Properties

Label 20.0.246...125.1
Degree $20$
Signature $[0, 10]$
Discriminant $2.463\times 10^{21}$
Root discriminant $11.74$
Ramified primes $5, 13, 41$
Class number $1$
Class group trivial
Galois group $C_4\times S_5$ (as 20T123)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - x^16 - 2*x^15 + 4*x^14 - x^13 + x^12 + x^11 + 4*x^10 - 8*x^9 + 2*x^8 - x^7 - 2*x^6 - x^5 + 8*x^4 - 5*x^3 + 3*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^20 - x^19 - x^16 - 2*x^15 + 4*x^14 - x^13 + x^12 + x^11 + 4*x^10 - 8*x^9 + 2*x^8 - x^7 - 2*x^6 - x^5 + 8*x^4 - 5*x^3 + 3*x^2 - 2*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 3, -5, 8, -1, -2, -1, 2, -8, 4, 1, 1, -1, 4, -2, -1, 0, 0, -1, 1]);
 

\( x^{20} - x^{19} - x^{16} - 2 x^{15} + 4 x^{14} - x^{13} + x^{12} + x^{11} + 4 x^{10} - 8 x^{9} + 2 x^{8} - x^{7} - 2 x^{6} - x^{5} + 8 x^{4} - 5 x^{3} + 3 x^{2} - 2 x + 1 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(2462968747589111328125\)\(\medspace = 5^{15}\cdot 13^{4}\cdot 41^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $11.74$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $5, 13, 41$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $4$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} + \frac{2}{5} a^{15} + \frac{2}{5} a^{14} - \frac{1}{5} a^{12} + \frac{2}{5} a^{11} + \frac{2}{5} a^{10} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{17} - \frac{2}{5} a^{15} + \frac{1}{5} a^{14} - \frac{1}{5} a^{13} - \frac{1}{5} a^{12} - \frac{2}{5} a^{11} + \frac{1}{5} a^{10} - \frac{2}{5} a^{9} + \frac{1}{5} a^{8} + \frac{2}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{18} - \frac{2}{5} a^{14} - \frac{1}{5} a^{13} + \frac{1}{5} a^{12} + \frac{2}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{25} a^{19} + \frac{1}{25} a^{18} + \frac{2}{25} a^{17} - \frac{1}{25} a^{16} - \frac{8}{25} a^{15} + \frac{7}{25} a^{14} + \frac{3}{25} a^{13} + \frac{2}{5} a^{12} + \frac{6}{25} a^{11} - \frac{12}{25} a^{10} - \frac{2}{5} a^{9} + \frac{7}{25} a^{8} - \frac{4}{25} a^{7} - \frac{4}{25} a^{6} + \frac{1}{5} a^{5} + \frac{9}{25} a^{4} + \frac{11}{25} a^{3} + \frac{2}{25} a^{2} + \frac{2}{25} a - \frac{3}{25}$

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

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{1}{5} a^{19} - \frac{1}{5} a^{18} + \frac{4}{5} a^{17} + \frac{1}{5} a^{16} + \frac{1}{5} a^{15} + \frac{4}{5} a^{14} + \frac{1}{5} a^{13} - \frac{11}{5} a^{12} - \frac{3}{5} a^{11} - \frac{2}{5} a^{10} - \frac{7}{5} a^{9} - \frac{1}{5} a^{8} + \frac{21}{5} a^{7} + a^{6} + \frac{2}{5} a^{5} + \frac{6}{5} a^{4} - \frac{4}{5} a^{3} - 2 a^{2} - \frac{1}{5} a + \frac{1}{5} \) (order $10$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 887.733037213 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{10}\cdot 887.733037213 \cdot 1}{10\sqrt{2462968747589111328125}}\approx 0.171534553169$

Galois group

$C_4\times S_5$ (as 20T123):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 480
The 28 conjugacy class representatives for $C_4\times S_5$
Character table for $C_4\times S_5$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.1.2665.1, 10.2.887778125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 sibling: data not computed
Degree 24 siblings: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $20$ $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ R $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.12.0.1$x^{12} + x^{2} - x + 2$$1$$12$$0$$C_{12}$$[\ ]^{12}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$