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Properties

Label 20.4.248...056.1
Degree $20$
Signature $[4, 8]$
Discriminant $2.489\times 10^{22}$
Root discriminant \(13.18\)
Ramified primes $2,11,23$
Class number $1$
Class group trivial
Galois group $C_2^5.C_2^8:C_{10}$ (as 20T751)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 5*x^18 + 8*x^16 + 2*x^14 - x^12 - 11*x^10 - 37*x^8 - 34*x^6 - 5*x^4 + 4*x^2 + 1)
 
gp: K = bnfinit(y^20 + 5*y^18 + 8*y^16 + 2*y^14 - y^12 - 11*y^10 - 37*y^8 - 34*y^6 - 5*y^4 + 4*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + 5*x^18 + 8*x^16 + 2*x^14 - x^12 - 11*x^10 - 37*x^8 - 34*x^6 - 5*x^4 + 4*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 + 5*x^18 + 8*x^16 + 2*x^14 - x^12 - 11*x^10 - 37*x^8 - 34*x^6 - 5*x^4 + 4*x^2 + 1)
 

\( x^{20} + 5x^{18} + 8x^{16} + 2x^{14} - x^{12} - 11x^{10} - 37x^{8} - 34x^{6} - 5x^{4} + 4x^{2} + 1 \) 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:  $[4, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(24890784868177585325056\) \(\medspace = 2^{10}\cdot 11^{16}\cdot 23^{2}\) 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:  \(13.18\)
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:  $2^{15/8}11^{4/5}23^{1/2}\approx 119.78689508462926$
Ramified primes:   \(2\), \(11\), \(23\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{86722}a^{18}+\frac{8467}{86722}a^{16}+\frac{7695}{43361}a^{14}-\frac{1}{2}a^{13}+\frac{17099}{86722}a^{12}-\frac{1}{2}a^{11}-\frac{1960}{43361}a^{10}-\frac{1}{2}a^{9}-\frac{43247}{86722}a^{8}-\frac{1}{2}a^{7}-\frac{16336}{43361}a^{6}+\frac{42599}{86722}a^{4}-\frac{1}{2}a^{3}+\frac{6370}{43361}a^{2}-\frac{1}{2}a-\frac{32923}{86722}$, $\frac{1}{86722}a^{19}+\frac{8467}{86722}a^{17}+\frac{7695}{43361}a^{15}-\frac{1}{2}a^{14}+\frac{17099}{86722}a^{13}-\frac{1}{2}a^{12}-\frac{1960}{43361}a^{11}-\frac{1}{2}a^{10}-\frac{43247}{86722}a^{9}-\frac{1}{2}a^{8}-\frac{16336}{43361}a^{7}+\frac{42599}{86722}a^{5}-\frac{1}{2}a^{4}+\frac{6370}{43361}a^{3}-\frac{1}{2}a^{2}-\frac{32923}{86722}a$ 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:   $a$, $\frac{45808}{43361}a^{18}+\frac{208996}{43361}a^{16}+\frac{282148}{43361}a^{14}-\frac{2112}{43361}a^{12}-\frac{9459}{43361}a^{10}-\frac{501540}{43361}a^{8}-\frac{1464974}{43361}a^{6}-\frac{997394}{43361}a^{4}+\frac{41582}{43361}a^{2}+\frac{88879}{43361}$, $\frac{110043}{43361}a^{19}+\frac{513245}{43361}a^{17}+\frac{704969}{43361}a^{15}-\frac{25338}{43361}a^{13}-\frac{100054}{43361}a^{11}-\frac{1157174}{43361}a^{9}-\frac{3689905}{43361}a^{7}-\frac{2464169}{43361}a^{5}+\frac{303495}{43361}a^{3}+\frac{256110}{43361}a$, $\frac{27476}{43361}a^{18}+\frac{137610}{43361}a^{16}+\frac{215973}{43361}a^{14}+\frac{39050}{43361}a^{12}-\frac{40557}{43361}a^{10}-\frac{293255}{43361}a^{8}-\frac{1033753}{43361}a^{6}-\frac{860569}{43361}a^{4}-\frac{52474}{43361}a^{2}+\frac{91556}{43361}$, $\frac{89111}{43361}a^{19}-\frac{48485}{86722}a^{18}+\frac{411686}{43361}a^{17}-\frac{240713}{86722}a^{16}+\frac{560275}{43361}a^{15}-\frac{187475}{43361}a^{14}-\frac{36463}{86722}a^{13}-\frac{69417}{86722}a^{12}-\frac{127891}{86722}a^{11}+\frac{26649}{43361}a^{10}-\frac{1926885}{86722}a^{9}+\frac{499889}{86722}a^{8}-\frac{5860951}{86722}a^{7}+\frac{886154}{43361}a^{6}-\frac{1950501}{43361}a^{5}+\frac{1519633}{86722}a^{4}+\frac{383125}{86722}a^{3}+\frac{54314}{43361}a^{2}+\frac{484585}{86722}a-\frac{106921}{86722}$, $\frac{89111}{43361}a^{19}+\frac{48485}{86722}a^{18}+\frac{411686}{43361}a^{17}+\frac{240713}{86722}a^{16}+\frac{560275}{43361}a^{15}+\frac{187475}{43361}a^{14}-\frac{36463}{86722}a^{13}+\frac{69417}{86722}a^{12}-\frac{127891}{86722}a^{11}-\frac{26649}{43361}a^{10}-\frac{1926885}{86722}a^{9}-\frac{499889}{86722}a^{8}-\frac{5860951}{86722}a^{7}-\frac{886154}{43361}a^{6}-\frac{1950501}{43361}a^{5}-\frac{1519633}{86722}a^{4}+\frac{383125}{86722}a^{3}-\frac{54314}{43361}a^{2}+\frac{484585}{86722}a+\frac{106921}{86722}$, $\frac{85868}{43361}a^{19}-\frac{202}{43361}a^{18}+\frac{400718}{43361}a^{17}+\frac{4851}{86722}a^{16}+\frac{559016}{43361}a^{15}+\frac{13212}{43361}a^{14}+\frac{10111}{43361}a^{13}+\frac{14882}{43361}a^{12}-\frac{112317}{86722}a^{11}-\frac{20677}{86722}a^{10}-\frac{921215}{43361}a^{9}-\frac{2695}{86722}a^{8}-\frac{5725483}{86722}a^{7}-\frac{34489}{43361}a^{6}-\frac{2037634}{43361}a^{5}-\frac{106242}{43361}a^{4}+\frac{90373}{43361}a^{3}+\frac{12999}{86722}a^{2}+\frac{426877}{86722}a+\frac{75787}{86722}$, $\frac{27321}{86722}a^{19}+\frac{123655}{86722}a^{18}+\frac{126055}{86722}a^{17}+\frac{277936}{43361}a^{16}+\frac{172017}{86722}a^{15}+\frac{358329}{43361}a^{14}-\frac{9635}{86722}a^{13}-\frac{78959}{86722}a^{12}-\frac{40011}{86722}a^{11}-\frac{19171}{43361}a^{10}-\frac{310925}{86722}a^{9}-\frac{669923}{43361}a^{8}-\frac{434693}{43361}a^{7}-\frac{3884197}{86722}a^{6}-\frac{306188}{43361}a^{5}-\frac{2343151}{86722}a^{4}+\frac{70438}{43361}a^{3}+\frac{276375}{86722}a^{2}+\frac{164745}{86722}a+\frac{244169}{86722}$, $\frac{50352}{43361}a^{18}+\frac{221837}{43361}a^{16}+\frac{273015}{43361}a^{14}-\frac{50529}{43361}a^{12}-\frac{568}{43361}a^{10}-\frac{547217}{43361}a^{8}-\frac{1545200}{43361}a^{6}-\frac{817598}{43361}a^{4}+\frac{175290}{43361}a^{2}+\frac{38856}{43361}$, $\frac{180439}{86722}a^{19}-\frac{27321}{86722}a^{18}+\frac{409699}{43361}a^{17}-\frac{126055}{86722}a^{16}+\frac{535856}{43361}a^{15}-\frac{172017}{86722}a^{14}-\frac{56108}{43361}a^{13}+\frac{9635}{86722}a^{12}-\frac{51485}{43361}a^{11}+\frac{40011}{86722}a^{10}-\frac{1934313}{86722}a^{9}+\frac{310925}{86722}a^{8}-\frac{5795183}{86722}a^{7}+\frac{434693}{43361}a^{6}-\frac{3552389}{86722}a^{5}+\frac{306188}{43361}a^{4}+\frac{286569}{43361}a^{3}-\frac{70438}{43361}a^{2}+\frac{384135}{86722}a-\frac{164745}{86722}$, $\frac{43303}{43361}a^{19}+\frac{43591}{86722}a^{18}+\frac{202690}{43361}a^{17}+\frac{169609}{86722}a^{16}+\frac{278127}{43361}a^{15}+\frac{78771}{43361}a^{14}-\frac{32239}{86722}a^{13}-\frac{99803}{86722}a^{12}-\frac{108973}{86722}a^{11}+\frac{26171}{43361}a^{10}-\frac{923805}{86722}a^{9}-\frac{537473}{86722}a^{8}-\frac{2931003}{86722}a^{7}-\frac{505205}{43361}a^{6}-\frac{953107}{43361}a^{5}-\frac{305343}{86722}a^{4}+\frac{299961}{86722}a^{3}+\frac{77548}{43361}a^{2}+\frac{393549}{86722}a+\frac{15885}{86722}$ 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:  \( 1235.36092177 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 1235.36092177 \cdot 1}{2\cdot\sqrt{24890784868177585325056}}\cr\approx \mathstrut & 0.152161050324 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 + 5*x^18 + 8*x^16 + 2*x^14 - x^12 - 11*x^10 - 37*x^8 - 34*x^6 - 5*x^4 + 4*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 + 5*x^18 + 8*x^16 + 2*x^14 - x^12 - 11*x^10 - 37*x^8 - 34*x^6 - 5*x^4 + 4*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 + 5*x^18 + 8*x^16 + 2*x^14 - x^12 - 11*x^10 - 37*x^8 - 34*x^6 - 5*x^4 + 4*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 + 5*x^18 + 8*x^16 + 2*x^14 - x^12 - 11*x^10 - 37*x^8 - 34*x^6 - 5*x^4 + 4*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

$C_2^5.C_2^8:C_{10}$ (as 20T751):

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 81920
The 332 conjugacy class representatives for $C_2^5.C_2^8:C_{10}$
Character table for $C_2^5.C_2^8:C_{10}$

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.4.4930254263.1

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

Degree 20 siblings: data not computed
Minimal sibling: 20.6.1108181030652775972732928.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.10.0.1}{10} }^{2}$ ${\href{/padicField/5.10.0.1}{10} }^{2}$ ${\href{/padicField/7.10.0.1}{10} }^{2}$ R ${\href{/padicField/13.10.0.1}{10} }^{2}$ ${\href{/padicField/17.10.0.1}{10} }^{2}$ ${\href{/padicField/19.10.0.1}{10} }^{2}$ R ${\href{/padicField/29.10.0.1}{10} }^{2}$ ${\href{/padicField/31.10.0.1}{10} }^{2}$ ${\href{/padicField/37.10.0.1}{10} }^{2}$ ${\href{/padicField/41.5.0.1}{5} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.5.0.1}{5} }^{4}$ ${\href{/padicField/53.10.0.1}{10} }^{2}$ ${\href{/padicField/59.10.0.1}{10} }^{2}$

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
\(2\) Copy content Toggle raw display 2.10.10.6$x^{10} - 6 x^{9} + 42 x^{8} - 104 x^{7} - 256 x^{6} - 112 x^{5} - 1568 x^{4} - 2016 x^{3} - 2832 x^{2} - 4960 x - 3616$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
2.10.0.1$x^{10} + x^{6} + x^{5} + x^{3} + x^{2} + x + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
\(11\) Copy content Toggle raw display 11.10.8.5$x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
\(23\) Copy content Toggle raw display 23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} + 115$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} + 115$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.0.1$x^{4} + 3 x^{2} + 19 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} + 3 x^{2} + 19 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$