Properties

Label 20.0.750...776.1
Degree $20$
Signature $[0, 10]$
Discriminant $7.507\times 10^{20}$
Root discriminant \(11.06\)
Ramified primes $2,41381$
Class number $1$
Class group trivial
Galois group $C_2\times C_4^4:S_5$ (as 20T673)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 + x^16 + 5*x^12 + 10*x^10 + 9*x^8 + 7*x^6 + 3*x^4 + 4*x^2 + 1)
 
gp: K = bnfinit(y^20 + y^16 + 5*y^12 + 10*y^10 + 9*y^8 + 7*y^6 + 3*y^4 + 4*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + x^16 + 5*x^12 + 10*x^10 + 9*x^8 + 7*x^6 + 3*x^4 + 4*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 + x^16 + 5*x^12 + 10*x^10 + 9*x^8 + 7*x^6 + 3*x^4 + 4*x^2 + 1)
 

\( x^{20} + x^{16} + 5x^{12} + 10x^{10} + 9x^{8} + 7x^{6} + 3x^{4} + 4x^{2} + 1 \) 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:  $[0, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(750661066024355819776\) \(\medspace = 2^{8}\cdot 41381^{4}\) 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:  \(11.06\)
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:  $2^{3/2}41381^{1/2}\approx 575.3677085134341$
Ramified primes:   \(2\), \(41381\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{8326}a^{18}+\frac{1123}{8326}a^{16}-\frac{259}{8326}a^{14}-\frac{1805}{4163}a^{12}-\frac{1}{2}a^{11}+\frac{737}{8326}a^{10}+\frac{3387}{8326}a^{8}-\frac{686}{4163}a^{6}-\frac{1}{2}a^{5}+\frac{1862}{4163}a^{4}-\frac{1}{2}a^{3}-\frac{880}{4163}a^{2}+\frac{949}{8326}$, $\frac{1}{8326}a^{19}+\frac{1123}{8326}a^{17}-\frac{259}{8326}a^{15}-\frac{1805}{4163}a^{13}-\frac{1}{2}a^{12}+\frac{737}{8326}a^{11}+\frac{3387}{8326}a^{9}-\frac{686}{4163}a^{7}-\frac{1}{2}a^{6}+\frac{1862}{4163}a^{5}-\frac{1}{2}a^{4}-\frac{880}{4163}a^{3}+\frac{949}{8326}a$ 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:  $9$
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:   $\frac{2032}{4163}a^{19}+\frac{612}{4163}a^{17}+\frac{2413}{4163}a^{15}-\frac{314}{4163}a^{13}+\frac{11393}{4163}a^{11}+\frac{21760}{4163}a^{9}+\frac{26284}{4163}a^{7}+\frac{19649}{4163}a^{5}+\frac{8023}{4163}a^{3}+\frac{9225}{4163}a$, $a$, $\frac{4545}{8326}a^{19}-\frac{611}{8326}a^{18}+\frac{197}{8326}a^{17}+\frac{371}{4163}a^{16}+\frac{5137}{8326}a^{15}+\frac{55}{8326}a^{14}-\frac{1067}{8326}a^{13}-\frac{340}{4163}a^{12}+\frac{11714}{4163}a^{11}-\frac{703}{8326}a^{10}+\frac{22467}{4163}a^{9}-\frac{4609}{8326}a^{8}+\frac{46227}{8326}a^{7}+\frac{2846}{4163}a^{6}+\frac{16063}{4163}a^{5}+\frac{1797}{8326}a^{4}+\frac{5206}{4163}a^{3}+\frac{653}{4163}a^{2}+\frac{6413}{4163}a-\frac{591}{4163}$, $\frac{4545}{8326}a^{19}+\frac{611}{8326}a^{18}+\frac{197}{8326}a^{17}-\frac{371}{4163}a^{16}+\frac{5137}{8326}a^{15}-\frac{55}{8326}a^{14}-\frac{1067}{8326}a^{13}+\frac{340}{4163}a^{12}+\frac{11714}{4163}a^{11}+\frac{703}{8326}a^{10}+\frac{22467}{4163}a^{9}+\frac{4609}{8326}a^{8}+\frac{46227}{8326}a^{7}-\frac{2846}{4163}a^{6}+\frac{16063}{4163}a^{5}-\frac{1797}{8326}a^{4}+\frac{5206}{4163}a^{3}-\frac{653}{4163}a^{2}+\frac{6413}{4163}a+\frac{591}{4163}$, $\frac{1632}{4163}a^{19}+\frac{1827}{8326}a^{18}-\frac{2131}{8326}a^{17}-\frac{319}{4163}a^{16}+\frac{1938}{4163}a^{15}+\frac{1389}{8326}a^{14}-\frac{875}{4163}a^{13}-\frac{639}{4163}a^{12}+\frac{8003}{4163}a^{11}+\frac{5088}{4163}a^{10}+\frac{11609}{4163}a^{9}+\frac{7160}{4163}a^{8}+\frac{4753}{4163}a^{7}+\frac{3904}{4163}a^{6}+\frac{11665}{8326}a^{5}+\frac{703}{4163}a^{4}+\frac{150}{4163}a^{3}-\frac{842}{4163}a^{2}+\frac{12753}{8326}a+\frac{10341}{8326}$, $\frac{1632}{4163}a^{19}-\frac{1827}{8326}a^{18}-\frac{2131}{8326}a^{17}+\frac{319}{4163}a^{16}+\frac{1938}{4163}a^{15}-\frac{1389}{8326}a^{14}-\frac{875}{4163}a^{13}+\frac{639}{4163}a^{12}+\frac{8003}{4163}a^{11}-\frac{5088}{4163}a^{10}+\frac{11609}{4163}a^{9}-\frac{7160}{4163}a^{8}+\frac{4753}{4163}a^{7}-\frac{3904}{4163}a^{6}+\frac{11665}{8326}a^{5}-\frac{703}{4163}a^{4}+\frac{150}{4163}a^{3}+\frac{842}{4163}a^{2}+\frac{12753}{8326}a-\frac{10341}{8326}$, $\frac{1590}{4163}a^{18}-\frac{357}{4163}a^{16}+\frac{327}{4163}a^{14}+\frac{877}{4163}a^{12}+\frac{6190}{4163}a^{10}+\frac{15060}{4163}a^{8}+\frac{4095}{4163}a^{6}+\frac{1374}{4163}a^{4}-\frac{864}{4163}a^{2}+\frac{1904}{4163}$, $\frac{104}{181}a^{19}-\frac{386}{4163}a^{18}-\frac{87}{362}a^{17}-\frac{526}{4163}a^{16}+\frac{247}{362}a^{15}+\frac{62}{4163}a^{14}-\frac{46}{181}a^{13}-\frac{1145}{4163}a^{12}+\frac{1075}{362}a^{11}-\frac{1398}{4163}a^{10}+\frac{1673}{362}a^{9}-\frac{12889}{8326}a^{8}+\frac{1147}{362}a^{7}-\frac{7435}{4163}a^{6}+\frac{499}{181}a^{5}-\frac{5392}{4163}a^{4}+\frac{445}{362}a^{3}-\frac{10907}{8326}a^{2}+\frac{645}{362}a-\frac{4103}{8326}$, $\frac{2018}{4163}a^{19}+\frac{1066}{4163}a^{18}-\frac{1079}{8326}a^{17}+\frac{511}{8326}a^{16}+\frac{1876}{4163}a^{15}+\frac{1491}{8326}a^{14}+\frac{270}{4163}a^{13}+\frac{867}{8326}a^{12}+\frac{9401}{4163}a^{11}+\frac{10159}{8326}a^{10}+\frac{36107}{8326}a^{9}+\frac{23257}{8326}a^{8}+\frac{12188}{4163}a^{7}+\frac{11150}{4163}a^{6}+\frac{22449}{8326}a^{5}+\frac{17379}{8326}a^{4}+\frac{11207}{8326}a^{3}+\frac{6869}{8326}a^{2}+\frac{8428}{4163}a+\frac{4213}{8326}$ 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:  \( 91.2702526158 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 91.2702526158 \cdot 1}{2\cdot\sqrt{750661066024355819776}}\cr\approx \mathstrut & 0.159726139411 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 + x^16 + 5*x^12 + 10*x^10 + 9*x^8 + 7*x^6 + 3*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 + x^16 + 5*x^12 + 10*x^10 + 9*x^8 + 7*x^6 + 3*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 + x^16 + 5*x^12 + 10*x^10 + 9*x^8 + 7*x^6 + 3*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 + x^16 + 5*x^12 + 10*x^10 + 9*x^8 + 7*x^6 + 3*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

$C_2\times C_4^4:S_5$ (as 20T673):

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 61440
The 126 conjugacy class representatives for $C_2\times C_4^4:S_5$ are not computed
Character table for $C_2\times C_4^4:S_5$ is not computed

Intermediate fields

5.1.41381.1, 10.2.1712387161.1

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 sibling: data not computed
Degree 40 siblings: 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 R $16{,}\,{\href{/padicField/3.4.0.1}{4} }$ ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}$ ${\href{/padicField/7.10.0.1}{10} }^{2}$ ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.8.0.1}{8} }$ $16{,}\,{\href{/padicField/13.4.0.1}{4} }$ ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}$ ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.8.0.1}{8} }$ ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.8.0.1}{8} }$ $16{,}\,{\href{/padicField/31.4.0.1}{4} }$ ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.8.0.1}{8} }$ ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ ${\href{/padicField/47.8.0.1}{8} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{3}$

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 2.8.8.5$x^{8} + 8 x^{7} + 56 x^{6} + 184 x^{5} + 576 x^{4} + 960 x^{3} + 1632 x^{2} + 1120 x + 304$$2$$4$$8$$C_8:C_2$$[2, 2]^{4}$
2.12.0.1$x^{12} + x^{7} + x^{6} + x^{5} + x^{3} + x + 1$$1$$12$$0$$C_{12}$$[\ ]^{12}$
\(41381\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$