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Properties

Label 20.0.715...000.1
Degree $20$
Signature $[0, 10]$
Discriminant $7.158\times 10^{20}$
Root discriminant \(11.03\)
Ramified primes $2,5,5783$
Class number $1$
Class group trivial
Galois group $C_2^9.C_2^4:S_5$ (as 20T964)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

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

\( x^{20} + 2x^{18} + 2x^{16} + 4x^{14} + 11x^{12} + 14x^{10} + 11x^{8} + 6x^{6} + 5x^{4} + 4x^{2} + 1 \) 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:  $[0, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(715801729191629440000\) \(\medspace = 2^{10}\cdot 5^{4}\cdot 5783^{4}\) 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:  \(11.03\)
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:  $2^{15/8}5^{1/2}5783^{1/2}\approx 623.724552629587$
Ramified primes:   \(2\), \(5\), \(5783\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{8}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{13}-\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{2}a^{9}+\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{14}-\frac{1}{4}a^{13}-\frac{1}{4}a^{11}+\frac{1}{4}a^{9}-\frac{1}{4}a^{8}+\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{4}a^{17}-\frac{1}{4}a^{14}-\frac{1}{4}a^{13}+\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{5}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{44}a^{18}+\frac{1}{11}a^{16}-\frac{1}{44}a^{14}-\frac{1}{4}a^{13}-\frac{9}{44}a^{12}+\frac{1}{11}a^{10}+\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}+\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{17}{44}a^{4}+\frac{1}{4}a^{3}+\frac{17}{44}a^{2}+\frac{5}{44}$, $\frac{1}{44}a^{19}+\frac{1}{11}a^{17}-\frac{1}{44}a^{15}-\frac{1}{4}a^{14}-\frac{9}{44}a^{13}+\frac{1}{11}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}+\frac{1}{4}a^{7}+\frac{1}{4}a^{6}+\frac{17}{44}a^{5}-\frac{1}{4}a^{4}+\frac{17}{44}a^{3}-\frac{1}{2}a^{2}+\frac{5}{44}a-\frac{1}{2}$ 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:   $a$, $\frac{31}{22}a^{19}+\frac{18}{11}a^{17}+\frac{12}{11}a^{15}+\frac{95}{22}a^{13}+\frac{128}{11}a^{11}+9a^{9}+\frac{11}{2}a^{7}+\frac{43}{22}a^{5}+\frac{49}{11}a^{3}+\frac{28}{11}a$, $\frac{43}{22}a^{19}+\frac{31}{11}a^{17}+\frac{28}{11}a^{15}+\frac{141}{22}a^{13}+\frac{196}{11}a^{11}+18a^{9}+\frac{25}{2}a^{7}+\frac{93}{22}a^{5}+\frac{74}{11}a^{3}+\frac{47}{11}a$, $\frac{21}{11}a^{19}+\frac{21}{44}a^{18}+\frac{127}{44}a^{17}+\frac{29}{44}a^{16}+\frac{103}{44}a^{15}+\frac{23}{44}a^{14}+\frac{139}{22}a^{13}+\frac{16}{11}a^{12}+\frac{787}{44}a^{11}+\frac{183}{44}a^{10}+\frac{71}{4}a^{9}+4a^{8}+\frac{23}{2}a^{7}+\frac{5}{2}a^{6}+\frac{229}{44}a^{5}+\frac{5}{44}a^{4}+\frac{153}{22}a^{3}+\frac{49}{44}a^{2}+\frac{189}{44}a+\frac{39}{44}$, $\frac{9}{11}a^{18}+\frac{14}{11}a^{16}+\frac{13}{11}a^{14}+\frac{69}{22}a^{12}+\frac{171}{22}a^{10}+\frac{17}{2}a^{8}+7a^{6}+\frac{43}{11}a^{4}+\frac{32}{11}a^{2}+\frac{57}{22}$, $\frac{31}{22}a^{19}+\frac{29}{11}a^{17}+\frac{23}{11}a^{15}+\frac{53}{11}a^{13}+\frac{161}{11}a^{11}+\frac{33}{2}a^{9}+10a^{7}+\frac{49}{11}a^{5}+\frac{109}{22}a^{3}+\frac{39}{11}a$, $\frac{1}{22}a^{19}-\frac{63}{44}a^{18}+\frac{19}{44}a^{17}-\frac{87}{44}a^{16}+\frac{5}{11}a^{15}-\frac{69}{44}a^{14}+\frac{15}{44}a^{13}-\frac{203}{44}a^{12}+\frac{63}{44}a^{11}-\frac{140}{11}a^{10}+\frac{13}{4}a^{9}-\frac{47}{4}a^{8}+\frac{9}{4}a^{7}-\frac{15}{2}a^{6}+\frac{17}{22}a^{5}-\frac{125}{44}a^{4}+\frac{3}{11}a^{3}-\frac{191}{44}a^{2}+\frac{8}{11}a-\frac{53}{22}$, $a^{19}-\frac{1}{22}a^{18}+\frac{7}{4}a^{17}-\frac{19}{44}a^{16}+\frac{7}{4}a^{15}-\frac{5}{11}a^{14}+\frac{15}{4}a^{13}-\frac{15}{44}a^{12}+\frac{41}{4}a^{11}-\frac{63}{44}a^{10}+12a^{9}-\frac{13}{4}a^{8}+\frac{37}{4}a^{7}-\frac{9}{4}a^{6}+5a^{5}-\frac{17}{22}a^{4}+\frac{19}{4}a^{3}-\frac{3}{11}a^{2}+\frac{11}{4}a-\frac{8}{11}$, $\frac{39}{22}a^{19}-\frac{14}{11}a^{18}+\frac{57}{22}a^{17}-\frac{23}{11}a^{16}+\frac{87}{44}a^{15}-\frac{19}{11}a^{14}+\frac{255}{44}a^{13}-\frac{189}{44}a^{12}+\frac{177}{11}a^{11}-\frac{543}{44}a^{10}+\frac{31}{2}a^{9}-\frac{53}{4}a^{8}+\frac{39}{4}a^{7}-\frac{17}{2}a^{6}+\frac{171}{44}a^{5}-\frac{40}{11}a^{4}+\frac{113}{22}a^{3}-\frac{113}{22}a^{2}+\frac{159}{44}a-\frac{137}{44}$ 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:  \( 87.5155752534 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 87.5155752534 \cdot 1}{2\cdot\sqrt{715801729191629440000}}\cr\approx \mathstrut & 0.156840301372 \end{aligned}\]

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

$C_2^9.C_2^4:S_5$ (as 20T964):

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 983040
The 155 conjugacy class representatives for $C_2^9.C_2^4:S_5$ are not computed
Character table for $C_2^9.C_2^4:S_5$ is not computed

Intermediate fields

5.3.5783.1, 10.2.836077225.1

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
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 ${\href{/padicField/3.5.0.1}{5} }^{4}$ R $16{,}\,{\href{/padicField/7.4.0.1}{4} }$ ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ $16{,}\,{\href{/padicField/17.4.0.1}{4} }$ ${\href{/padicField/19.3.0.1}{3} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ ${\href{/padicField/23.10.0.1}{10} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.4.0.1}{4} }^{2}$ ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.10.0.1}{10} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ $16{,}\,{\href{/padicField/47.4.0.1}{4} }$ ${\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.8.0.1}{8} }$

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 2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.3$x^{10} + 10 x^{9} + 74 x^{8} + 320 x^{7} + 1104 x^{6} + 2752 x^{5} + 6176 x^{4} + 12096 x^{3} + 17712 x^{2} + 15968 x + 8416$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
\(5\) Copy content Toggle raw display 5.6.0.1$x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
5.6.0.1$x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(5783\) Copy content Toggle raw display Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $8$$2$$4$$4$