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Properties

Label 20.4.781...000.1
Degree $20$
Signature $[4, 8]$
Discriminant $7.812\times 10^{24}$
Root discriminant \(17.56\)
Ramified primes $2,5$
Class number $1$
Class group trivial
Galois group $C_4\times D_5$ (as 20T6)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 22*x^15 - 6*x^10 + 22*x^5 + 1)
 
gp: K = bnfinit(y^20 - 22*y^15 - 6*y^10 + 22*y^5 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 22*x^15 - 6*x^10 + 22*x^5 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 22*x^15 - 6*x^10 + 22*x^5 + 1)
 

\( x^{20} - 22x^{15} - 6x^{10} + 22x^{5} + 1 \) 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:  $[4, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(7812500000000000000000000\) \(\medspace = 2^{20}\cdot 5^{27}\) 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:  \(17.56\)
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:  $2\cdot 5^{27/20}\approx 17.564650049460273$
Ramified primes:   \(2\), \(5\) 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:  \(\Q(\sqrt{5}) \)
$\card{ \Aut(K/\Q) }$:  $4$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5}a^{7}-\frac{1}{5}a^{6}-\frac{1}{5}a^{5}+\frac{2}{5}a^{2}-\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{8}-\frac{2}{5}a^{6}-\frac{1}{5}a^{5}+\frac{2}{5}a^{3}+\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{9}+\frac{2}{5}a^{6}-\frac{2}{5}a^{5}+\frac{2}{5}a^{4}-\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{10}a^{10}+\frac{2}{5}a^{5}-\frac{1}{10}$, $\frac{1}{10}a^{11}+\frac{2}{5}a^{6}-\frac{1}{10}a$, $\frac{1}{10}a^{12}+\frac{2}{5}a^{6}+\frac{2}{5}a^{5}+\frac{1}{10}a^{2}-\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{10}a^{13}-\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{1}{10}a^{3}-\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{50}a^{14}-\frac{1}{25}a^{13}-\frac{1}{50}a^{12}+\frac{1}{25}a^{11}+\frac{1}{50}a^{10}+\frac{2}{25}a^{9}+\frac{1}{25}a^{8}-\frac{2}{25}a^{7}-\frac{6}{25}a^{6}-\frac{3}{25}a^{5}-\frac{21}{50}a^{4}+\frac{6}{25}a^{3}+\frac{21}{50}a^{2}+\frac{9}{25}a+\frac{9}{50}$, $\frac{1}{50}a^{15}+\frac{1}{50}a^{10}+\frac{17}{50}a^{5}+\frac{13}{50}$, $\frac{1}{50}a^{16}+\frac{1}{50}a^{11}+\frac{17}{50}a^{6}+\frac{13}{50}a$, $\frac{1}{50}a^{17}+\frac{1}{50}a^{12}-\frac{3}{50}a^{7}+\frac{2}{5}a^{6}+\frac{2}{5}a^{5}+\frac{23}{50}a^{2}-\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{50}a^{18}+\frac{1}{50}a^{13}-\frac{3}{50}a^{8}-\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{23}{50}a^{3}-\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{50}a^{19}+\frac{1}{25}a^{13}+\frac{1}{50}a^{12}-\frac{1}{25}a^{11}-\frac{1}{50}a^{10}+\frac{3}{50}a^{9}-\frac{1}{25}a^{8}+\frac{2}{25}a^{7}-\frac{4}{25}a^{6}-\frac{12}{25}a^{5}+\frac{7}{25}a^{4}-\frac{6}{25}a^{3}-\frac{21}{50}a^{2}-\frac{4}{25}a-\frac{19}{50}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

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:  $11$
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:   $a$, $\frac{9}{50}a^{17}-\frac{201}{50}a^{12}+\frac{13}{50}a^{7}+\frac{177}{50}a^{2}$, $\frac{1}{10}a^{15}-\frac{11}{5}a^{10}-\frac{1}{2}a^{5}+\frac{3}{5}$, $\frac{4}{25}a^{17}+\frac{1}{50}a^{16}+\frac{1}{50}a^{15}-\frac{177}{50}a^{12}-\frac{12}{25}a^{11}-\frac{12}{25}a^{10}-\frac{12}{25}a^{7}+\frac{37}{50}a^{6}+\frac{37}{50}a^{5}+\frac{149}{50}a^{2}+\frac{14}{25}a+\frac{14}{25}$, $\frac{1}{10}a^{19}-\frac{3}{25}a^{18}-\frac{1}{25}a^{17}-\frac{1}{25}a^{16}+\frac{1}{25}a^{15}-\frac{111}{50}a^{14}+\frac{131}{50}a^{13}+\frac{22}{25}a^{12}+\frac{23}{25}a^{11}-\frac{22}{25}a^{10}-\frac{9}{50}a^{9}+\frac{28}{25}a^{8}+\frac{1}{5}a^{7}-\frac{16}{25}a^{6}-\frac{1}{5}a^{5}+\frac{141}{50}a^{4}-\frac{17}{10}a^{3}-\frac{11}{25}a^{2}-\frac{32}{25}a+\frac{11}{25}$, $\frac{14}{25}a^{19}-\frac{4}{25}a^{17}+\frac{1}{25}a^{15}-\frac{617}{50}a^{14}+\frac{177}{50}a^{12}-\frac{43}{50}a^{10}-\frac{72}{25}a^{9}+\frac{12}{25}a^{7}-\frac{18}{25}a^{5}+\frac{579}{50}a^{4}-\frac{149}{50}a^{2}+\frac{41}{50}$, $\frac{11}{25}a^{19}-\frac{6}{25}a^{18}-\frac{3}{50}a^{17}+\frac{3}{50}a^{16}+\frac{1}{50}a^{15}-\frac{243}{25}a^{14}+\frac{132}{25}a^{13}+\frac{13}{10}a^{12}-\frac{63}{50}a^{11}-\frac{11}{25}a^{10}-\frac{44}{25}a^{9}+\frac{7}{5}a^{8}+\frac{41}{50}a^{7}-\frac{83}{50}a^{6}-\frac{1}{10}a^{5}+\frac{247}{25}a^{4}-\frac{106}{25}a^{3}-\frac{77}{50}a^{2}+\frac{7}{10}a-\frac{7}{25}$, $\frac{11}{50}a^{19}+\frac{2}{25}a^{18}+\frac{3}{25}a^{17}-\frac{1}{10}a^{16}-\frac{3}{50}a^{15}-\frac{121}{25}a^{14}-\frac{17}{10}a^{13}-\frac{131}{50}a^{12}+\frac{109}{50}a^{11}+\frac{32}{25}a^{10}-\frac{13}{10}a^{9}-\frac{44}{25}a^{8}-\frac{28}{25}a^{7}+\frac{51}{50}a^{6}+\frac{57}{50}a^{5}+\frac{113}{25}a^{4}+\frac{21}{50}a^{3}+\frac{17}{10}a^{2}-\frac{49}{50}a-\frac{8}{25}$, $\frac{11}{25}a^{19}-\frac{1}{10}a^{18}+\frac{7}{50}a^{17}-\frac{2}{25}a^{16}+\frac{1}{50}a^{15}-\frac{243}{25}a^{14}+\frac{111}{50}a^{13}-\frac{31}{10}a^{12}+\frac{9}{5}a^{11}-\frac{11}{25}a^{10}-\frac{44}{25}a^{9}+\frac{9}{50}a^{8}-\frac{19}{50}a^{7}-\frac{11}{25}a^{6}-\frac{1}{10}a^{5}+\frac{247}{25}a^{4}-\frac{141}{50}a^{3}+\frac{163}{50}a^{2}-\frac{43}{25}a+\frac{18}{25}$, $\frac{9}{50}a^{19}-\frac{1}{10}a^{18}+\frac{4}{25}a^{17}+\frac{1}{25}a^{16}+\frac{1}{50}a^{15}-\frac{199}{50}a^{14}+\frac{111}{50}a^{13}-\frac{87}{25}a^{12}-\frac{22}{25}a^{11}-\frac{11}{25}a^{10}-\frac{29}{50}a^{9}+\frac{9}{50}a^{8}-\frac{46}{25}a^{7}-\frac{1}{5}a^{6}-\frac{1}{10}a^{5}+\frac{27}{10}a^{4}-\frac{141}{50}a^{3}+\frac{88}{25}a^{2}+\frac{11}{25}a-\frac{7}{25}$, $\frac{11}{25}a^{19}-\frac{1}{10}a^{18}-\frac{1}{25}a^{17}+\frac{1}{10}a^{16}+\frac{1}{50}a^{15}-\frac{243}{25}a^{14}+\frac{111}{50}a^{13}+\frac{23}{25}a^{12}-\frac{111}{50}a^{11}-\frac{11}{25}a^{10}-\frac{44}{25}a^{9}+\frac{9}{50}a^{8}-\frac{16}{25}a^{7}-\frac{9}{50}a^{6}-\frac{1}{10}a^{5}+\frac{247}{25}a^{4}-\frac{141}{50}a^{3}-\frac{32}{25}a^{2}+\frac{141}{50}a+\frac{18}{25}$ 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:  \( 44054.0373872 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 44054.0373872 \cdot 1}{2\cdot\sqrt{7812500000000000000000000}}\cr\approx \mathstrut & 0.306280702620 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 - 22*x^15 - 6*x^10 + 22*x^5 + 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 - 22*x^15 - 6*x^10 + 22*x^5 + 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 - 22*x^15 - 6*x^10 + 22*x^5 + 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 - 22*x^15 - 6*x^10 + 22*x^5 + 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

$C_4\times D_5$ (as 20T6):

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 solvable group of order 40
The 16 conjugacy class representatives for $C_4\times D_5$
Character table for $C_4\times D_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 5.1.250000.1, 10.2.312500000000.1

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]
 

Sibling fields

Galois closure: deg 40
Degree 20 sibling: 20.0.488281250000000000000000.1
Minimal sibling: 20.0.488281250000000000000000.1

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 $20$ ${\href{/padicField/11.2.0.1}{2} }^{10}$ ${\href{/padicField/13.4.0.1}{4} }^{5}$ ${\href{/padicField/17.4.0.1}{4} }^{5}$ ${\href{/padicField/19.2.0.1}{2} }^{8}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ $20$ ${\href{/padicField/29.10.0.1}{10} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{10}$ ${\href{/padicField/37.4.0.1}{4} }^{5}$ ${\href{/padicField/41.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/padicField/53.4.0.1}{4} }^{5}$ ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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

# 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 2.4.4.2$x^{4} + 4 x^{3} + 4 x^{2} + 12$$2$$2$$4$$C_4$$[2]^{2}$
2.8.8.1$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
\(5\) Copy content Toggle raw display Deg $20$$20$$1$$27$