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Properties

Label 20.20.186...072.1
Degree $20$
Signature $(20, 0)$
Discriminant $1.867\times 10^{64}$
Root discriminant \(1635.13\)
Ramified primes $2,31,53$
Class number not computed
Class group not computed
Galois group $C_{20}:C_4$ (as 20T18)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^20 - 1276*x^18 - 16*x^17 + 658002*x^16 + 8416*x^15 - 174997424*x^14 - 2239984*x^13 + 25314437011*x^12 + 628091280*x^11 - 1921782848456*x^10 - 102900137840*x^9 + 68574729140642*x^8 + 5077647112400*x^7 - 1097427171073028*x^6 + 52349328823712*x^5 + 6801945730533617*x^4 - 3019425270314576*x^3 - 10867727001230072*x^2 + 5179746228111552*x + 2300893286748984)
 
Copy content gp:K = bnfinit(y^20 - 1276*y^18 - 16*y^17 + 658002*y^16 + 8416*y^15 - 174997424*y^14 - 2239984*y^13 + 25314437011*y^12 + 628091280*y^11 - 1921782848456*y^10 - 102900137840*y^9 + 68574729140642*y^8 + 5077647112400*y^7 - 1097427171073028*y^6 + 52349328823712*y^5 + 6801945730533617*y^4 - 3019425270314576*y^3 - 10867727001230072*y^2 + 5179746228111552*y + 2300893286748984, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 1276*x^18 - 16*x^17 + 658002*x^16 + 8416*x^15 - 174997424*x^14 - 2239984*x^13 + 25314437011*x^12 + 628091280*x^11 - 1921782848456*x^10 - 102900137840*x^9 + 68574729140642*x^8 + 5077647112400*x^7 - 1097427171073028*x^6 + 52349328823712*x^5 + 6801945730533617*x^4 - 3019425270314576*x^3 - 10867727001230072*x^2 + 5179746228111552*x + 2300893286748984);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 1276*x^18 - 16*x^17 + 658002*x^16 + 8416*x^15 - 174997424*x^14 - 2239984*x^13 + 25314437011*x^12 + 628091280*x^11 - 1921782848456*x^10 - 102900137840*x^9 + 68574729140642*x^8 + 5077647112400*x^7 - 1097427171073028*x^6 + 52349328823712*x^5 + 6801945730533617*x^4 - 3019425270314576*x^3 - 10867727001230072*x^2 + 5179746228111552*x + 2300893286748984)
 

\( x^{20} - 1276 x^{18} - 16 x^{17} + 658002 x^{16} + 8416 x^{15} - 174997424 x^{14} + \cdots + 23\!\cdots\!84 \) 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:  $(20, 0)$
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:   \(18666260658981574277800445093752394630489110554050541457686659072\) \(\medspace = 2^{59}\cdot 31^{15}\cdot 53^{14}\) 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:  \(1635.13\)
Copy content comment:Root discriminant
 
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:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  not computed
Ramified primes:   \(2\), \(31\), \(53\) 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:  \(\Q(\sqrt{62}) \)
$\Aut(K/\Q)$:   $C_2$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{7}-\frac{1}{2}a^{5}-\frac{1}{4}a^{3}$, $\frac{1}{424}a^{12}+\frac{5}{212}a^{11}-\frac{6}{53}a^{10}-\frac{7}{106}a^{9}+\frac{93}{424}a^{8}-\frac{9}{106}a^{7}+\frac{99}{212}a^{5}+\frac{185}{424}a^{4}-\frac{21}{53}a^{3}-\frac{13}{106}a^{2}-\frac{5}{53}a-\frac{23}{53}$, $\frac{1}{424}a^{13}-\frac{21}{212}a^{11}+\frac{7}{106}a^{10}-\frac{51}{424}a^{9}+\frac{47}{212}a^{8}+\frac{21}{212}a^{7}-\frac{7}{212}a^{6}-\frac{99}{424}a^{5}-\frac{55}{212}a^{4}+\frac{19}{212}a^{3}+\frac{7}{53}a^{2}-\frac{26}{53}a+\frac{18}{53}$, $\frac{1}{848}a^{14}-\frac{1}{848}a^{13}-\frac{1}{848}a^{12}-\frac{25}{424}a^{11}+\frac{73}{848}a^{10}+\frac{57}{848}a^{9}+\frac{157}{848}a^{8}-\frac{77}{424}a^{7}-\frac{85}{848}a^{6}+\frac{263}{848}a^{5}+\frac{101}{848}a^{4}+\frac{29}{106}a^{3}+\frac{45}{106}a^{2}+\frac{51}{106}a-\frac{7}{106}$, $\frac{1}{2544}a^{15}+\frac{1}{1272}a^{13}-\frac{1}{848}a^{12}-\frac{89}{2544}a^{11}+\frac{45}{424}a^{10}+\frac{25}{424}a^{9}+\frac{201}{848}a^{8}+\frac{533}{2544}a^{7}+\frac{91}{424}a^{6}-\frac{17}{636}a^{5}-\frac{397}{848}a^{4}+\frac{46}{159}a^{3}-\frac{45}{106}a^{2}+\frac{62}{159}a+\frac{7}{106}$, $\frac{1}{5088}a^{16}-\frac{1}{5088}a^{14}-\frac{1}{848}a^{13}+\frac{1}{1272}a^{12}+\frac{25}{424}a^{11}+\frac{177}{1696}a^{10}+\frac{21}{848}a^{9}-\frac{25}{318}a^{8}-\frac{101}{424}a^{7}+\frac{907}{5088}a^{6}-\frac{229}{848}a^{5}+\frac{1843}{5088}a^{4}-\frac{51}{212}a^{3}+\frac{79}{159}a^{2}+\frac{35}{106}a+\frac{23}{212}$, $\frac{1}{5088}a^{17}-\frac{1}{5088}a^{15}-\frac{1}{2544}a^{13}-\frac{1}{848}a^{12}-\frac{75}{1696}a^{11}-\frac{25}{424}a^{10}-\frac{281}{2544}a^{9}+\frac{181}{848}a^{8}+\frac{607}{5088}a^{7}-\frac{51}{424}a^{6}+\frac{2533}{5088}a^{5}-\frac{237}{848}a^{4}-\frac{47}{636}a^{3}-\frac{19}{106}a^{2}-\frac{11}{212}a-\frac{23}{106}$, $\frac{1}{93\cdots 36}a^{18}+\frac{22\cdots 97}{31\cdots 12}a^{17}-\frac{15\cdots 01}{15\cdots 56}a^{16}-\frac{72\cdots 97}{93\cdots 36}a^{15}+\frac{17\cdots 69}{31\cdots 12}a^{14}-\frac{33\cdots 97}{46\cdots 68}a^{13}-\frac{13\cdots 73}{93\cdots 36}a^{12}+\frac{94\cdots 35}{93\cdots 36}a^{11}-\frac{56\cdots 49}{93\cdots 36}a^{10}-\frac{10\cdots 79}{15\cdots 56}a^{9}-\frac{11\cdots 71}{31\cdots 12}a^{8}-\frac{15\cdots 77}{93\cdots 36}a^{7}+\frac{49\cdots 05}{46\cdots 68}a^{6}+\frac{84\cdots 35}{93\cdots 36}a^{5}+\frac{98\cdots 31}{31\cdots 12}a^{4}-\frac{16\cdots 65}{14\cdots 49}a^{3}-\frac{20\cdots 57}{11\cdots 92}a^{2}-\frac{40\cdots 39}{11\cdots 92}a+\frac{18\cdots 29}{39\cdots 64}$, $\frac{1}{44\cdots 12}a^{19}+\frac{24\cdots 95}{55\cdots 64}a^{18}-\frac{45\cdots 55}{14\cdots 04}a^{17}-\frac{86\cdots 49}{44\cdots 12}a^{16}-\frac{39\cdots 57}{37\cdots 76}a^{15}-\frac{10\cdots 85}{44\cdots 12}a^{14}-\frac{15\cdots 67}{44\cdots 12}a^{13}+\frac{61\cdots 41}{74\cdots 52}a^{12}-\frac{13\cdots 09}{11\cdots 28}a^{11}+\frac{37\cdots 19}{44\cdots 12}a^{10}+\frac{13\cdots 51}{14\cdots 04}a^{9}+\frac{52\cdots 71}{22\cdots 56}a^{8}+\frac{10\cdots 03}{44\cdots 12}a^{7}-\frac{14\cdots 41}{44\cdots 12}a^{6}+\frac{18\cdots 17}{74\cdots 52}a^{5}-\frac{21\cdots 53}{44\cdots 12}a^{4}-\frac{13\cdots 93}{55\cdots 64}a^{3}+\frac{12\cdots 21}{46\cdots 22}a^{2}-\frac{91\cdots 33}{92\cdots 44}a+\frac{59\cdots 39}{18\cdots 88}$ 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:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  not computed
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:  not computed
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:  $19$
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:  not computed
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:  not computed
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 
Unit signature rank:  \( 15 \)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot R \cdot h}{2\cdot\sqrt{18666260658981574277800445093752394630489110554050541457686659072}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

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 - 1276*x^18 - 16*x^17 + 658002*x^16 + 8416*x^15 - 174997424*x^14 - 2239984*x^13 + 25314437011*x^12 + 628091280*x^11 - 1921782848456*x^10 - 102900137840*x^9 + 68574729140642*x^8 + 5077647112400*x^7 - 1097427171073028*x^6 + 52349328823712*x^5 + 6801945730533617*x^4 - 3019425270314576*x^3 - 10867727001230072*x^2 + 5179746228111552*x + 2300893286748984) 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 - 1276*x^18 - 16*x^17 + 658002*x^16 + 8416*x^15 - 174997424*x^14 - 2239984*x^13 + 25314437011*x^12 + 628091280*x^11 - 1921782848456*x^10 - 102900137840*x^9 + 68574729140642*x^8 + 5077647112400*x^7 - 1097427171073028*x^6 + 52349328823712*x^5 + 6801945730533617*x^4 - 3019425270314576*x^3 - 10867727001230072*x^2 + 5179746228111552*x + 2300893286748984, 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 - 1276*x^18 - 16*x^17 + 658002*x^16 + 8416*x^15 - 174997424*x^14 - 2239984*x^13 + 25314437011*x^12 + 628091280*x^11 - 1921782848456*x^10 - 102900137840*x^9 + 68574729140642*x^8 + 5077647112400*x^7 - 1097427171073028*x^6 + 52349328823712*x^5 + 6801945730533617*x^4 - 3019425270314576*x^3 - 10867727001230072*x^2 + 5179746228111552*x + 2300893286748984); 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 = polynomial_ring(QQ); K, a = number_field(x^20 - 1276*x^18 - 16*x^17 + 658002*x^16 + 8416*x^15 - 174997424*x^14 - 2239984*x^13 + 25314437011*x^12 + 628091280*x^11 - 1921782848456*x^10 - 102900137840*x^9 + 68574729140642*x^8 + 5077647112400*x^7 - 1097427171073028*x^6 + 52349328823712*x^5 + 6801945730533617*x^4 - 3019425270314576*x^3 - 10867727001230072*x^2 + 5179746228111552*x + 2300893286748984); 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

$C_{20}:C_4$ (as 20T18):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 80
The 14 conjugacy class representatives for $C_{20}:C_4$
Character table for $C_{20}:C_4$

Intermediate fields

\(\Q(\sqrt{3286}) \), \(\Q(\sqrt{124 +2 \sqrt{3286}})\), 5.5.2382032.1, deg 10

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(L)]
 
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 40 siblings: data not computed
Minimal sibling: 20.20.352193597339274986373593303655705559065832274604727197314842624.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ $20$ $20$ ${\href{/padicField/13.10.0.1}{10} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{5}$ ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{10}$ R ${\href{/padicField/37.2.0.1}{2} }^{8}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ $20$ ${\href{/padicField/47.4.0.1}{4} }^{5}$ R ${\href{/padicField/59.5.0.1}{5} }^{4}$

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

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 2.4.11.14$x^{4} + 8 x + 10$$4$$1$$11$$D_{4}$$$[3, 4]^{2}$$
2.16.48.9803$x^{16} + 8 x^{15} + 40 x^{14} + 144 x^{13} + 406 x^{12} + 928 x^{11} + 1760 x^{10} + 2800 x^{9} + 3761 x^{8} + 4272 x^{7} + 4092 x^{6} + 3280 x^{5} + 2164 x^{4} + 1152 x^{3} + 472 x^{2} + 160 x + 25$$8$$2$$48$$C_4:C_4$$$[2, 3, 4]^{2}$$
\(31\) Copy content Toggle raw display 31.4.3.1$x^{4} + 31$$4$$1$$3$$D_{4}$$$[\ ]_{4}^{2}$$
31.16.12.1$x^{16} + 12 x^{14} + 64 x^{13} + 66 x^{12} + 576 x^{11} + 1752 x^{10} + 2304 x^{9} + 9675 x^{8} + 21568 x^{7} + 23688 x^{6} + 56064 x^{5} + 93778 x^{4} + 54336 x^{3} + 14148 x^{2} + 1728 x + 112$$4$$4$$12$$C_4:C_4$$$[\ ]_{4}^{4}$$
\(53\) Copy content Toggle raw display 53.4.2.1$x^{4} + 98 x^{3} + 2405 x^{2} + 196 x + 57$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
53.8.6.1$x^{8} + 196 x^{7} + 14414 x^{6} + 471772 x^{5} + 5822449 x^{4} + 943544 x^{3} + 57656 x^{2} + 1568 x + 69$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
53.8.6.1$x^{8} + 196 x^{7} + 14414 x^{6} + 471772 x^{5} + 5822449 x^{4} + 943544 x^{3} + 57656 x^{2} + 1568 x + 69$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$

Spectrum of ring of integers

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