$ 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:  \( 2189.13991382 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 2189.13991382 \cdot 1}{22\cdot\sqrt{2941196258837390453449}}\cr\approx \mathstrut & 0.175949290411 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 24*x^18 - 72*x^17 + 177*x^16 - 363*x^15 + 637*x^14 - 973*x^13 + 1307*x^12 - 1556*x^11 + 1649*x^10 - 1556*x^9 + 1307*x^8 - 973*x^7 + 637*x^6 - 363*x^5 + 177*x^4 - 72*x^3 + 24*x^2 - 6*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 - 6*x^19 + 24*x^18 - 72*x^17 + 177*x^16 - 363*x^15 + 637*x^14 - 973*x^13 + 1307*x^12 - 1556*x^11 + 1649*x^10 - 1556*x^9 + 1307*x^8 - 973*x^7 + 637*x^6 - 363*x^5 + 177*x^4 - 72*x^3 + 24*x^2 - 6*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 - 6*x^19 + 24*x^18 - 72*x^17 + 177*x^16 - 363*x^15 + 637*x^14 - 973*x^13 + 1307*x^12 - 1556*x^11 + 1649*x^10 - 1556*x^9 + 1307*x^8 - 973*x^7 + 637*x^6 - 363*x^5 + 177*x^4 - 72*x^3 + 24*x^2 - 6*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 - 6*x^19 + 24*x^18 - 72*x^17 + 177*x^16 - 363*x^15 + 637*x^14 - 973*x^13 + 1307*x^12 - 1556*x^11 + 1649*x^10 - 1556*x^9 + 1307*x^8 - 973*x^7 + 637*x^6 - 363*x^5 + 177*x^4 - 72*x^3 + 24*x^2 - 6*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

$C_2^5:C_{10}$ (as 20T86):

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 320
The 32 conjugacy class representatives for $C_2^5:C_{10}$
Character table for $C_2^5:C_{10}$ is not computed

Intermediate fields

\(\Q(\sqrt{-11}) \), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{11})\), 10.6.54232796893.1, 10.4.4930254263.1

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: 20.8.823067302269314181883621609.4

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.10.0.1}{10} }^{2}$ ${\href{/padicField/3.5.0.1}{5} }^{4}$ ${\href{/padicField/5.10.0.1}{10} }^{2}$ ${\href{/padicField/7.10.0.1}{10} }^{2}$ R ${\href{/padicField/13.10.0.1}{10} }^{2}$ ${\href{/padicField/17.10.0.1}{10} }^{2}$ ${\href{/padicField/19.10.0.1}{10} }^{2}$ R ${\href{/padicField/29.10.0.1}{10} }^{2}$ ${\href{/padicField/31.5.0.1}{5} }^{4}$ ${\href{/padicField/37.10.0.1}{10} }^{2}$ ${\href{/padicField/41.10.0.1}{10} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{10}$ ${\href{/padicField/47.5.0.1}{5} }^{4}$ ${\href{/padicField/53.10.0.1}{10} }^{2}$ ${\href{/padicField/59.5.0.1}{5} }^{4}$

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
\(11\) Copy content Toggle raw display 11.20.18.1$x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$$10$$2$$18$20T3$[\ ]_{10}^{2}$
\(23\) Copy content Toggle raw display $\Q_{23}$$x + 18$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 18$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 18$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 18$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} + 115$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} + 115$$2$$1$$1$$C_2$$[\ ]_{2}$