Properties

Label 20.0.233...125.2
Degree $20$
Signature $[0, 10]$
Discriminant $2.338\times 10^{31}$
Root discriminant \(37.02\)
Ramified primes $5,11$
Class number $5$ (GRH)
Class group [5] (GRH)
Galois group $C_5:F_5$ (as 20T26)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 54*x^15 + 781*x^10 - 2904*x^5 + 161051)
 
gp: K = bnfinit(y^20 - 54*y^15 + 781*y^10 - 2904*y^5 + 161051, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 54*x^15 + 781*x^10 - 2904*x^5 + 161051);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 54*x^15 + 781*x^10 - 2904*x^5 + 161051)
 

\( x^{20} - 54x^{15} + 781x^{10} - 2904x^{5} + 161051 \) 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:   \(23383113568432629108428955078125\) \(\medspace = 5^{27}\cdot 11^{12}\) 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:  \(37.02\)
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))
 
Ramified primes:   \(5\), \(11\) 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:  \(\Q(\sqrt{5}) \)
$\card{ \Aut(K/\Q) }$:  $5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5}a^{7}-\frac{2}{5}a^{6}+\frac{1}{5}a^{5}-\frac{1}{5}a^{2}+\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{8}+\frac{2}{5}a^{6}+\frac{2}{5}a^{5}-\frac{1}{5}a^{3}-\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{9}+\frac{1}{5}a^{6}-\frac{2}{5}a^{5}-\frac{1}{5}a^{4}-\frac{1}{5}a+\frac{2}{5}$, $\frac{1}{110}a^{10}+\frac{23}{110}a^{5}+\frac{1}{10}$, $\frac{1}{110}a^{11}+\frac{23}{110}a^{6}+\frac{1}{10}a$, $\frac{1}{1210}a^{12}-\frac{21}{1210}a^{7}-\frac{1}{5}a^{6}-\frac{2}{5}a^{5}-\frac{5}{22}a^{2}+\frac{1}{5}a+\frac{2}{5}$, $\frac{1}{1210}a^{13}-\frac{21}{1210}a^{8}+\frac{1}{5}a^{6}+\frac{1}{5}a^{5}-\frac{5}{22}a^{3}-\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{66550}a^{14}+\frac{1}{6050}a^{13}+\frac{1}{6050}a^{12}+\frac{1}{550}a^{11}+\frac{1}{550}a^{10}-\frac{5587}{66550}a^{9}+\frac{463}{6050}a^{8}+\frac{463}{6050}a^{7}-\frac{87}{550}a^{6}-\frac{87}{550}a^{5}+\frac{1801}{6050}a^{4}+\frac{151}{550}a^{3}+\frac{151}{550}a^{2}+\frac{1}{50}a+\frac{1}{50}$, $\frac{1}{732050}a^{15}-\frac{263}{732050}a^{10}-\frac{28207}{66550}a^{5}+\frac{2}{25}$, $\frac{1}{732050}a^{16}-\frac{263}{732050}a^{11}-\frac{28207}{66550}a^{6}+\frac{2}{25}a$, $\frac{1}{732050}a^{17}-\frac{263}{732050}a^{12}-\frac{1587}{66550}a^{7}+\frac{1}{5}a^{6}+\frac{2}{5}a^{5}-\frac{8}{25}a^{2}-\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{732050}a^{18}-\frac{263}{732050}a^{13}-\frac{1587}{66550}a^{8}-\frac{1}{5}a^{6}-\frac{1}{5}a^{5}-\frac{8}{25}a^{3}+\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{732050}a^{19}+\frac{1}{732050}a^{14}-\frac{1}{6050}a^{13}-\frac{1}{6050}a^{12}-\frac{1}{550}a^{11}-\frac{1}{550}a^{10}-\frac{103}{2662}a^{9}-\frac{463}{6050}a^{8}-\frac{463}{6050}a^{7}-\frac{243}{550}a^{6}+\frac{197}{550}a^{5}-\frac{531}{3025}a^{4}-\frac{151}{550}a^{3}-\frac{151}{550}a^{2}-\frac{21}{50}a-\frac{11}{50}$ 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

$C_{5}$, which has order $5$ (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:   \( \frac{53}{732050} a^{15} - \frac{629}{732050} a^{10} - \frac{3041}{66550} a^{5} + \frac{11}{25} \)  (order $10$) 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:   $\frac{6}{73205}a^{15}-\frac{247}{73205}a^{10}-\frac{84}{6655}a^{5}$, $\frac{3}{73205}a^{19}-\frac{2}{366025}a^{18}-\frac{9}{146410}a^{17}+\frac{59}{366025}a^{16}-\frac{1}{29282}a^{15}-\frac{876}{366025}a^{14}+\frac{81}{73205}a^{13}+\frac{1913}{732050}a^{12}-\frac{3538}{366025}a^{11}+\frac{3913}{732050}a^{10}+\frac{1312}{33275}a^{9}-\frac{1919}{33275}a^{8}-\frac{151}{66550}a^{7}+\frac{4619}{33275}a^{6}-\frac{19131}{66550}a^{5}-\frac{276}{3025}a^{4}+\frac{201}{275}a^{3}-\frac{361}{275}a^{2}-\frac{1}{5}a+\frac{79}{25}$, $\frac{4}{366025}a^{19}+\frac{1}{33275}a^{18}+\frac{7}{146410}a^{17}+\frac{61}{732050}a^{16}+\frac{14}{366025}a^{15}-\frac{29}{366025}a^{14}-\frac{3}{2662}a^{13}-\frac{307}{366025}a^{12}-\frac{701}{366025}a^{11}+\frac{311}{366025}a^{10}-\frac{194}{33275}a^{9}+\frac{79}{6050}a^{8}-\frac{36}{33275}a^{7}+\frac{178}{33275}a^{6}-\frac{559}{33275}a^{5}+\frac{29}{3025}a^{4}-\frac{101}{550}a^{3}-\frac{59}{550}a^{2}-\frac{1}{2}a-\frac{11}{25}$, $\frac{1}{146410}a^{19}-\frac{2}{366025}a^{18}-\frac{1}{73205}a^{17}+\frac{61}{732050}a^{16}-\frac{38}{366025}a^{15}-\frac{102}{366025}a^{14}+\frac{284}{366025}a^{13}-\frac{137}{366025}a^{12}-\frac{701}{366025}a^{11}+\frac{677}{366025}a^{10}+\frac{54}{33275}a^{9}-\frac{357}{33275}a^{8}+\frac{59}{33275}a^{7}+\frac{178}{33275}a^{6}+\frac{179}{6655}a^{5}-\frac{809}{6050}a^{4}-\frac{1}{55}a^{3}+\frac{3}{275}a^{2}-\frac{1}{2}a+\frac{1}{25}$, $\frac{1}{366025}a^{19}-\frac{3}{732050}a^{18}-\frac{7}{146410}a^{17}-\frac{1}{366025}a^{16}+\frac{38}{366025}a^{15}+\frac{123}{732050}a^{14}+\frac{63}{732050}a^{13}+\frac{307}{366025}a^{12}-\frac{161}{146410}a^{11}-\frac{677}{366025}a^{10}-\frac{57}{66550}a^{9}+\frac{823}{66550}a^{8}+\frac{36}{33275}a^{7}+\frac{391}{66550}a^{6}-\frac{179}{6655}a^{5}-\frac{463}{6050}a^{4}-\frac{24}{275}a^{3}+\frac{59}{550}a^{2}-\frac{9}{50}a-\frac{1}{25}$, $\frac{7}{366025}a^{19}-\frac{1}{33275}a^{18}+\frac{1}{73205}a^{17}-\frac{1}{366025}a^{16}-\frac{47}{366025}a^{15}-\frac{503}{732050}a^{14}+\frac{3}{2662}a^{13}+\frac{137}{366025}a^{12}-\frac{161}{146410}a^{11}+\frac{1713}{366025}a^{10}+\frac{269}{66550}a^{9}-\frac{79}{6050}a^{8}-\frac{59}{33275}a^{7}+\frac{391}{66550}a^{6}-\frac{183}{6655}a^{5}-\frac{59}{1210}a^{4}+\frac{101}{550}a^{3}-\frac{3}{275}a^{2}-\frac{9}{50}a+\frac{14}{25}$, $\frac{4}{73205}a^{19}-\frac{3}{73205}a^{18}-\frac{13}{146410}a^{17}+\frac{151}{732050}a^{16}-\frac{13}{73205}a^{15}-\frac{1146}{366025}a^{14}+\frac{2203}{732050}a^{13}+\frac{2333}{732050}a^{12}-\frac{182}{14641}a^{11}+\frac{11563}{732050}a^{10}+\frac{1667}{33275}a^{9}-\frac{5421}{66550}a^{8}+\frac{899}{66550}a^{7}+\frac{5356}{33275}a^{6}-\frac{27341}{66550}a^{5}+\frac{19}{275}a^{4}+\frac{493}{550}a^{3}-\frac{411}{275}a^{2}+\frac{37}{50}a+\frac{123}{50}$, $\frac{3}{73205}a^{19}-\frac{16}{366025}a^{18}-\frac{89}{732050}a^{17}+\frac{37}{366025}a^{16}+\frac{271}{732050}a^{15}-\frac{876}{366025}a^{14}+\frac{457}{366025}a^{13}+\frac{1003}{146410}a^{12}-\frac{2159}{732050}a^{11}-\frac{1404}{73205}a^{10}+\frac{1312}{33275}a^{9}+\frac{609}{33275}a^{8}-\frac{7323}{66550}a^{7}-\frac{1399}{66550}a^{6}+\frac{11171}{33275}a^{5}-\frac{276}{3025}a^{4}-\frac{43}{55}a^{3}+\frac{96}{275}a^{2}+\frac{89}{50}a-\frac{123}{50}$, $\frac{23}{732050}a^{19}+\frac{16}{366025}a^{18}-\frac{9}{146410}a^{17}-\frac{59}{366025}a^{16}-\frac{119}{732050}a^{15}-\frac{1407}{732050}a^{14}-\frac{457}{366025}a^{13}+\frac{1501}{366025}a^{12}+\frac{3538}{366025}a^{11}+\frac{7339}{732050}a^{10}+\frac{317}{13310}a^{9}-\frac{609}{33275}a^{8}-\frac{3777}{33275}a^{7}-\frac{4619}{33275}a^{6}-\frac{7651}{66550}a^{5}+\frac{797}{3025}a^{4}+\frac{43}{55}a^{3}+\frac{527}{550}a^{2}+\frac{1}{5}a-\frac{87}{25}$ 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:  \( 69020367.6146 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 69020367.6146 \cdot 5}{10\cdot\sqrt{23383113568432629108428955078125}}\cr\approx \mathstrut & 0.684376024294 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 - 54*x^15 + 781*x^10 - 2904*x^5 + 161051)
 
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 - 54*x^15 + 781*x^10 - 2904*x^5 + 161051, 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 - 54*x^15 + 781*x^10 - 2904*x^5 + 161051);
 
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 - 54*x^15 + 781*x^10 - 2904*x^5 + 161051);
 
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

$C_5:F_5$ (as 20T26):

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 100
The 13 conjugacy class representatives for $C_5:F_5$
Character table for $C_5:F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\)

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 25 sibling: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{5}$ ${\href{/padicField/3.4.0.1}{4} }^{5}$ R ${\href{/padicField/7.4.0.1}{4} }^{5}$ R ${\href{/padicField/13.4.0.1}{4} }^{5}$ ${\href{/padicField/17.4.0.1}{4} }^{5}$ ${\href{/padicField/19.10.0.1}{10} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{5}$ ${\href{/padicField/29.10.0.1}{10} }^{2}$ ${\href{/padicField/31.5.0.1}{5} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{5}$ ${\href{/padicField/37.4.0.1}{4} }^{5}$ ${\href{/padicField/41.5.0.1}{5} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{5}$ ${\href{/padicField/43.4.0.1}{4} }^{5}$ ${\href{/padicField/47.4.0.1}{4} }^{5}$ ${\href{/padicField/53.4.0.1}{4} }^{5}$ ${\href{/padicField/59.10.0.1}{10} }^{2}$

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
\(5\) Copy content Toggle raw display Deg $20$$20$$1$$27$
\(11\) Copy content Toggle raw display $\Q_{11}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 9$$1$$1$$0$Trivial$[\ ]$
11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$