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Properties

Label 20.0.516...984.1
Degree $20$
Signature $[0, 10]$
Discriminant $5.165\times 10^{44}$
Root discriminant \(172.05\)
Ramified primes $2,3,61$
Class number $10120400$ (GRH)
Class group [2, 2, 2530100] (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 183*x^18 + 12627*x^16 + 421632*x^14 + 7258329*x^12 + 64153944*x^10 + 284912883*x^8 + 601532163*x^6 + 511482438*x^4 + 81645084*x^2 + 3601989)
 
gp: K = bnfinit(y^20 + 183*y^18 + 12627*y^16 + 421632*y^14 + 7258329*y^12 + 64153944*y^10 + 284912883*y^8 + 601532163*y^6 + 511482438*y^4 + 81645084*y^2 + 3601989, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + 183*x^18 + 12627*x^16 + 421632*x^14 + 7258329*x^12 + 64153944*x^10 + 284912883*x^8 + 601532163*x^6 + 511482438*x^4 + 81645084*x^2 + 3601989);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 + 183*x^18 + 12627*x^16 + 421632*x^14 + 7258329*x^12 + 64153944*x^10 + 284912883*x^8 + 601532163*x^6 + 511482438*x^4 + 81645084*x^2 + 3601989)
 

\( x^{20} + 183 x^{18} + 12627 x^{16} + 421632 x^{14} + 7258329 x^{12} + 64153944 x^{10} + 284912883 x^{8} + \cdots + 3601989 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(516510703469799849283535395746427778167209984\) \(\medspace = 2^{20}\cdot 3^{10}\cdot 61^{19}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(172.05\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2\cdot 3^{1/2}61^{19/20}\approx 172.04971846021803$
Ramified primes:   \(2\), \(3\), \(61\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{61}) \)
$\card{ \Gal(K/\Q) }$:  $20$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(732=2^{2}\cdot 3\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{732}(1,·)$, $\chi_{732}(11,·)$, $\chi_{732}(325,·)$, $\chi_{732}(455,·)$, $\chi_{732}(587,·)$, $\chi_{732}(529,·)$, $\chi_{732}(599,·)$, $\chi_{732}(601,·)$, $\chi_{732}(155,·)$, $\chi_{732}(419,·)$, $\chi_{732}(613,·)$, $\chi_{732}(647,·)$, $\chi_{732}(253,·)$, $\chi_{732}(241,·)$, $\chi_{732}(695,·)$, $\chi_{732}(23,·)$, $\chi_{732}(121,·)$, $\chi_{732}(217,·)$, $\chi_{732}(637,·)$, $\chi_{732}(191,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{512}$

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

$1$, $a$, $\frac{1}{3}a^{2}$, $\frac{1}{3}a^{3}$, $\frac{1}{9}a^{4}$, $\frac{1}{9}a^{5}$, $\frac{1}{27}a^{6}$, $\frac{1}{27}a^{7}$, $\frac{1}{81}a^{8}$, $\frac{1}{81}a^{9}$, $\frac{1}{3159}a^{10}-\frac{1}{1053}a^{8}+\frac{1}{351}a^{6}-\frac{1}{117}a^{4}+\frac{1}{39}a^{2}-\frac{1}{13}$, $\frac{1}{3159}a^{11}-\frac{1}{1053}a^{9}+\frac{1}{351}a^{7}-\frac{1}{117}a^{5}+\frac{1}{39}a^{3}-\frac{1}{13}a$, $\frac{1}{9477}a^{12}-\frac{1}{13}$, $\frac{1}{9477}a^{13}-\frac{1}{13}a$, $\frac{1}{369603}a^{14}-\frac{1}{123201}a^{12}+\frac{5}{41067}a^{10}-\frac{70}{13689}a^{8}+\frac{19}{1521}a^{6}-\frac{2}{169}a^{4}+\frac{43}{507}a^{2}-\frac{30}{169}$, $\frac{1}{369603}a^{15}-\frac{1}{123201}a^{13}+\frac{5}{41067}a^{11}-\frac{70}{13689}a^{9}+\frac{19}{1521}a^{7}-\frac{2}{169}a^{5}+\frac{43}{507}a^{3}-\frac{30}{169}a$, $\frac{1}{52114023}a^{16}-\frac{17}{17371341}a^{14}+\frac{190}{5790447}a^{12}-\frac{11}{643383}a^{10}+\frac{3257}{643383}a^{8}+\frac{877}{214461}a^{6}+\frac{1397}{71487}a^{4}-\frac{313}{7943}a^{2}+\frac{1377}{7943}$, $\frac{1}{52114023}a^{17}-\frac{17}{17371341}a^{15}+\frac{190}{5790447}a^{13}-\frac{11}{643383}a^{11}+\frac{3257}{643383}a^{9}+\frac{877}{214461}a^{7}+\frac{1397}{71487}a^{5}-\frac{313}{7943}a^{3}+\frac{1377}{7943}a$, $\frac{1}{85\!\cdots\!41}a^{18}+\frac{94544462}{28\!\cdots\!47}a^{16}+\frac{4736255416}{94\!\cdots\!49}a^{14}+\frac{131691240659}{31\!\cdots\!83}a^{12}-\frac{23798126569}{351648048785787}a^{10}+\frac{96905103688}{39072005420643}a^{8}-\frac{1858616811560}{117216016261929}a^{6}+\frac{1849417613825}{39072005420643}a^{4}-\frac{1451954339066}{13024001806881}a^{2}-\frac{1026470992862}{4341333935627}$, $\frac{1}{85\!\cdots\!41}a^{19}+\frac{94544462}{28\!\cdots\!47}a^{17}+\frac{4736255416}{94\!\cdots\!49}a^{15}+\frac{131691240659}{31\!\cdots\!83}a^{13}-\frac{23798126569}{351648048785787}a^{11}+\frac{96905103688}{39072005420643}a^{9}-\frac{1858616811560}{117216016261929}a^{7}+\frac{1849417613825}{39072005420643}a^{5}-\frac{1451954339066}{13024001806881}a^{3}-\frac{1026470992862}{4341333935627}a$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $13$

Class group and class number

$C_{2}\times C_{2}\times C_{2530100}$, which has order $10120400$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{13035332716}{28\!\cdots\!47}a^{18}+\frac{793469941963}{94\!\cdots\!49}a^{16}+\frac{18186785240248}{31\!\cdots\!83}a^{14}+\frac{4282057866640}{22445620135263}a^{12}+\frac{34\!\cdots\!80}{10\!\cdots\!61}a^{10}+\frac{33\!\cdots\!14}{117216016261929}a^{8}+\frac{14\!\cdots\!04}{117216016261929}a^{6}+\frac{10\!\cdots\!08}{4341333935627}a^{4}+\frac{798806651890594}{4341333935627}a^{2}+\frac{64304991827036}{4341333935627}$, $\frac{29597819405}{85\!\cdots\!41}a^{18}+\frac{1803427483418}{28\!\cdots\!47}a^{16}+\frac{41401517481722}{94\!\cdots\!49}a^{14}+\frac{153116645524693}{10\!\cdots\!61}a^{12}+\frac{26\!\cdots\!86}{10\!\cdots\!61}a^{10}+\frac{76\!\cdots\!96}{351648048785787}a^{8}+\frac{12\!\cdots\!94}{13024001806881}a^{6}+\frac{24\!\cdots\!01}{13024001806881}a^{4}+\frac{18\!\cdots\!20}{13024001806881}a^{2}+\frac{27141051753640}{4341333935627}$, $\frac{3140234503}{85\!\cdots\!41}a^{18}+\frac{186363936665}{28\!\cdots\!47}a^{16}+\frac{1364610283825}{31\!\cdots\!83}a^{14}+\frac{42037312338995}{31\!\cdots\!83}a^{12}+\frac{68878484876785}{351648048785787}a^{10}+\frac{430746715209866}{351648048785787}a^{8}+\frac{208896163033084}{117216016261929}a^{6}-\frac{31568880683205}{4341333935627}a^{4}-\frac{225560557747919}{13024001806881}a^{2}-\frac{6461423379342}{4341333935627}$, $\frac{49811189014}{85\!\cdots\!41}a^{18}+\frac{3037360978546}{28\!\cdots\!47}a^{16}+\frac{23272022713205}{31\!\cdots\!83}a^{14}+\frac{258747082693990}{10\!\cdots\!61}a^{12}+\frac{44\!\cdots\!61}{10\!\cdots\!61}a^{10}+\frac{43\!\cdots\!43}{117216016261929}a^{8}+\frac{19\!\cdots\!01}{117216016261929}a^{6}+\frac{13\!\cdots\!51}{39072005420643}a^{4}+\frac{34\!\cdots\!09}{13024001806881}a^{2}+\frac{84718368488293}{4341333935627}$, $\frac{21868963172}{85\!\cdots\!41}a^{18}+\frac{444989415640}{94\!\cdots\!49}a^{16}+\frac{30740105772971}{94\!\cdots\!49}a^{14}+\frac{12696706588634}{117216016261929}a^{12}+\frac{19\!\cdots\!10}{10\!\cdots\!61}a^{10}+\frac{72195539752957}{4341333935627}a^{8}+\frac{87\!\cdots\!11}{117216016261929}a^{6}+\frac{686500529231853}{4341333935627}a^{4}+\frac{562214132083543}{4341333935627}a^{2}+\frac{51190605077197}{4341333935627}$, $\frac{8068412672}{85\!\cdots\!41}a^{18}+\frac{165511185430}{94\!\cdots\!49}a^{16}+\frac{11581399455386}{94\!\cdots\!49}a^{14}+\frac{14648916507607}{351648048785787}a^{12}+\frac{87219781751195}{117216016261929}a^{10}+\frac{272889377717968}{39072005420643}a^{8}+\frac{39\!\cdots\!24}{117216016261929}a^{6}+\frac{340958074772753}{4341333935627}a^{4}+\frac{300096731414878}{4341333935627}a^{2}+\frac{31353615634252}{4341333935627}$, $\frac{1107830188}{28\!\cdots\!47}a^{18}+\frac{197919270844}{28\!\cdots\!47}a^{16}+\frac{1457919574175}{31\!\cdots\!83}a^{14}+\frac{45417554080447}{31\!\cdots\!83}a^{12}+\frac{76235614707229}{351648048785787}a^{10}+\frac{510389909020853}{351648048785787}a^{8}+\frac{40851935792098}{13024001806881}a^{6}-\frac{36941146780219}{13024001806881}a^{4}-\frac{123712658711734}{13024001806881}a^{2}-\frac{3272348671578}{4341333935627}$, $\frac{50975629706}{85\!\cdots\!41}a^{18}+\frac{1036099719100}{94\!\cdots\!49}a^{16}+\frac{23814917953024}{31\!\cdots\!83}a^{14}+\frac{794332563755462}{31\!\cdots\!83}a^{12}+\frac{505436172418876}{117216016261929}a^{10}+\frac{13\!\cdots\!07}{351648048785787}a^{8}+\frac{65\!\cdots\!72}{39072005420643}a^{6}+\frac{45\!\cdots\!04}{13024001806881}a^{4}+\frac{12\!\cdots\!40}{4341333935627}a^{2}+\frac{109401919494280}{4341333935627}$, $\frac{40426739218}{85\!\cdots\!41}a^{18}+\frac{2467866990256}{28\!\cdots\!47}a^{16}+\frac{56829101315786}{94\!\cdots\!49}a^{14}+\frac{23473430482066}{117216016261929}a^{12}+\frac{12\!\cdots\!50}{351648048785787}a^{10}+\frac{10\!\cdots\!66}{351648048785787}a^{8}+\frac{16\!\cdots\!10}{117216016261929}a^{6}+\frac{11\!\cdots\!96}{39072005420643}a^{4}+\frac{31\!\cdots\!51}{13024001806881}a^{2}+\frac{80423961984290}{4341333935627}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 36549838.47150319 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 36549838.47150319 \cdot 10120400}{2\cdot\sqrt{516510703469799849283535395746427778167209984}}\cr\approx \mathstrut & 0.780390843371090 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 + 183*x^18 + 12627*x^16 + 421632*x^14 + 7258329*x^12 + 64153944*x^10 + 284912883*x^8 + 601532163*x^6 + 511482438*x^4 + 81645084*x^2 + 3601989)
 
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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^20 + 183*x^18 + 12627*x^16 + 421632*x^14 + 7258329*x^12 + 64153944*x^10 + 284912883*x^8 + 601532163*x^6 + 511482438*x^4 + 81645084*x^2 + 3601989, 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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^20 + 183*x^18 + 12627*x^16 + 421632*x^14 + 7258329*x^12 + 64153944*x^10 + 284912883*x^8 + 601532163*x^6 + 511482438*x^4 + 81645084*x^2 + 3601989);
 
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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 + 183*x^18 + 12627*x^16 + 421632*x^14 + 7258329*x^12 + 64153944*x^10 + 284912883*x^8 + 601532163*x^6 + 511482438*x^4 + 81645084*x^2 + 3601989);
 
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):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A cyclic group of order 20
The 20 conjugacy class representatives for $C_{20}$
Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{61}) \), 4.0.32685264.1, 5.5.13845841.1, 10.10.11694146092834141.1

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
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 R ${\href{/padicField/5.5.0.1}{5} }^{4}$ $20$ ${\href{/padicField/11.4.0.1}{4} }^{5}$ ${\href{/padicField/13.1.0.1}{1} }^{20}$ $20$ ${\href{/padicField/19.5.0.1}{5} }^{4}$ $20$ ${\href{/padicField/29.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/padicField/41.5.0.1}{5} }^{4}$ $20$ ${\href{/padicField/47.1.0.1}{1} }^{20}$ $20$ $20$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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 Deg $20$$2$$10$$20$
\(3\) Copy content Toggle raw display 3.10.5.2$x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
\(61\) Copy content Toggle raw display 61.20.19.6$x^{20} + 61$$20$$1$$19$20T1$[\ ]_{20}$