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Properties

Label 20.0.59492328979...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{27}\cdot 41^{8}$
Root discriminant $38.79$
Ramified primes $5, 41$
Class number $10$ (GRH)
Class group $[10]$ (GRH)
Galois group $C_4\times A_5$ (as 20T63)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, 0, 500, 0, 10900, 0, 30650, 0, 37710, 0, 25525, 0, 10325, 0, 2550, 0, 375, 0, 30, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 30*x^18 + 375*x^16 + 2550*x^14 + 10325*x^12 + 25525*x^10 + 37710*x^8 + 30650*x^6 + 10900*x^4 + 500*x^2 + 5)
 
gp: K = bnfinit(x^20 + 30*x^18 + 375*x^16 + 2550*x^14 + 10325*x^12 + 25525*x^10 + 37710*x^8 + 30650*x^6 + 10900*x^4 + 500*x^2 + 5, 1)
 

Normalized defining polynomial

\( x^{20} + 30 x^{18} + 375 x^{16} + 2550 x^{14} + 10325 x^{12} + 25525 x^{10} + 37710 x^{8} + 30650 x^{6} + 10900 x^{4} + 500 x^{2} + 5 \)

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:  \(59492328979976475238800048828125=5^{27}\cdot 41^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.79$
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:  $5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{637102} a^{18} - \frac{77499}{637102} a^{16} - \frac{44308}{318551} a^{14} + \frac{122997}{637102} a^{12} + \frac{97}{637102} a^{10} - \frac{1}{2} a^{9} + \frac{75218}{318551} a^{8} - \frac{1}{2} a^{7} - \frac{7171}{637102} a^{6} - \frac{198937}{637102} a^{4} - \frac{1}{2} a^{3} + \frac{56903}{318551} a^{2} - \frac{1}{2} a - \frac{19638}{318551}$, $\frac{1}{637102} a^{19} - \frac{77499}{637102} a^{17} - \frac{44308}{318551} a^{15} + \frac{122997}{637102} a^{13} + \frac{97}{637102} a^{11} + \frac{75218}{318551} a^{9} - \frac{7171}{637102} a^{7} - \frac{1}{2} a^{6} + \frac{59807}{318551} a^{5} - \frac{1}{2} a^{4} + \frac{56903}{318551} a^{3} - \frac{1}{2} a^{2} - \frac{19638}{318551} a - \frac{1}{2}$

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

Class group and class number

$C_{10}$, which has order $10$ (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:  \( \frac{63545}{637102} a^{19} - \frac{183093}{318551} a^{18} + \frac{858630}{318551} a^{17} - \frac{9897755}{637102} a^{16} + \frac{9358408}{318551} a^{15} - \frac{53942465}{318551} a^{14} + \frac{106273063}{637102} a^{13} - \frac{612427125}{637102} a^{12} + \frac{169524830}{318551} a^{11} - \frac{1952241409}{637102} a^{10} + \frac{304086260}{318551} a^{9} - \frac{3494718585}{637102} a^{8} + \frac{570053025}{637102} a^{7} - \frac{1629814235}{318551} a^{6} + \frac{110659582}{318551} a^{5} - \frac{620642200}{318551} a^{4} + \frac{5125550}{318551} a^{3} - \frac{26487625}{637102} a^{2} + \frac{3239175}{637102} a + \frac{190394}{318551} \) (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:  \( 15507096.7142 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times A_5$ (as 20T63):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 240
The 20 conjugacy class representatives for $C_4\times A_5$
Character table for $C_4\times A_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.5.26265625.1, 10.10.3449415283203125.1

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

Sibling fields

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 $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ $20$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.3.2.1$x^{3} - 41$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
41.3.2.1$x^{3} - 41$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
41.3.2.1$x^{3} - 41$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
41.3.2.1$x^{3} - 41$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$