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Properties

Label 20.0.39815113037...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 601^{4}$
Root discriminant $12.02$
Ramified primes $5, 601$
Class number $1$
Class group Trivial
Galois group $D_5^2:C_4$ (as 20T93)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 19, -39, 59, -65, 50, -16, -21, 49, -63, 61, -43, 17, 5, -17, 19, -15, 9, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 9*x^18 - 15*x^17 + 19*x^16 - 17*x^15 + 5*x^14 + 17*x^13 - 43*x^12 + 61*x^11 - 63*x^10 + 49*x^9 - 21*x^8 - 16*x^7 + 50*x^6 - 65*x^5 + 59*x^4 - 39*x^3 + 19*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 9*x^18 - 15*x^17 + 19*x^16 - 17*x^15 + 5*x^14 + 17*x^13 - 43*x^12 + 61*x^11 - 63*x^10 + 49*x^9 - 21*x^8 - 16*x^7 + 50*x^6 - 65*x^5 + 59*x^4 - 39*x^3 + 19*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 9 x^{18} - 15 x^{17} + 19 x^{16} - 17 x^{15} + 5 x^{14} + 17 x^{13} - 43 x^{12} + 61 x^{11} - 63 x^{10} + 49 x^{9} - 21 x^{8} - 16 x^{7} + 50 x^{6} - 65 x^{5} + 59 x^{4} - 39 x^{3} + 19 x^{2} - 6 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3981511303741455078125=5^{15}\cdot 601^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.02$
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:  $5, 601$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{176639} a^{19} + \frac{16856}{176639} a^{18} - \frac{19982}{176639} a^{17} - \frac{45962}{176639} a^{16} - \frac{4008}{176639} a^{15} + \frac{77840}{176639} a^{14} - \frac{45365}{176639} a^{13} - \frac{7013}{176639} a^{12} - \frac{67732}{176639} a^{11} + \frac{9676}{176639} a^{10} - \frac{77139}{176639} a^{9} + \frac{29466}{176639} a^{8} + \frac{87871}{176639} a^{7} + \frac{33751}{176639} a^{6} + \frac{87691}{176639} a^{5} + \frac{1765}{176639} a^{4} + \frac{82607}{176639} a^{3} - \frac{44534}{176639} a^{2} + \frac{49168}{176639} a + \frac{5647}{176639}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{246477}{176639} a^{19} - \frac{992802}{176639} a^{18} + \frac{2241491}{176639} a^{17} - \frac{3719667}{176639} a^{16} + \frac{4654725}{176639} a^{15} - \frac{4114641}{176639} a^{14} + \frac{1056068}{176639} a^{13} + \frac{4462028}{176639} a^{12} - \frac{10826614}{176639} a^{11} + \frac{15122628}{176639} a^{10} - \frac{15288214}{176639} a^{9} + \frac{11483693}{176639} a^{8} - \frac{4473215}{176639} a^{7} - \frac{4561092}{176639} a^{6} + \frac{12807936}{176639} a^{5} - \frac{16104101}{176639} a^{4} + \frac{14032407}{176639} a^{3} - \frac{8914569}{176639} a^{2} + \frac{3995321}{176639} a - \frac{942896}{176639} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1218.45033926 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5^2:C_4$ (as 20T93):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 28 conjugacy class representatives for $D_5^2:C_4$
Character table for $D_5^2:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 10.2.1128753125.1

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 $20$ $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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
601Data not computed