↑

Properties

Label 20.0.200...608.1
Degree $20$
Signature $(0, 10)$
Discriminant $2.003\times 10^{35}$
Root discriminant \(58.22\)
Ramified primes $2,11$
Class number $2050$ (GRH)
Class group [2050] (GRH)
Galois group $C_{20}$ (as 20T1)

Related objects

Downloads

Learn more

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 + 44*x^18 + 770*x^16 + 6952*x^14 + 35552*x^12 + 107712*x^10 + 196504*x^8 + 212960*x^6 + 129712*x^4 + 38720*x^2 + 3872)
 
Copy content gp:K = bnfinit(y^20 + 44*y^18 + 770*y^16 + 6952*y^14 + 35552*y^12 + 107712*y^10 + 196504*y^8 + 212960*y^6 + 129712*y^4 + 38720*y^2 + 3872, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + 44*x^18 + 770*x^16 + 6952*x^14 + 35552*x^12 + 107712*x^10 + 196504*x^8 + 212960*x^6 + 129712*x^4 + 38720*x^2 + 3872);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 + 44*x^18 + 770*x^16 + 6952*x^14 + 35552*x^12 + 107712*x^10 + 196504*x^8 + 212960*x^6 + 129712*x^4 + 38720*x^2 + 3872)
 

\( x^{20} + 44 x^{18} + 770 x^{16} + 6952 x^{14} + 35552 x^{12} + 107712 x^{10} + 196504 x^{8} + \cdots + 3872 \) 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:   \(200317132330035063121671003054276608\) \(\medspace = 2^{55}\cdot 11^{18}\) 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:  \(58.22\)
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:  $2^{11/4}11^{9/10}\approx 58.22183708777889$
Ramified primes:   \(2\), \(11\) 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:  \(\Q(\sqrt{2}) \)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $C_{20}$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(176=2^{4}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{176}(1,·)$, $\chi_{176}(9,·)$, $\chi_{176}(13,·)$, $\chi_{176}(173,·)$, $\chi_{176}(81,·)$, $\chi_{176}(149,·)$, $\chi_{176}(25,·)$, $\chi_{176}(89,·)$, $\chi_{176}(29,·)$, $\chi_{176}(97,·)$, $\chi_{176}(101,·)$, $\chi_{176}(49,·)$, $\chi_{176}(169,·)$, $\chi_{176}(109,·)$, $\chi_{176}(113,·)$, $\chi_{176}(117,·)$, $\chi_{176}(137,·)$, $\chi_{176}(61,·)$, $\chi_{176}(85,·)$, $\chi_{176}(21,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{512}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{4}a^{9}$, $\frac{1}{44}a^{10}$, $\frac{1}{44}a^{11}$, $\frac{1}{88}a^{12}$, $\frac{1}{88}a^{13}$, $\frac{1}{88}a^{14}$, $\frac{1}{88}a^{15}$, $\frac{1}{4048}a^{16}+\frac{1}{2024}a^{14}+\frac{5}{2024}a^{12}+\frac{7}{1012}a^{10}-\frac{2}{23}a^{8}+\frac{3}{23}a^{6}-\frac{7}{46}a^{4}-\frac{4}{23}a^{2}+\frac{5}{23}$, $\frac{1}{4048}a^{17}+\frac{1}{2024}a^{15}+\frac{5}{2024}a^{13}+\frac{7}{1012}a^{11}-\frac{2}{23}a^{9}+\frac{3}{23}a^{7}-\frac{7}{46}a^{5}-\frac{4}{23}a^{3}+\frac{5}{23}a$, $\frac{1}{805552}a^{18}-\frac{45}{805552}a^{16}+\frac{945}{201388}a^{14}-\frac{87}{36616}a^{12}-\frac{340}{50347}a^{10}+\frac{396}{4577}a^{8}-\frac{197}{9154}a^{6}+\frac{126}{4577}a^{4}-\frac{129}{4577}a^{2}+\frac{1950}{4577}$, $\frac{1}{805552}a^{19}-\frac{45}{805552}a^{17}+\frac{945}{201388}a^{15}-\frac{87}{36616}a^{13}-\frac{340}{50347}a^{11}+\frac{396}{4577}a^{9}-\frac{197}{9154}a^{7}+\frac{126}{4577}a^{5}-\frac{129}{4577}a^{3}+\frac{1950}{4577}a$ 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:  $C_{2050}$, which has order $2050$ (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:  $C_{2050}$, which has order $2050$ (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:   $2050$ (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:   $\frac{80}{50347}a^{18}+\frac{53243}{805552}a^{16}+\frac{53650}{50347}a^{14}+\frac{429353}{50347}a^{12}+\frac{168120}{4577}a^{10}+\frac{408773}{4577}a^{8}+\frac{586340}{4577}a^{6}+\frac{498784}{4577}a^{4}+\frac{215368}{4577}a^{2}+\frac{24818}{4577}$, $\frac{2155}{402776}a^{18}+\frac{44943}{201388}a^{16}+\frac{726367}{201388}a^{14}+\frac{2905351}{100694}a^{12}+\frac{1124049}{9154}a^{10}+\frac{5205787}{18308}a^{8}+\frac{1625762}{4577}a^{6}+\frac{2072165}{9154}a^{4}+\frac{292546}{4577}a^{2}+\frac{25207}{4577}$, $\frac{1515}{402776}a^{18}+\frac{126529}{805552}a^{16}+\frac{511767}{201388}a^{14}+\frac{2046645}{100694}a^{12}+\frac{787809}{9154}a^{10}+\frac{3570695}{18308}a^{8}+\frac{1039422}{4577}a^{6}+\frac{1074597}{9154}a^{4}+\frac{77178}{4577}a^{2}+\frac{4966}{4577}$, $\frac{147}{50347}a^{18}+\frac{6519}{50347}a^{16}+\frac{115074}{50347}a^{14}+\frac{8370363}{402776}a^{12}+\frac{971265}{9154}a^{10}+\frac{2866739}{9154}a^{8}+\frac{2417217}{4577}a^{6}+\frac{4380669}{9154}a^{4}+\frac{915666}{4577}a^{2}+\frac{112681}{4577}$, $\frac{9863}{805552}a^{18}+\frac{52901}{100694}a^{16}+\frac{1783211}{201388}a^{14}+\frac{3799442}{50347}a^{12}+\frac{71105401}{201388}a^{10}+\frac{4277082}{4577}a^{8}+\frac{6359953}{4577}a^{6}+\frac{5020952}{4577}a^{4}+\frac{1813765}{4577}a^{2}+\frac{188749}{4577}$, $\frac{315}{805552}a^{18}+\frac{10501}{805552}a^{16}+\frac{26239}{201388}a^{14}+\frac{14159}{402776}a^{12}-\frac{28841}{4378}a^{10}-\frac{176347}{4577}a^{8}-\frac{801937}{9154}a^{6}-\frac{793633}{9154}a^{4}-\frac{148493}{4577}a^{2}-\frac{8421}{4577}$, $\frac{3505}{402776}a^{18}+\frac{292893}{805552}a^{16}+\frac{1186775}{201388}a^{14}+\frac{19080835}{402776}a^{12}+\frac{40961655}{201388}a^{10}+\frac{8759023}{18308}a^{8}+\frac{121963}{199}a^{6}+\frac{3687069}{9154}a^{4}+\frac{529903}{4577}a^{2}+\frac{46358}{4577}$, $\frac{4863}{805552}a^{18}+\frac{209413}{805552}a^{16}+\frac{443585}{100694}a^{14}+\frac{693213}{18308}a^{12}+\frac{18079093}{100694}a^{10}+\frac{8886047}{18308}a^{8}+\frac{3416917}{4577}a^{6}+\frac{2853279}{4577}a^{4}+\frac{49487}{199}a^{2}+\frac{149551}{4577}$, $\frac{7511}{805552}a^{18}+\frac{39863}{100694}a^{16}+\frac{120265}{18308}a^{14}+\frac{22025173}{402776}a^{12}+\frac{49737571}{201388}a^{10}+\frac{5687425}{9154}a^{8}+\frac{3942736}{4577}a^{6}+\frac{5661235}{9154}a^{4}+\frac{898099}{4577}a^{2}+\frac{76068}{4577}$ 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:  \( 530208.250733 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 530208.250733 \cdot 2050}{2\cdot\sqrt{200317132330035063121671003054276608}}\cr\approx \mathstrut & 0.116442087724 \end{aligned}\] (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 + 44*x^18 + 770*x^16 + 6952*x^14 + 35552*x^12 + 107712*x^10 + 196504*x^8 + 212960*x^6 + 129712*x^4 + 38720*x^2 + 3872) 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 + 44*x^18 + 770*x^16 + 6952*x^14 + 35552*x^12 + 107712*x^10 + 196504*x^8 + 212960*x^6 + 129712*x^4 + 38720*x^2 + 3872, 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 + 44*x^18 + 770*x^16 + 6952*x^14 + 35552*x^12 + 107712*x^10 + 196504*x^8 + 212960*x^6 + 129712*x^4 + 38720*x^2 + 3872); 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 + 44*x^18 + 770*x^16 + 6952*x^14 + 35552*x^12 + 107712*x^10 + 196504*x^8 + 212960*x^6 + 129712*x^4 + 38720*x^2 + 3872); 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

$C_{20}$ (as 20T1):

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 cyclic group of order 20
The 20 conjugacy class representatives for $C_{20}$
Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-22 +11 \sqrt{2}})\), \(\Q(\zeta_{11})^+\), 10.10.7024111812608.1

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $20$ $20$ ${\href{/padicField/7.5.0.1}{5} }^{4}$ R $20$ ${\href{/padicField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/padicField/23.2.0.1}{2} }^{10}$ $20$ ${\href{/padicField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/padicField/41.5.0.1}{5} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{5}$ ${\href{/padicField/47.5.0.1}{5} }^{4}$ $20$ $20$

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 2.5.4.55a1.6634$x^{20} + 4 x^{17} + 12 x^{15} + 6 x^{14} + 36 x^{12} + 4 x^{11} + 34 x^{10} + 36 x^{9} + x^{8} + 68 x^{7} + 12 x^{6} + 36 x^{5} + 34 x^{4} + 36 x^{2} + 31$$4$$5$$55$20T1not computed
\(11\) Copy content Toggle raw display 11.2.10.18a1.7$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} + 1996241653 x^{10} + 2355135020 x^{9} + 1953240660 x^{8} + 1157466240 x^{7} + 496075680 x^{6} + 154293888 x^{5} + 34538880 x^{4} + 5429760 x^{3} + 569600 x^{2} + 35950 x + 1057$$10$$2$$18$20T1$$[\ ]_{10}^{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)