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Properties

Label 20.0.697...000.1
Degree $20$
Signature $[0, 10]$
Discriminant $6.976\times 10^{30}$
Root discriminant \(34.85\)
Ramified primes $2,5,67$
Class number $4$ (GRH)
Class group [4] (GRH)
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 + 11*x^18 + 11*x^16 - 50*x^14 - 23*x^12 + 101*x^10 - 23*x^8 - 50*x^6 + 11*x^4 + 11*x^2 + 1)
 
gp: K = bnfinit(y^20 + 11*y^18 + 11*y^16 - 50*y^14 - 23*y^12 + 101*y^10 - 23*y^8 - 50*y^6 + 11*y^4 + 11*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + 11*x^18 + 11*x^16 - 50*x^14 - 23*x^12 + 101*x^10 - 23*x^8 - 50*x^6 + 11*x^4 + 11*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 + 11*x^18 + 11*x^16 - 50*x^14 - 23*x^12 + 101*x^10 - 23*x^8 - 50*x^6 + 11*x^4 + 11*x^2 + 1)
 

\( x^{20} + 11x^{18} + 11x^{16} - 50x^{14} - 23x^{12} + 101x^{10} - 23x^{8} - 50x^{6} + 11x^{4} + 11x^{2} + 1 \) 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:   \(6976189580273785130450944000000\) \(\medspace = 2^{40}\cdot 5^{6}\cdot 67^{8}\) 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:  \(34.85\)
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:  not computed
Ramified primes:   \(2\), \(5\), \(67\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{8}a^{12}+\frac{1}{8}a^{10}-\frac{1}{8}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}+\frac{1}{8}a^{2}-\frac{1}{2}a-\frac{3}{8}$, $\frac{1}{8}a^{13}+\frac{1}{8}a^{11}-\frac{1}{8}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}+\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{3}{8}a$, $\frac{1}{8}a^{14}-\frac{1}{8}a^{10}-\frac{1}{8}a^{8}-\frac{1}{2}a^{7}+\frac{1}{8}a^{6}+\frac{1}{8}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{3}{8}$, $\frac{1}{8}a^{15}-\frac{1}{8}a^{11}-\frac{1}{8}a^{9}+\frac{1}{8}a^{7}+\frac{1}{8}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{3}{8}a-\frac{1}{2}$, $\frac{1}{8}a^{16}+\frac{1}{8}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{3}{8}$, $\frac{1}{8}a^{17}+\frac{1}{8}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{3}{8}a$, $\frac{1}{48}a^{18}-\frac{1}{16}a^{17}+\frac{1}{48}a^{16}-\frac{1}{24}a^{14}-\frac{1}{4}a^{11}+\frac{7}{48}a^{10}+\frac{3}{16}a^{9}-\frac{5}{48}a^{8}+\frac{1}{4}a^{7}+\frac{1}{8}a^{6}+\frac{11}{24}a^{4}+\frac{13}{48}a^{2}+\frac{7}{16}a+\frac{7}{48}$, $\frac{1}{48}a^{19}-\frac{1}{24}a^{17}-\frac{1}{16}a^{16}-\frac{1}{24}a^{15}-\frac{5}{48}a^{11}-\frac{1}{4}a^{10}+\frac{1}{12}a^{9}+\frac{3}{16}a^{8}-\frac{1}{8}a^{7}+\frac{1}{4}a^{6}+\frac{11}{24}a^{5}-\frac{1}{2}a^{4}+\frac{13}{48}a^{3}-\frac{1}{2}a^{2}+\frac{1}{12}a-\frac{1}{16}$ 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

$C_{4}$, which has order $4$ (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:   \( -\frac{3}{4} a^{19} - \frac{61}{8} a^{17} - \frac{3}{2} a^{15} + \frac{173}{4} a^{13} - \frac{25}{2} a^{11} - \frac{637}{8} a^{9} + \frac{295}{4} a^{7} + \frac{5}{2} a^{5} - \frac{37}{2} a^{3} + \frac{9}{8} a \)  (order $4$) 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:   $\frac{5}{8}a^{18}+\frac{27}{4}a^{16}+\frac{23}{4}a^{14}-\frac{119}{4}a^{12}-\frac{31}{8}a^{10}+\frac{113}{2}a^{8}-35a^{6}-\frac{41}{4}a^{4}+\frac{75}{8}a^{2}+\frac{3}{4}$, $\frac{7}{8}a^{19}-\frac{59}{48}a^{18}+\frac{157}{16}a^{17}-\frac{653}{48}a^{16}+\frac{49}{4}a^{15}-\frac{361}{24}a^{14}-\frac{281}{8}a^{13}+\frac{107}{2}a^{12}-\frac{75}{4}a^{11}+\frac{1003}{48}a^{10}+\frac{1045}{16}a^{9}-\frac{4931}{48}a^{8}-\frac{183}{8}a^{7}+\frac{341}{8}a^{6}-\frac{49}{4}a^{5}+\frac{659}{24}a^{4}+\frac{15}{4}a^{3}-\frac{575}{48}a^{2}-\frac{9}{16}a-\frac{119}{48}$, $\frac{17}{48}a^{19}-\frac{25}{24}a^{18}+\frac{91}{24}a^{17}-\frac{527}{48}a^{16}+\frac{67}{24}a^{15}-\frac{77}{12}a^{14}-\frac{147}{8}a^{13}+\frac{439}{8}a^{12}-\frac{103}{48}a^{11}-\frac{13}{6}a^{10}+\frac{449}{12}a^{9}-\frac{5063}{48}a^{8}-\frac{79}{4}a^{7}+\frac{593}{8}a^{6}-\frac{365}{24}a^{5}+\frac{259}{12}a^{4}+\frac{395}{48}a^{3}-\frac{133}{6}a^{2}+\frac{73}{24}a-\frac{161}{48}$, $\frac{67}{48}a^{19}+\frac{25}{48}a^{18}+\frac{715}{48}a^{17}+\frac{67}{12}a^{16}+\frac{31}{3}a^{15}+\frac{49}{12}a^{14}-\frac{299}{4}a^{13}-\frac{225}{8}a^{12}-\frac{509}{48}a^{11}-\frac{257}{48}a^{10}+\frac{7123}{48}a^{9}+\frac{697}{12}a^{8}-\frac{289}{4}a^{7}-\frac{191}{8}a^{6}-\frac{155}{3}a^{5}-\frac{305}{12}a^{4}+\frac{1243}{48}a^{3}+\frac{535}{48}a^{2}+\frac{451}{48}a+\frac{95}{24}$, $\frac{31}{24}a^{19}+\frac{307}{24}a^{17}-\frac{7}{12}a^{15}-\frac{287}{4}a^{13}+\frac{1123}{24}a^{11}+\frac{2929}{24}a^{9}-175a^{7}+\frac{557}{12}a^{5}+\frac{877}{24}a^{3}-\frac{437}{24}a$, $\frac{43}{24}a^{18}+\frac{463}{24}a^{16}+\frac{91}{6}a^{14}-\frac{375}{4}a^{12}-\frac{491}{24}a^{10}+\frac{4495}{24}a^{8}-\frac{325}{4}a^{6}-\frac{217}{3}a^{4}+\frac{769}{24}a^{2}+\frac{337}{24}$, $\frac{103}{24}a^{19}+\frac{1093}{24}a^{17}+\frac{359}{12}a^{15}-\frac{443}{2}a^{13}-\frac{95}{24}a^{11}+\frac{10183}{24}a^{9}-\frac{1117}{4}a^{7}-\frac{1009}{12}a^{5}+\frac{1999}{24}a^{3}+\frac{199}{24}a$, $\frac{139}{12}a^{19}+\frac{367}{3}a^{17}+\frac{443}{6}a^{15}-\frac{2441}{4}a^{13}+\frac{29}{6}a^{11}+\frac{3502}{3}a^{9}-\frac{3145}{4}a^{7}-\frac{710}{3}a^{5}+\frac{703}{3}a^{3}+\frac{325}{12}a$, $\frac{1}{12}a^{18}+\frac{23}{24}a^{16}+\frac{19}{12}a^{14}-\frac{5}{4}a^{12}+\frac{1}{12}a^{10}-\frac{31}{24}a^{8}-9a^{6}+\frac{109}{12}a^{4}+\frac{11}{6}a^{2}-\frac{49}{24}$ 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:  \( 17447203.7272 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 17447203.7272 \cdot 4}{4\cdot\sqrt{6976189580273785130450944000000}}\cr\approx \mathstrut & 0.633454431552 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 + 11*x^18 + 11*x^16 - 50*x^14 - 23*x^12 + 101*x^10 - 23*x^8 - 50*x^6 + 11*x^4 + 11*x^2 + 1)
 
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 + 11*x^18 + 11*x^16 - 50*x^14 - 23*x^12 + 101*x^10 - 23*x^8 - 50*x^6 + 11*x^4 + 11*x^2 + 1, 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 + 11*x^18 + 11*x^16 - 50*x^14 - 23*x^12 + 101*x^10 - 23*x^8 - 50*x^6 + 11*x^4 + 11*x^2 + 1);
 
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 + 11*x^18 + 11*x^16 - 50*x^14 - 23*x^12 + 101*x^10 - 23*x^8 - 50*x^6 + 11*x^4 + 11*x^2 + 1);
 
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 $C_2\wr S_5$

Intermediate fields

\(\Q(\sqrt{-1}) \), 5.5.5745920.1, 10.4.2641247731712000.2, 10.0.528249546342400.1, 10.6.165077983232000.2

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: 10.4.2641247731712000.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.8.0.1}{8} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ R ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ ${\href{/padicField/19.10.0.1}{10} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.5.0.1}{5} }^{4}$ ${\href{/padicField/31.10.0.1}{10} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ ${\href{/padicField/59.10.0.1}{10} }^{2}$

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
\(2\) Copy content Toggle raw display 2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
Deg $16$$8$$2$$36$
\(5\) Copy content Toggle raw display 5.3.0.1$x^{3} + 3 x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
5.3.0.1$x^{3} + 3 x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
5.3.0.1$x^{3} + 3 x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
5.3.0.1$x^{3} + 3 x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
5.4.3.2$x^{4} + 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} + 5$$4$$1$$3$$C_4$$[\ ]_{4}$
\(67\) Copy content Toggle raw display 67.2.0.1$x^{2} + 63 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} + 63 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} + 63 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} + 63 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
67.6.4.1$x^{6} + 189 x^{5} + 11913 x^{4} + 250937 x^{3} + 36489 x^{2} + 797721 x + 16732320$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
67.6.4.1$x^{6} + 189 x^{5} + 11913 x^{4} + 250937 x^{3} + 36489 x^{2} + 797721 x + 16732320$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$