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Properties

Label 20.4.35543731469...5625.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{10}\cdot 11^{16}\cdot 89^{2}$
Root discriminant $23.85$
Ramified primes $5, 11, 89$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2\times C_2^4:C_5$ (as 20T86)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, 5, 16, 15, 18, 5, -31, -76, -67, -231, 67, -76, 31, 5, -18, 15, -16, 5, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 5*x^18 - 16*x^17 + 15*x^16 - 18*x^15 + 5*x^14 + 31*x^13 - 76*x^12 + 67*x^11 - 231*x^10 - 67*x^9 - 76*x^8 - 31*x^7 + 5*x^6 + 18*x^5 + 15*x^4 + 16*x^3 + 5*x^2 + 4*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 5*x^18 - 16*x^17 + 15*x^16 - 18*x^15 + 5*x^14 + 31*x^13 - 76*x^12 + 67*x^11 - 231*x^10 - 67*x^9 - 76*x^8 - 31*x^7 + 5*x^6 + 18*x^5 + 15*x^4 + 16*x^3 + 5*x^2 + 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 5 x^{18} - 16 x^{17} + 15 x^{16} - 18 x^{15} + 5 x^{14} + 31 x^{13} - 76 x^{12} + 67 x^{11} - 231 x^{10} - 67 x^{9} - 76 x^{8} - 31 x^{7} + 5 x^{6} + 18 x^{5} + 15 x^{4} + 16 x^{3} + 5 x^{2} + 4 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3554373146966358274228515625=5^{10}\cdot 11^{16}\cdot 89^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.85$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $5, 11, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{1630687741} a^{18} + \frac{226592350}{1630687741} a^{17} + \frac{492415087}{1630687741} a^{16} - \frac{796983389}{1630687741} a^{15} - \frac{695848150}{1630687741} a^{14} - \frac{757531008}{1630687741} a^{13} - \frac{312854551}{1630687741} a^{12} - \frac{418743360}{1630687741} a^{11} - \frac{752360488}{1630687741} a^{10} + \frac{41178111}{1630687741} a^{9} + \frac{752360488}{1630687741} a^{8} - \frac{418743360}{1630687741} a^{7} + \frac{312854551}{1630687741} a^{6} - \frac{757531008}{1630687741} a^{5} + \frac{695848150}{1630687741} a^{4} - \frac{796983389}{1630687741} a^{3} - \frac{492415087}{1630687741} a^{2} + \frac{226592350}{1630687741} a - \frac{1}{1630687741}$, $\frac{1}{177744963769} a^{19} - \frac{37}{177744963769} a^{18} + \frac{16982768571}{177744963769} a^{17} + \frac{283771873}{177744963769} a^{16} + \frac{21900803716}{177744963769} a^{15} - \frac{83828697622}{177744963769} a^{14} - \frac{76799964823}{177744963769} a^{13} + \frac{51964550472}{177744963769} a^{12} - \frac{63136647574}{177744963769} a^{11} + \frac{74960472782}{177744963769} a^{10} + \frac{14195639028}{177744963769} a^{9} - \frac{19894654837}{177744963769} a^{8} - \frac{53917993830}{177744963769} a^{7} - \frac{51510137099}{177744963769} a^{6} + \frac{48141006194}{177744963769} a^{5} + \frac{17046673585}{177744963769} a^{4} - \frac{1307729171}{7728041903} a^{3} + \frac{31759591908}{177744963769} a^{2} - \frac{12862025855}{177744963769} a - \frac{81307794663}{177744963769}$

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

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 331811.045684 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_2^4:C_5$ (as 20T86):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 32 conjugacy class representatives for $C_2^2\times C_2^4:C_5$
Character table for $C_2^2\times C_2^4:C_5$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{11})^+\), 10.2.19077940409.1, 10.2.59618563778125.1, 10.10.669871503125.1

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

Sibling fields

Degree 20 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
$89$$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$