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Properties

Label 20.20.3359647770...3125.1
Degree $20$
Signature $[20, 0]$
Discriminant $5^{5}\cdot 401^{10}$
Root discriminant $29.94$
Ramified primes $5, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5:D_4$ (as 20T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, -63, -151, 510, 1796, -914, -6081, 827, 9965, -1650, -8464, 2469, 3339, -1245, -639, 272, 58, -27, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 27*x^18 + 58*x^17 + 272*x^16 - 639*x^15 - 1245*x^14 + 3339*x^13 + 2469*x^12 - 8464*x^11 - 1650*x^10 + 9965*x^9 + 827*x^8 - 6081*x^7 - 914*x^6 + 1796*x^5 + 510*x^4 - 151*x^3 - 63*x^2 - x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 - 27*x^18 + 58*x^17 + 272*x^16 - 639*x^15 - 1245*x^14 + 3339*x^13 + 2469*x^12 - 8464*x^11 - 1650*x^10 + 9965*x^9 + 827*x^8 - 6081*x^7 - 914*x^6 + 1796*x^5 + 510*x^4 - 151*x^3 - 63*x^2 - x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 27 x^{18} + 58 x^{17} + 272 x^{16} - 639 x^{15} - 1245 x^{14} + 3339 x^{13} + 2469 x^{12} - 8464 x^{11} - 1650 x^{10} + 9965 x^{9} + 827 x^{8} - 6081 x^{7} - 914 x^{6} + 1796 x^{5} + 510 x^{4} - 151 x^{3} - 63 x^{2} - x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(335964777096080824022512503125=5^{5}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.94$
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:  $5, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{173152322983} a^{19} + \frac{70189390319}{173152322983} a^{18} - \frac{36541051166}{173152322983} a^{17} + \frac{23563338952}{173152322983} a^{16} + \frac{3640122728}{9113280157} a^{15} + \frac{70607839831}{173152322983} a^{14} + \frac{46927796743}{173152322983} a^{13} - \frac{84548642267}{173152322983} a^{12} - \frac{85285031087}{173152322983} a^{11} - \frac{82832494331}{173152322983} a^{10} + \frac{39578592047}{173152322983} a^{9} - \frac{41597935655}{173152322983} a^{8} + \frac{3243526887}{9113280157} a^{7} + \frac{34440905236}{173152322983} a^{6} - \frac{76937945147}{173152322983} a^{5} - \frac{3116755619}{173152322983} a^{4} + \frac{12827894237}{173152322983} a^{3} - \frac{38840177249}{173152322983} a^{2} - \frac{11242246859}{173152322983} a - \frac{16112254428}{173152322983}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $19$
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:  \( 151466349.639 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5:D_4$ (as 20T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 13 conjugacy class representatives for $C_5:D_4$
Character table for $C_5:D_4$

Intermediate fields

\(\Q(\sqrt{401}) \), 4.4.804005.1, 5.5.160801.1 x5, 10.10.10368641602001.1

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

Sibling fields

Galois closure: data not computed
Degree 20 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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$5$5.5.0.1$x^{5} - x + 2$$1$$5$$0$$C_5$$[\ ]^{5}$
5.5.0.1$x^{5} - x + 2$$1$$5$$0$$C_5$$[\ ]^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
401Data not computed