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Properties

Label 20.0.91340287376...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{35}\cdot 11^{12}$
Root discriminant $70.47$
Ramified primes $5, 11$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $C_5\times F_5$ (as 20T29)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![161051, 0, 0, 0, 0, -40172, 0, 0, 0, 0, 4169, 0, 0, 0, 0, -118, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 118*x^15 + 4169*x^10 - 40172*x^5 + 161051)
 
gp: K = bnfinit(x^20 - 118*x^15 + 4169*x^10 - 40172*x^5 + 161051, 1)
 

Normalized defining polynomial

\( x^{20} - 118 x^{15} + 4169 x^{10} - 40172 x^{5} + 161051 \)

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:  \(9134028737668995745480060577392578125=5^{35}\cdot 11^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{22} a^{10} + \frac{3}{22} a^{5} - \frac{1}{2}$, $\frac{1}{22} a^{11} + \frac{3}{22} a^{6} - \frac{1}{2} a$, $\frac{1}{22} a^{12} + \frac{3}{22} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{22} a^{13} + \frac{3}{22} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{22} a^{14} + \frac{3}{22} a^{9} - \frac{1}{2} a^{4}$, $\frac{1}{2033042} a^{15} + \frac{28515}{2033042} a^{10} + \frac{18891}{184822} a^{5} + \frac{944}{8401}$, $\frac{1}{2033042} a^{16} + \frac{28515}{2033042} a^{11} + \frac{18891}{184822} a^{6} + \frac{944}{8401} a$, $\frac{1}{22363462} a^{17} - \frac{216770}{11181731} a^{12} - \frac{145973}{1016521} a^{7} + \frac{27091}{184822} a^{2}$, $\frac{1}{22363462} a^{18} - \frac{216770}{11181731} a^{13} - \frac{145973}{1016521} a^{8} + \frac{27091}{184822} a^{3}$, $\frac{1}{245998082} a^{19} + \frac{4649065}{245998082} a^{14} + \frac{9226387}{22363462} a^{9} - \frac{217482}{1016521} a^{4}$

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:  \( \frac{81}{2033042} a^{15} - \frac{280}{1016521} a^{10} - \frac{12007}{92411} a^{5} + \frac{26913}{16802} \) (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:  \( 25950246651.417282 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times F_5$ (as 20T29):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 100
The 25 conjugacy class representatives for $C_5\times F_5$
Character table for $C_5\times F_5$ is not computed

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ $20$ R $20$ R $20$ $20$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ $20$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{5}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.5$x^{5} - 99$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.2$x^{5} - 891$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$