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Properties

Label 20.4.526...000.1
Degree $20$
Signature $[4, 8]$
Discriminant $5.268\times 10^{68}$
Root discriminant \(2729.49\)
Ramified primes $2,5,13,197$
Class number not computed
Class group not computed
Galois group $C_4\times F_5$ (as 20T20)

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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 - x^19 - 253*x^18 - 24544*x^17 + 398685*x^16 + 8721545*x^15 - 70361823*x^14 - 3938000908*x^13 - 11285455377*x^12 + 2955494285207*x^11 - 38784617114891*x^10 - 385201843870298*x^9 + 14936778594591960*x^8 - 124575614095119552*x^7 - 656667047355929392*x^6 + 20992220260103943024*x^5 - 141973587074569077668*x^4 - 247833350982037177072*x^3 + 9459373257275866071148*x^2 - 56140694999572599084636*x + 117438049640833114893196)
 
Copy content gp:K = bnfinit(y^20 - y^19 - 253*y^18 - 24544*y^17 + 398685*y^16 + 8721545*y^15 - 70361823*y^14 - 3938000908*y^13 - 11285455377*y^12 + 2955494285207*y^11 - 38784617114891*y^10 - 385201843870298*y^9 + 14936778594591960*y^8 - 124575614095119552*y^7 - 656667047355929392*y^6 + 20992220260103943024*y^5 - 141973587074569077668*y^4 - 247833350982037177072*y^3 + 9459373257275866071148*y^2 - 56140694999572599084636*y + 117438049640833114893196, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - x^19 - 253*x^18 - 24544*x^17 + 398685*x^16 + 8721545*x^15 - 70361823*x^14 - 3938000908*x^13 - 11285455377*x^12 + 2955494285207*x^11 - 38784617114891*x^10 - 385201843870298*x^9 + 14936778594591960*x^8 - 124575614095119552*x^7 - 656667047355929392*x^6 + 20992220260103943024*x^5 - 141973587074569077668*x^4 - 247833350982037177072*x^3 + 9459373257275866071148*x^2 - 56140694999572599084636*x + 117438049640833114893196);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - x^19 - 253*x^18 - 24544*x^17 + 398685*x^16 + 8721545*x^15 - 70361823*x^14 - 3938000908*x^13 - 11285455377*x^12 + 2955494285207*x^11 - 38784617114891*x^10 - 385201843870298*x^9 + 14936778594591960*x^8 - 124575614095119552*x^7 - 656667047355929392*x^6 + 20992220260103943024*x^5 - 141973587074569077668*x^4 - 247833350982037177072*x^3 + 9459373257275866071148*x^2 - 56140694999572599084636*x + 117438049640833114893196)
 

\( x^{20} - x^{19} - 253 x^{18} - 24544 x^{17} + 398685 x^{16} + 8721545 x^{15} - 70361823 x^{14} + \cdots + 11\!\cdots\!96 \) 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:  $[4, 8]$
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:   \(526792882835395388103720314122737484676904390310681994000000000000000\) \(\medspace = 2^{16}\cdot 5^{15}\cdot 13^{15}\cdot 197^{16}\) 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:  \(2729.49\)
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:  $2^{4/5}5^{3/4}13^{3/4}197^{4/5}\approx 2729.493063361952$
Ramified primes:   \(2\), \(5\), \(13\), \(197\) 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:  \(\Q(\sqrt{65}) \)
$\Aut(K/\Q)$:   $C_4$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}$, $\frac{1}{10}a^{12}-\frac{1}{5}a^{11}+\frac{1}{5}a^{9}-\frac{1}{5}a^{8}-\frac{1}{5}a^{7}-\frac{1}{2}a^{6}+\frac{1}{5}a^{4}-\frac{2}{5}a^{3}+\frac{2}{5}a^{2}-\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{10}a^{13}+\frac{1}{10}a^{11}+\frac{1}{5}a^{10}+\frac{1}{5}a^{9}+\frac{2}{5}a^{8}+\frac{1}{10}a^{7}-\frac{3}{10}a^{5}-\frac{2}{5}a^{3}+\frac{2}{5}a^{2}-\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{60}a^{14}+\frac{1}{30}a^{13}+\frac{1}{30}a^{12}+\frac{1}{30}a^{11}-\frac{3}{20}a^{10}-\frac{1}{6}a^{9}-\frac{7}{15}a^{8}+\frac{7}{60}a^{6}-\frac{1}{10}a^{5}-\frac{1}{30}a^{4}+\frac{1}{30}a^{3}-\frac{1}{3}a^{2}-\frac{1}{10}a-\frac{1}{3}$, $\frac{1}{60}a^{15}-\frac{1}{30}a^{13}-\frac{1}{30}a^{12}-\frac{13}{60}a^{11}+\frac{2}{15}a^{10}-\frac{2}{15}a^{9}-\frac{1}{15}a^{8}+\frac{7}{60}a^{7}-\frac{1}{3}a^{6}+\frac{1}{6}a^{5}+\frac{1}{10}a^{4}-\frac{2}{5}a^{3}-\frac{13}{30}a^{2}-\frac{2}{15}a-\frac{1}{3}$, $\frac{1}{180}a^{16}-\frac{1}{180}a^{15}-\frac{1}{90}a^{13}+\frac{1}{36}a^{12}-\frac{7}{36}a^{11}-\frac{4}{45}a^{10}-\frac{17}{90}a^{9}-\frac{7}{20}a^{8}-\frac{29}{60}a^{7}-\frac{23}{90}a^{6}+\frac{1}{90}a^{5}-\frac{7}{18}a^{4}-\frac{41}{90}a^{3}-\frac{29}{90}a^{2}-\frac{1}{9}$, $\frac{1}{180}a^{17}-\frac{1}{180}a^{15}+\frac{1}{180}a^{14}-\frac{1}{20}a^{13}-\frac{1}{30}a^{12}-\frac{1}{20}a^{11}-\frac{23}{180}a^{10}-\frac{37}{180}a^{9}-\frac{2}{5}a^{8}+\frac{83}{180}a^{7}-\frac{23}{180}a^{6}-\frac{8}{45}a^{5}+\frac{29}{90}a^{4}+\frac{23}{90}a^{3}+\frac{31}{90}a^{2}-\frac{37}{90}a-\frac{2}{45}$, $\frac{1}{69\cdots 60}a^{18}+\frac{5079369877093}{34\cdots 30}a^{17}+\frac{1248953290175}{13\cdots 72}a^{16}-\frac{16065740067407}{23\cdots 20}a^{15}-\frac{39521807986933}{69\cdots 60}a^{14}+\frac{69823220663306}{17\cdots 65}a^{13}-\frac{5833881731459}{69\cdots 60}a^{12}+\frac{202604489547439}{23\cdots 20}a^{11}-\frac{13\cdots 09}{69\cdots 60}a^{10}-\frac{169649607759241}{17\cdots 65}a^{9}+\frac{509444267825635}{13\cdots 72}a^{8}+\frac{28\cdots 37}{69\cdots 60}a^{7}+\frac{138095687281997}{34\cdots 30}a^{6}+\frac{109801049810827}{695672919946386}a^{5}+\frac{340830820751693}{34\cdots 30}a^{4}-\frac{487999879588151}{34\cdots 30}a^{3}+\frac{86035073769061}{386484955525770}a^{2}+\frac{228432525092221}{579727433288655}a-\frac{153854688531982}{347836459973193}$, $\frac{1}{91\cdots 60}a^{19}-\frac{17\cdots 03}{50\cdots 70}a^{18}-\frac{11\cdots 11}{10\cdots 40}a^{17}+\frac{54\cdots 55}{18\cdots 72}a^{16}+\frac{14\cdots 13}{45\cdots 30}a^{15}+\frac{12\cdots 69}{91\cdots 60}a^{14}-\frac{10\cdots 81}{20\cdots 08}a^{13}+\frac{23\cdots 09}{91\cdots 60}a^{12}+\frac{32\cdots 71}{45\cdots 30}a^{11}-\frac{21\cdots 13}{91\cdots 60}a^{10}-\frac{27\cdots 97}{91\cdots 60}a^{9}+\frac{32\cdots 33}{91\cdots 60}a^{8}-\frac{10\cdots 91}{30\cdots 20}a^{7}+\frac{29\cdots 49}{91\cdots 60}a^{6}-\frac{14\cdots 21}{91\cdots 86}a^{5}-\frac{66\cdots 87}{15\cdots 10}a^{4}+\frac{10\cdots 23}{22\cdots 15}a^{3}-\frac{18\cdots 87}{50\cdots 70}a^{2}+\frac{10\cdots 59}{45\cdots 30}a-\frac{85\cdots 77}{22\cdots 15}$ 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:  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:  $11$
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)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot R \cdot h}{2\cdot\sqrt{526792882835395388103720314122737484676904390310681994000000000000000}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

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 - x^19 - 253*x^18 - 24544*x^17 + 398685*x^16 + 8721545*x^15 - 70361823*x^14 - 3938000908*x^13 - 11285455377*x^12 + 2955494285207*x^11 - 38784617114891*x^10 - 385201843870298*x^9 + 14936778594591960*x^8 - 124575614095119552*x^7 - 656667047355929392*x^6 + 20992220260103943024*x^5 - 141973587074569077668*x^4 - 247833350982037177072*x^3 + 9459373257275866071148*x^2 - 56140694999572599084636*x + 117438049640833114893196) 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^19 - 253*x^18 - 24544*x^17 + 398685*x^16 + 8721545*x^15 - 70361823*x^14 - 3938000908*x^13 - 11285455377*x^12 + 2955494285207*x^11 - 38784617114891*x^10 - 385201843870298*x^9 + 14936778594591960*x^8 - 124575614095119552*x^7 - 656667047355929392*x^6 + 20992220260103943024*x^5 - 141973587074569077668*x^4 - 247833350982037177072*x^3 + 9459373257275866071148*x^2 - 56140694999572599084636*x + 117438049640833114893196, 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^19 - 253*x^18 - 24544*x^17 + 398685*x^16 + 8721545*x^15 - 70361823*x^14 - 3938000908*x^13 - 11285455377*x^12 + 2955494285207*x^11 - 38784617114891*x^10 - 385201843870298*x^9 + 14936778594591960*x^8 - 124575614095119552*x^7 - 656667047355929392*x^6 + 20992220260103943024*x^5 - 141973587074569077668*x^4 - 247833350982037177072*x^3 + 9459373257275866071148*x^2 - 56140694999572599084636*x + 117438049640833114893196); 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 - x^19 - 253*x^18 - 24544*x^17 + 398685*x^16 + 8721545*x^15 - 70361823*x^14 - 3938000908*x^13 - 11285455377*x^12 + 2955494285207*x^11 - 38784617114891*x^10 - 385201843870298*x^9 + 14936778594591960*x^8 - 124575614095119552*x^7 - 656667047355929392*x^6 + 20992220260103943024*x^5 - 141973587074569077668*x^4 - 247833350982037177072*x^3 + 9459373257275866071148*x^2 - 56140694999572599084636*x + 117438049640833114893196); 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

$C_4\times F_5$ (as 20T20):

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); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_4\times F_5$
Character table for $C_4\times F_5$

Intermediate fields

\(\Q(\sqrt{65}) \), 4.4.274625.2, 5.1.509074806578000.1, 10.2.16845215315007827284555460000000.1

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(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed
Minimal sibling: 20.0.4214343062683163104829762512981899877415235122485455952000000000000.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{5}$ R ${\href{/padicField/7.4.0.1}{4} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{5}$ R ${\href{/padicField/17.4.0.1}{4} }^{5}$ $20$ ${\href{/padicField/23.4.0.1}{4} }^{5}$ ${\href{/padicField/29.2.0.1}{2} }^{10}$ ${\href{/padicField/31.4.0.1}{4} }^{5}$ ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{5}$ ${\href{/padicField/43.4.0.1}{4} }^{5}$ ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{5}$ $20$

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 2.2.5.8a1.1$x^{10} + 5 x^{9} + 15 x^{8} + 30 x^{7} + 45 x^{6} + 51 x^{5} + 45 x^{4} + 30 x^{3} + 15 x^{2} + 5 x + 3$$5$$2$$8$$F_5$$$[\ ]_{5}^{4}$$
2.2.5.8a1.1$x^{10} + 5 x^{9} + 15 x^{8} + 30 x^{7} + 45 x^{6} + 51 x^{5} + 45 x^{4} + 30 x^{3} + 15 x^{2} + 5 x + 3$$5$$2$$8$$F_5$$$[\ ]_{5}^{4}$$
\(5\) Copy content Toggle raw display 5.1.4.3a1.2$x^{4} + 10$$4$$1$$3$$C_4$$$[\ ]_{4}$$
5.4.4.12a1.4$x^{16} + 16 x^{14} + 16 x^{13} + 104 x^{12} + 192 x^{11} + 448 x^{10} + 864 x^{9} + 1432 x^{8} + 2048 x^{7} + 2624 x^{6} + 2752 x^{5} + 2208 x^{4} + 1280 x^{3} + 512 x^{2} + 128 x + 21$$4$$4$$12$$C_4^2$$$[\ ]_{4}^{4}$$
\(13\) Copy content Toggle raw display 13.1.4.3a1.4$x^{4} + 104$$4$$1$$3$$C_4$$$[\ ]_{4}$$
13.4.4.12a1.4$x^{16} + 12 x^{14} + 48 x^{13} + 62 x^{12} + 432 x^{11} + 1044 x^{10} + 1584 x^{9} + 5505 x^{8} + 9936 x^{7} + 11592 x^{6} + 23904 x^{5} + 31352 x^{4} + 15552 x^{3} + 3552 x^{2} + 384 x + 29$$4$$4$$12$$C_4^2$$$[\ ]_{4}^{4}$$
\(197\) Copy content Toggle raw display 197.2.5.8a1.1$x^{10} + 960 x^{9} + 368650 x^{8} + 70786560 x^{7} + 6796984360 x^{6} + 261202401792 x^{5} + 13593968720 x^{4} + 283146240 x^{3} + 2949200 x^{2} + 15360 x + 229$$5$$2$$8$$F_5$$$[\ ]_{5}^{4}$$
197.2.5.8a1.1$x^{10} + 960 x^{9} + 368650 x^{8} + 70786560 x^{7} + 6796984360 x^{6} + 261202401792 x^{5} + 13593968720 x^{4} + 283146240 x^{3} + 2949200 x^{2} + 15360 x + 229$$5$$2$$8$$F_5$$$[\ ]_{5}^{4}$$

Spectrum of ring of integers

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