↑

Properties

Label 20.0.23383113568...8125.2
Degree $20$
Signature $[0, 10]$
Discriminant $5^{27}\cdot 11^{12}$
Root discriminant $37.02$
Ramified primes $5, 11$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $D_5.D_5$ (as 20T26)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![161051, 0, 0, 0, 0, -2904, 0, 0, 0, 0, 781, 0, 0, 0, 0, -54, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 54*x^15 + 781*x^10 - 2904*x^5 + 161051)
 
gp: K = bnfinit(x^20 - 54*x^15 + 781*x^10 - 2904*x^5 + 161051, 1)
 

Normalized defining polynomial

\( x^{20} - 54 x^{15} + 781 x^{10} - 2904 x^{5} + 161051 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(23383113568432629108428955078125=5^{27}\cdot 11^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.02$
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:  $5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{3} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{110} a^{10} + \frac{23}{110} a^{5} + \frac{1}{10}$, $\frac{1}{110} a^{11} + \frac{23}{110} a^{6} + \frac{1}{10} a$, $\frac{1}{1210} a^{12} - \frac{21}{1210} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{5}{22} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{1210} a^{13} - \frac{21}{1210} a^{8} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{5}{22} a^{3} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{66550} a^{14} + \frac{1}{6050} a^{13} + \frac{1}{6050} a^{12} + \frac{1}{550} a^{11} + \frac{1}{550} a^{10} - \frac{5587}{66550} a^{9} + \frac{463}{6050} a^{8} + \frac{463}{6050} a^{7} - \frac{87}{550} a^{6} - \frac{87}{550} a^{5} + \frac{1801}{6050} a^{4} + \frac{151}{550} a^{3} + \frac{151}{550} a^{2} + \frac{1}{50} a + \frac{1}{50}$, $\frac{1}{732050} a^{15} - \frac{263}{732050} a^{10} - \frac{28207}{66550} a^{5} + \frac{2}{25}$, $\frac{1}{732050} a^{16} - \frac{263}{732050} a^{11} - \frac{28207}{66550} a^{6} + \frac{2}{25} a$, $\frac{1}{732050} a^{17} - \frac{263}{732050} a^{12} - \frac{1587}{66550} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{8}{25} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{732050} a^{18} - \frac{263}{732050} a^{13} - \frac{1587}{66550} a^{8} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{8}{25} a^{3} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{732050} a^{19} + \frac{1}{732050} a^{14} - \frac{1}{6050} a^{13} - \frac{1}{6050} a^{12} - \frac{1}{550} a^{11} - \frac{1}{550} a^{10} - \frac{103}{2662} a^{9} - \frac{463}{6050} a^{8} - \frac{463}{6050} a^{7} - \frac{243}{550} a^{6} + \frac{197}{550} a^{5} - \frac{531}{3025} a^{4} - \frac{151}{550} a^{3} - \frac{151}{550} a^{2} - \frac{21}{50} a - \frac{11}{50}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{53}{732050} a^{15} - \frac{629}{732050} a^{10} - \frac{3041}{66550} a^{5} + \frac{11}{25} \) (order $10$)
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:  \( 69020367.6146 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5.D_5$ (as 20T26):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 100
The 13 conjugacy class representatives for $D_5.D_5$
Character table for $D_5.D_5$

Intermediate fields

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

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

Sibling fields

Degree 25 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
5Data not computed
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$