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Properties

Label 20.0.200...569.1
Degree $20$
Signature $[0, 10]$
Discriminant $2.004\times 10^{24}$
Root discriminant \(16.41\)
Ramified primes $19,83$
Class number $1$
Class group trivial
Galois group $C_2\wr S_5$ (as 20T288)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 2*x^18 + 11*x^16 + 31*x^14 + 67*x^12 + 115*x^10 + 171*x^8 + 194*x^6 + 161*x^4 + 84*x^2 + 16)
 
gp: K = bnfinit(y^20 + 2*y^18 + 11*y^16 + 31*y^14 + 67*y^12 + 115*y^10 + 171*y^8 + 194*y^6 + 161*y^4 + 84*y^2 + 16, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + 2*x^18 + 11*x^16 + 31*x^14 + 67*x^12 + 115*x^10 + 171*x^8 + 194*x^6 + 161*x^4 + 84*x^2 + 16);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 + 2*x^18 + 11*x^16 + 31*x^14 + 67*x^12 + 115*x^10 + 171*x^8 + 194*x^6 + 161*x^4 + 84*x^2 + 16)
 

\( x^{20} + 2x^{18} + 11x^{16} + 31x^{14} + 67x^{12} + 115x^{10} + 171x^{8} + 194x^{6} + 161x^{4} + 84x^{2} + 16 \) 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:   \(2004493091475536548901569\) \(\medspace = 19^{10}\cdot 83^{6}\) 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:  \(16.41\)
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:  $19^{1/2}83^{3/4}\approx 119.86306251014346$
Ramified primes:   \(19\), \(83\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $4$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{206}a^{16}-\frac{7}{103}a^{14}-\frac{7}{206}a^{12}+\frac{29}{206}a^{10}-\frac{21}{103}a^{8}+\frac{26}{103}a^{6}-\frac{1}{2}a^{5}-\frac{38}{103}a^{4}-\frac{1}{2}a^{3}+\frac{53}{206}a^{2}+\frac{46}{103}$, $\frac{1}{206}a^{17}-\frac{7}{103}a^{15}-\frac{7}{206}a^{13}+\frac{29}{206}a^{11}-\frac{21}{103}a^{9}+\frac{26}{103}a^{7}-\frac{1}{2}a^{6}-\frac{38}{103}a^{5}-\frac{1}{2}a^{4}+\frac{53}{206}a^{3}+\frac{46}{103}a$, $\frac{1}{35432}a^{18}-\frac{1}{412}a^{17}-\frac{7}{17716}a^{16}+\frac{7}{206}a^{15}-\frac{625}{35432}a^{14}+\frac{7}{412}a^{13}+\frac{4355}{35432}a^{12}-\frac{29}{412}a^{11}+\frac{7271}{35432}a^{10}-\frac{61}{412}a^{9}-\frac{17149}{35432}a^{8}+\frac{51}{412}a^{7}+\frac{221}{824}a^{6}+\frac{179}{412}a^{5}-\frac{849}{17716}a^{4}-\frac{39}{103}a^{3}+\frac{4109}{35432}a^{2}+\frac{11}{412}a+\frac{31}{86}$, $\frac{1}{70864}a^{19}+\frac{79}{35432}a^{17}+\frac{14683}{70864}a^{15}-\frac{14565}{70864}a^{13}-\frac{5457}{70864}a^{11}-\frac{6657}{70864}a^{9}-\frac{395}{1648}a^{7}-\frac{1}{2}a^{6}+\frac{10331}{35432}a^{5}-\frac{1}{2}a^{4}+\frac{13225}{70864}a^{3}-\frac{1}{2}a^{2}-\frac{3069}{8858}a-\frac{1}{2}$ 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:  $2$

Class group and class number

Trivial group, which has order $1$

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:   $\frac{7739}{70864}a^{19}+\frac{131}{8858}a^{18}+\frac{5081}{35432}a^{17}+\frac{72}{4429}a^{16}+\frac{76649}{70864}a^{15}+\frac{579}{4429}a^{14}+\frac{1823}{688}a^{13}+\frac{1517}{4429}a^{12}+\frac{376117}{70864}a^{11}+\frac{2240}{4429}a^{10}+\frac{615365}{70864}a^{9}+\frac{8917}{8858}a^{8}+\frac{20231}{1648}a^{7}+\frac{237}{206}a^{6}+\frac{423969}{35432}a^{5}+\frac{4064}{4429}a^{4}+\frac{593419}{70864}a^{3}+\frac{5337}{8858}a^{2}+\frac{12951}{4429}a+\frac{8215}{8858}$, $\frac{7355}{70864}a^{19}-\frac{695}{35432}a^{18}+\frac{6307}{35432}a^{17}-\frac{295}{17716}a^{16}+\frac{74129}{70864}a^{15}-\frac{5773}{35432}a^{14}+\frac{201221}{70864}a^{13}-\frac{13629}{35432}a^{12}+\frac{404681}{70864}a^{11}-\frac{20109}{35432}a^{10}+\frac{644457}{70864}a^{9}-\frac{31489}{35432}a^{8}+\frac{21235}{1648}a^{7}-\frac{863}{824}a^{6}+\frac{462487}{35432}a^{5}-\frac{9881}{17716}a^{4}+\frac{614475}{70864}a^{3}-\frac{1243}{35432}a^{2}+\frac{47279}{17716}a+\frac{800}{4429}$, $\frac{11105}{70864}a^{19}+\frac{7491}{35432}a^{17}+\frac{115675}{70864}a^{15}+\frac{274859}{70864}a^{13}+\frac{597615}{70864}a^{11}+\frac{980015}{70864}a^{9}+\frac{33421}{1648}a^{7}+\frac{747719}{35432}a^{5}+\frac{1197513}{70864}a^{3}+\frac{28894}{4429}a+\frac{1}{2}$, $\frac{1807}{17716}a^{19}+\frac{4155}{35432}a^{18}+\frac{2179}{17716}a^{17}+\frac{2563}{17716}a^{16}+\frac{18381}{17716}a^{15}+\frac{42465}{35432}a^{14}+\frac{20775}{8858}a^{13}+\frac{95397}{35432}a^{12}+\frac{22619}{4429}a^{11}+\frac{210937}{35432}a^{10}+\frac{35176}{4429}a^{9}+\frac{317513}{35432}a^{8}+\frac{1204}{103}a^{7}+\frac{10943}{824}a^{6}+\frac{206981}{17716}a^{5}+\frac{223467}{17716}a^{4}+\frac{151685}{17716}a^{3}+\frac{319923}{35432}a^{2}+\frac{55987}{17716}a+\frac{18443}{8858}$, $\frac{6725}{70864}a^{19}+\frac{3579}{35432}a^{17}+\frac{65399}{70864}a^{15}+\frac{144039}{70864}a^{13}+\frac{299563}{70864}a^{11}+\frac{443939}{70864}a^{9}+\frac{15121}{1648}a^{7}+\frac{290867}{35432}a^{5}+\frac{369477}{70864}a^{3}-\frac{1}{2}a^{2}+\frac{5927}{4429}a-\frac{1}{2}$, $\frac{1893}{70864}a^{19}+\frac{507}{35432}a^{18}+\frac{1369}{35432}a^{17}+\frac{751}{17716}a^{16}+\frac{19671}{70864}a^{15}+\frac{5625}{35432}a^{14}+\frac{49419}{70864}a^{13}+\frac{21865}{35432}a^{12}+\frac{102559}{70864}a^{11}+\frac{38277}{35432}a^{10}+\frac{181855}{70864}a^{9}+\frac{85713}{35432}a^{8}+\frac{5869}{1648}a^{7}+\frac{2555}{824}a^{6}+\frac{142857}{35432}a^{5}+\frac{66551}{17716}a^{4}+\frac{239749}{70864}a^{3}+\frac{111971}{35432}a^{2}+\frac{37937}{17716}a+\frac{7024}{4429}$, $\frac{213}{4429}a^{19}+\frac{2469}{17716}a^{18}+\frac{357}{8858}a^{17}+\frac{1981}{8858}a^{16}+\frac{4005}{8858}a^{15}+\frac{25687}{17716}a^{14}+\frac{4190}{4429}a^{13}+\frac{65791}{17716}a^{12}+\frac{8291}{4429}a^{11}+\frac{139865}{17716}a^{10}+\frac{24779}{8858}a^{9}+\frac{224647}{17716}a^{8}+\frac{425}{103}a^{7}+\frac{7613}{412}a^{6}+\frac{13759}{4429}a^{5}+\frac{168973}{8858}a^{4}+\frac{8552}{4429}a^{3}+\frac{255379}{17716}a^{2}+\frac{3393}{8858}a+\frac{22386}{4429}$, $\frac{6349}{70864}a^{19}-\frac{2069}{17716}a^{18}+\frac{4491}{35432}a^{17}-\frac{1513}{8858}a^{16}+\frac{65103}{70864}a^{15}-\frac{21729}{17716}a^{14}+\frac{160511}{70864}a^{13}-\frac{52563}{17716}a^{12}+\frac{335899}{70864}a^{11}-\frac{115647}{17716}a^{10}+\frac{552387}{70864}a^{9}-\frac{183779}{17716}a^{8}+\frac{18505}{1648}a^{7}-\frac{6275}{412}a^{6}+\frac{404207}{35432}a^{5}-\frac{136881}{8858}a^{4}+\frac{590933}{70864}a^{3}-\frac{214131}{17716}a^{2}+\frac{31931}{8858}a-\frac{37563}{8858}$, $\frac{9559}{35432}a^{19}-\frac{505}{17716}a^{18}+\frac{7907}{17716}a^{17}-\frac{1109}{8858}a^{16}+\frac{97741}{35432}a^{15}-\frac{6101}{17716}a^{14}+\frac{260349}{35432}a^{13}-\frac{26055}{17716}a^{12}+\frac{534241}{35432}a^{11}-\frac{52545}{17716}a^{10}+\frac{871241}{35432}a^{9}-\frac{91115}{17716}a^{8}+\frac{29559}{824}a^{7}-\frac{2945}{412}a^{6}+\frac{654345}{17716}a^{5}-\frac{77537}{8858}a^{4}+\frac{952375}{35432}a^{3}-\frac{104785}{17716}a^{2}+\frac{47522}{4429}a-\frac{5770}{4429}$ Copy content Toggle raw display
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:  \( 14407.5933023 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 14407.5933023 \cdot 1}{2\cdot\sqrt{2004493091475536548901569}}\cr\approx \mathstrut & 0.487930365750 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 + 2*x^18 + 11*x^16 + 31*x^14 + 67*x^12 + 115*x^10 + 171*x^8 + 194*x^6 + 161*x^4 + 84*x^2 + 16)
 
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 + 2*x^18 + 11*x^16 + 31*x^14 + 67*x^12 + 115*x^10 + 171*x^8 + 194*x^6 + 161*x^4 + 84*x^2 + 16, 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 + 2*x^18 + 11*x^16 + 31*x^14 + 67*x^12 + 115*x^10 + 171*x^8 + 194*x^6 + 161*x^4 + 84*x^2 + 16);
 
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 + 2*x^18 + 11*x^16 + 31*x^14 + 67*x^12 + 115*x^10 + 171*x^8 + 194*x^6 + 161*x^4 + 84*x^2 + 16);
 
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

$C_2\wr S_5$ (as 20T288):

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 non-solvable group of order 3840
The 36 conjugacy class representatives for $C_2\wr S_5$
Character table for $C_2\wr S_5$ is not computed

Intermediate fields

\(\Q(\sqrt{-19}) \), 5.3.29963.1, 10.2.1415801218913.1, 10.0.17057846011.1, 10.4.74515853627.2

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]
 

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 32 siblings: data not computed
Degree 40 siblings: data not computed
Minimal sibling: 10.2.1415801218913.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.6.0.1}{6} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }^{4}$ ${\href{/padicField/3.10.0.1}{10} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ ${\href{/padicField/7.5.0.1}{5} }^{4}$ ${\href{/padicField/11.5.0.1}{5} }^{4}$ ${\href{/padicField/13.8.0.1}{8} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{6}$ R ${\href{/padicField/23.5.0.1}{5} }^{4}$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ ${\href{/padicField/31.10.0.1}{10} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{10}$ ${\href{/padicField/41.10.0.1}{10} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ ${\href{/padicField/47.8.0.1}{8} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.10.0.1}{10} }^{2}$

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
\(19\) Copy content Toggle raw display 19.4.2.1$x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.8.4.1$x^{8} + 80 x^{6} + 22 x^{5} + 2250 x^{4} - 792 x^{3} + 25817 x^{2} - 22946 x + 107924$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
19.8.4.1$x^{8} + 80 x^{6} + 22 x^{5} + 2250 x^{4} - 792 x^{3} + 25817 x^{2} - 22946 x + 107924$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(83\) Copy content Toggle raw display 83.2.0.1$x^{2} + 82 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} + 82 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.4.3.2$x^{4} + 166$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
83.4.0.1$x^{4} + 4 x^{2} + 42 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
83.4.0.1$x^{4} + 4 x^{2} + 42 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
83.4.3.2$x^{4} + 166$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$