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Properties

Label 20.2.129...000.1
Degree $20$
Signature $[2, 9]$
Discriminant $-1.292\times 10^{24}$
Root discriminant \(16.05\)
Ramified primes $2,5,149$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $A_5^2:D_4$ (as 20T546)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^20 - x^18 + x^16 + 6*x^14 - 15*x^12 + 21*x^10 - 21*x^8 + 14*x^6 - x^4 - 5*x^2 - 1)
 
Copy content gp:K = bnfinit(y^20 - y^18 + y^16 + 6*y^14 - 15*y^12 + 21*y^10 - 21*y^8 + 14*y^6 - y^4 - 5*y^2 - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - x^18 + x^16 + 6*x^14 - 15*x^12 + 21*x^10 - 21*x^8 + 14*x^6 - x^4 - 5*x^2 - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - x^18 + x^16 + 6*x^14 - 15*x^12 + 21*x^10 - 21*x^8 + 14*x^6 - x^4 - 5*x^2 - 1)
 

\( x^{20} - x^{18} + x^{16} + 6x^{14} - 15x^{12} + 21x^{10} - 21x^{8} + 14x^{6} - x^{4} - 5x^{2} - 1 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $20$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[2, 9]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(-1292066884157440000000000\) \(\medspace = -\,2^{28}\cdot 5^{10}\cdot 149^{4}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(16.05\)
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  not computed
Ramified primes:   \(2\), \(5\), \(149\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-1}) \)
$\Aut(K/\Q)$:   $C_2$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{10}+\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{11}+\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{4}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{8}-\frac{1}{2}a^{4}-\frac{1}{4}$, $\frac{1}{4}a^{17}-\frac{1}{4}a^{9}-\frac{1}{2}a^{5}-\frac{1}{4}a$, $\frac{1}{212}a^{18}+\frac{25}{212}a^{16}+\frac{15}{212}a^{14}+\frac{25}{212}a^{12}-\frac{1}{212}a^{10}+\frac{12}{53}a^{8}-\frac{1}{2}a^{7}+\frac{2}{53}a^{6}-\frac{1}{2}a^{5}+\frac{63}{212}a^{4}-\frac{1}{2}a^{3}+\frac{25}{53}a^{2}-\frac{55}{212}$, $\frac{1}{212}a^{19}+\frac{25}{212}a^{17}+\frac{15}{212}a^{15}+\frac{25}{212}a^{13}-\frac{1}{212}a^{11}+\frac{12}{53}a^{9}+\frac{2}{53}a^{7}-\frac{1}{2}a^{6}+\frac{63}{212}a^{5}+\frac{25}{53}a^{3}-\frac{55}{212}a-\frac{1}{2}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $10$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{85}{106}a^{19}-\frac{101}{106}a^{17}+\frac{109}{106}a^{15}+\frac{241}{53}a^{13}-\frac{1357}{106}a^{11}+\frac{1033}{53}a^{9}-\frac{1144}{53}a^{7}+\frac{1751}{106}a^{5}-\frac{308}{53}a^{3}-\frac{138}{53}a$, $\frac{85}{106}a^{18}-\frac{101}{106}a^{16}+\frac{109}{106}a^{14}+\frac{241}{53}a^{12}-\frac{1357}{106}a^{10}+\frac{1033}{53}a^{8}-\frac{1144}{53}a^{6}+\frac{1751}{106}a^{4}-\frac{308}{53}a^{2}-\frac{138}{53}$, $\frac{85}{106}a^{18}-\frac{101}{106}a^{16}+\frac{109}{106}a^{14}+\frac{241}{53}a^{12}-\frac{1357}{106}a^{10}+\frac{1033}{53}a^{8}-\frac{1144}{53}a^{6}+\frac{1751}{106}a^{4}-\frac{255}{53}a^{2}-\frac{138}{53}$, $\frac{177}{212}a^{19}-\frac{239}{212}a^{17}+\frac{135}{106}a^{15}+\frac{245}{53}a^{13}-\frac{3039}{212}a^{11}+\frac{4839}{212}a^{9}-\frac{1342}{53}a^{7}+\frac{2051}{106}a^{5}-\frac{1221}{212}a^{3}-\frac{513}{212}a$, $\frac{29}{212}a^{18}-\frac{35}{106}a^{16}+\frac{16}{53}a^{14}+\frac{71}{106}a^{12}-\frac{665}{212}a^{10}+\frac{295}{53}a^{8}-\frac{679}{106}a^{6}+\frac{675}{106}a^{4}-\frac{545}{212}a^{2}-\frac{29}{106}$, $\frac{179}{212}a^{19}-\frac{21}{212}a^{18}-\frac{121}{106}a^{17}+\frac{5}{212}a^{16}+\frac{247}{212}a^{15}+\frac{3}{212}a^{14}+\frac{977}{212}a^{13}-\frac{77}{106}a^{12}-\frac{3041}{212}a^{11}+\frac{45}{53}a^{10}+\frac{4723}{212}a^{9}-\frac{40}{53}a^{8}-\frac{1338}{53}a^{7}-\frac{9}{212}a^{6}+\frac{4281}{212}a^{5}-\frac{51}{212}a^{4}-\frac{401}{53}a^{3}-\frac{33}{212}a^{2}-\frac{285}{106}a+\frac{21}{106}$, $\frac{307}{212}a^{19}+\frac{21}{106}a^{18}-\frac{381}{212}a^{17}-\frac{63}{212}a^{16}+\frac{209}{106}a^{15}+\frac{47}{212}a^{14}+\frac{1739}{212}a^{13}+\frac{255}{212}a^{12}-\frac{1256}{53}a^{11}-\frac{196}{53}a^{10}+\frac{7793}{212}a^{9}+\frac{531}{106}a^{8}-\frac{8515}{212}a^{7}-\frac{521}{106}a^{6}+\frac{3231}{106}a^{5}+\frac{579}{212}a^{4}-\frac{921}{106}a^{3}-\frac{93}{212}a^{2}-\frac{519}{106}a-\frac{243}{212}$, $\frac{99}{106}a^{19}+\frac{23}{53}a^{18}-\frac{61}{53}a^{17}-\frac{69}{106}a^{16}+\frac{267}{212}a^{15}+\frac{161}{212}a^{14}+\frac{567}{106}a^{13}+\frac{249}{106}a^{12}-\frac{3219}{212}a^{11}-\frac{1629}{212}a^{10}+\frac{4999}{212}a^{9}+\frac{2773}{212}a^{8}-\frac{5359}{212}a^{7}-\frac{3133}{212}a^{6}+\frac{4153}{212}a^{5}+\frac{2563}{212}a^{4}-\frac{297}{53}a^{3}-\frac{541}{106}a^{2}-\frac{767}{212}a-\frac{237}{212}$, $\frac{99}{106}a^{19}+\frac{63}{212}a^{18}-\frac{61}{53}a^{17}-\frac{17}{53}a^{16}+\frac{267}{212}a^{15}+\frac{97}{212}a^{14}+\frac{567}{106}a^{13}+\frac{89}{53}a^{12}-\frac{3219}{212}a^{11}-\frac{241}{53}a^{10}+\frac{4999}{212}a^{9}+\frac{1593}{212}a^{8}-\frac{5359}{212}a^{7}-\frac{1775}{212}a^{6}+\frac{4153}{212}a^{5}+\frac{1425}{212}a^{4}-\frac{297}{53}a^{3}-\frac{537}{212}a^{2}-\frac{767}{212}a-\frac{179}{212}$, $\frac{11}{53}a^{19}-\frac{87}{212}a^{18}-\frac{33}{106}a^{17}+\frac{26}{53}a^{16}+\frac{77}{212}a^{15}-\frac{43}{106}a^{14}+\frac{199}{212}a^{13}-\frac{479}{212}a^{12}-\frac{393}{106}a^{11}+\frac{353}{53}a^{10}+\frac{1317}{212}a^{9}-\frac{975}{106}a^{8}-\frac{442}{53}a^{7}+\frac{2219}{212}a^{6}+\frac{1553}{212}a^{5}-\frac{403}{53}a^{4}-\frac{741}{212}a^{3}+\frac{104}{53}a^{2}+\frac{9}{106}a+\frac{439}{212}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 11903.1740533 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{9}\cdot 11903.1740533 \cdot 1}{2\cdot\sqrt{1292066884157440000000000}}\cr\approx \mathstrut & 0.319645869770 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^20 - x^18 + x^16 + 6*x^14 - 15*x^12 + 21*x^10 - 21*x^8 + 14*x^6 - x^4 - 5*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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^20 - x^18 + x^16 + 6*x^14 - 15*x^12 + 21*x^10 - 21*x^8 + 14*x^6 - x^4 - 5*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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - x^18 + x^16 + 6*x^14 - 15*x^12 + 21*x^10 - 21*x^8 + 14*x^6 - x^4 - 5*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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - x^18 + x^16 + 6*x^14 - 15*x^12 + 21*x^10 - 21*x^8 + 14*x^6 - x^4 - 5*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

$A_5^2:D_4$ (as 20T546):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A non-solvable group of order 28800
The 65 conjugacy class representatives for $A_5^2:D_4$
Character table for $A_5^2:D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.400.1, 10.2.17760800000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 20 sibling: data not computed
Degree 24 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 $20$ R ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.5.0.1}{5} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ $20$ ${\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{5}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ $20$ $20$ ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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 2.2.2.4a2.2$x^{4} + 4 x^{3} + 5 x^{2} + 4 x + 7$$2$$2$$4$$D_{4}$$$[2, 2]^{2}$$
2.2.8.24b2.5$x^{16} + 8 x^{15} + 38 x^{14} + 126 x^{13} + 322 x^{12} + 660 x^{11} + 1116 x^{10} + 1578 x^{9} + 1881 x^{8} + 1892 x^{7} + 1600 x^{6} + 1126 x^{5} + 650 x^{4} + 300 x^{3} + 108 x^{2} + 28 x + 7$$8$$2$$24$16T193$$[\frac{4}{3}, \frac{4}{3}, 2, 2]_{3}^{2}$$
\(5\) Copy content Toggle raw display 5.10.2.10a1.2$x^{20} + 6 x^{15} + 6 x^{14} + 4 x^{13} + 8 x^{12} + 2 x^{11} + 13 x^{10} + 18 x^{9} + 21 x^{8} + 36 x^{7} + 34 x^{6} + 34 x^{5} + 32 x^{4} + 16 x^{3} + 17 x^{2} + 4 x + 9$$2$$10$$10$20T3not computed
\(149\) Copy content Toggle raw display 149.2.1.0a1.1$x^{2} + 145 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
149.2.1.0a1.1$x^{2} + 145 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
149.2.3.4a1.2$x^{6} + 435 x^{5} + 63081 x^{4} + 3050365 x^{3} + 126162 x^{2} + 1740 x + 157$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
149.10.1.0a1.1$x^{10} + 74 x^{5} + 42 x^{4} + 148 x^{3} + 143 x^{2} + 51 x + 2$$1$$10$$0$$C_{10}$$$[\ ]^{10}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)