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Properties

Label 20.0.992...000.1
Degree $20$
Signature $[0, 10]$
Discriminant $9.923\times 10^{34}$
Root discriminant \(56.21\)
Ramified primes $2,5,7,23,431$
Class number $1904$ (GRH)
Class group [2, 2, 476] (GRH)
Galois group $C_2^8.(C_4\times S_5)$ (as 20T802)

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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 + 25*x^18 + 268*x^16 + 1613*x^14 + 5989*x^12 + 14177*x^10 + 21323*x^8 + 19642*x^6 + 10226*x^4 + 2590*x^2 + 245)
 
Copy content gp:K = bnfinit(y^20 + 25*y^18 + 268*y^16 + 1613*y^14 + 5989*y^12 + 14177*y^10 + 21323*y^8 + 19642*y^6 + 10226*y^4 + 2590*y^2 + 245, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + 25*x^18 + 268*x^16 + 1613*x^14 + 5989*x^12 + 14177*x^10 + 21323*x^8 + 19642*x^6 + 10226*x^4 + 2590*x^2 + 245);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 + 25*x^18 + 268*x^16 + 1613*x^14 + 5989*x^12 + 14177*x^10 + 21323*x^8 + 19642*x^6 + 10226*x^4 + 2590*x^2 + 245)
 

\( x^{20} + 25 x^{18} + 268 x^{16} + 1613 x^{14} + 5989 x^{12} + 14177 x^{10} + 21323 x^{8} + 19642 x^{6} + \cdots + 245 \) 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:  $[0, 10]$
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:   \(99230781955913713241292800000000000\) \(\medspace = 2^{32}\cdot 5^{11}\cdot 7^{2}\cdot 23^{4}\cdot 431^{4}\) 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:  \(56.21\)
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:   \(2\), \(5\), \(7\), \(23\), \(431\) 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:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$:   $C_2$
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 a CM field.
Reflex fields:  unavailable$^{512}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{49}a^{18}-\frac{3}{49}a^{16}+\frac{9}{49}a^{14}-\frac{11}{49}a^{12}-\frac{24}{49}a^{10}+\frac{2}{49}a^{8}+\frac{1}{49}a^{6}+\frac{2}{7}a^{4}-\frac{15}{49}a^{2}+\frac{3}{7}$, $\frac{1}{49}a^{19}-\frac{3}{49}a^{17}+\frac{9}{49}a^{15}-\frac{11}{49}a^{13}-\frac{24}{49}a^{11}+\frac{2}{49}a^{9}+\frac{1}{49}a^{7}+\frac{2}{7}a^{5}-\frac{15}{49}a^{3}+\frac{3}{7}a$ 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:  $C_{2}\times C_{2}\times C_{476}$, which has order $1904$ (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:  $C_{2}\times C_{2}\times C_{476}$, which has order $1904$ (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);
 
Relative class number:   $1904$ (assuming GRH)

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:  $9$
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:   $\frac{6}{49}a^{18}+\frac{129}{49}a^{16}+\frac{1132}{49}a^{14}+\frac{5177}{49}a^{12}+\frac{12890}{49}a^{10}+\frac{15741}{49}a^{8}+\frac{3632}{49}a^{6}-\frac{1535}{7}a^{4}-\frac{9008}{49}a^{2}-\frac{234}{7}$, $a^{2}+2$, $\frac{12}{49}a^{18}+\frac{258}{49}a^{16}+\frac{2313}{49}a^{14}+\frac{11285}{49}a^{12}+\frac{32836}{49}a^{10}+\frac{58726}{49}a^{8}+\frac{63957}{49}a^{6}+\frac{5743}{7}a^{4}+\frac{12854}{49}a^{2}+\frac{218}{7}$, $\frac{66}{49}a^{18}+\frac{1468}{49}a^{16}+\frac{13628}{49}a^{14}+\frac{68609}{49}a^{12}+\frac{203530}{49}a^{10}+\frac{361262}{49}a^{8}+\frac{370947}{49}a^{6}+\frac{28650}{7}a^{4}+\frac{47079}{49}a^{2}+\frac{527}{7}$, $\frac{13}{49}a^{18}+\frac{255}{49}a^{16}+\frac{1979}{49}a^{14}+\frac{7550}{49}a^{12}+\frac{13408}{49}a^{10}+\frac{3015}{49}a^{8}-\frac{26937}{49}a^{6}-\frac{5868}{7}a^{4}-\frac{22392}{49}a^{2}-\frac{528}{7}$, $\frac{36}{49}a^{18}+\frac{774}{49}a^{16}+\frac{6890}{49}a^{14}+\frac{32924}{49}a^{12}+\frac{91501}{49}a^{10}+\frac{149620}{49}a^{8}+\frac{138608}{49}a^{6}+\frac{9466}{7}a^{4}+\frac{13964}{49}a^{2}+\frac{143}{7}$, $\frac{53}{49}a^{18}+\frac{1164}{49}a^{16}+\frac{10669}{49}a^{14}+\frac{53121}{49}a^{12}+\frac{156655}{49}a^{10}+\frac{279749}{49}a^{8}+\frac{296503}{49}a^{6}+\frac{24956}{7}a^{4}+\frac{49920}{49}a^{2}+\frac{754}{7}$, $\frac{12}{7}a^{18}+\frac{272}{7}a^{16}+\frac{2593}{7}a^{14}+\frac{13560}{7}a^{12}+\frac{42531}{7}a^{10}+\frac{82120}{7}a^{8}+\frac{96248}{7}a^{6}+9250a^{4}+\frac{22059}{7}a^{2}+394$, $\frac{15}{49}a^{18}+\frac{298}{49}a^{16}+\frac{2340}{49}a^{14}+\frac{8949}{49}a^{12}+\frac{15075}{49}a^{10}-\frac{2616}{49}a^{8}-\frac{50112}{49}a^{6}-\frac{10442}{7}a^{4}-\frac{40062}{49}a^{2}-\frac{886}{7}$ 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:  \( 1397389.2024 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 1397389.2024 \cdot 1904}{2\cdot\sqrt{99230781955913713241292800000000000}}\cr\approx \mathstrut & 0.40497648656 \end{aligned}\] (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 + 25*x^18 + 268*x^16 + 1613*x^14 + 5989*x^12 + 14177*x^10 + 21323*x^8 + 19642*x^6 + 10226*x^4 + 2590*x^2 + 245) 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 + 25*x^18 + 268*x^16 + 1613*x^14 + 5989*x^12 + 14177*x^10 + 21323*x^8 + 19642*x^6 + 10226*x^4 + 2590*x^2 + 245, 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 + 25*x^18 + 268*x^16 + 1613*x^14 + 5989*x^12 + 14177*x^10 + 21323*x^8 + 19642*x^6 + 10226*x^4 + 2590*x^2 + 245); 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 + 25*x^18 + 268*x^16 + 1613*x^14 + 5989*x^12 + 14177*x^10 + 21323*x^8 + 19642*x^6 + 10226*x^4 + 2590*x^2 + 245); 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

$C_2^8.(C_4\times S_5)$ (as 20T802):

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 non-solvable group of order 122880
The 138 conjugacy class representatives for $C_2^8.(C_4\times S_5)$
Character table for $C_2^8.(C_4\times S_5)$

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.396520.1, 10.10.19653513800000.1

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: 20.0.777969330534363511811735552000000000.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $20$ R R ${\href{/padicField/11.10.0.1}{10} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.10.0.1}{10} }^{2}$ R ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}$ ${\href{/padicField/41.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/padicField/53.4.0.1}{4} }^{5}$ ${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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 2.2.4.20a1.2$x^{8} + 4 x^{7} + 14 x^{6} + 36 x^{5} + 59 x^{4} + 68 x^{3} + 50 x^{2} + 24 x + 7$$4$$2$$20$$Q_8:C_2$$$[2, 3, \frac{7}{2}]^{2}$$
2.6.2.12a9.2$x^{12} + 2 x^{11} + 4 x^{10} + 4 x^{9} + 7 x^{8} + 6 x^{7} + 7 x^{6} + 8 x^{5} + 6 x^{4} + 4 x^{3} + 7 x^{2} + 2 x + 3$$2$$6$$12$12T29$$[2, 2]^{12}$$
\(5\) Copy content Toggle raw display 5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.1.4.3a1.4$x^{4} + 20$$4$$1$$3$$C_4$$$[\ ]_{4}$$
5.3.2.3a1.2$x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 14$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
5.3.2.3a1.2$x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 14$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
\(7\) Copy content Toggle raw display 7.2.2.2a1.2$x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
7.4.1.0a1.1$x^{4} + 5 x^{2} + 4 x + 3$$1$$4$$0$$C_4$$$[\ ]^{4}$$
7.6.1.0a1.1$x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$$1$$6$$0$$C_6$$$[\ ]^{6}$$
7.6.1.0a1.1$x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$$1$$6$$0$$C_6$$$[\ ]^{6}$$
\(23\) Copy content Toggle raw display 23.2.1.0a1.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
23.2.1.0a1.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
23.2.1.0a1.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
23.2.1.0a1.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
23.4.1.0a1.1$x^{4} + 3 x^{2} + 19 x + 5$$1$$4$$0$$C_4$$$[\ ]^{4}$$
23.2.2.2a1.2$x^{4} + 42 x^{3} + 451 x^{2} + 210 x + 48$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
23.2.2.2a1.2$x^{4} + 42 x^{3} + 451 x^{2} + 210 x + 48$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(431\) Copy content Toggle raw display $\Q_{431}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{431}$$x$$1$$1$$0$Trivial$$[\ ]$$
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$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $4$$1$$4$$0$$C_4$$$[\ ]^{4}$$

Spectrum of ring of integers

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