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Properties

Label 20.0.25000000000...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{28}\cdot 5^{30}$
Root discriminant $29.51$
Ramified primes $2, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_5\wr C_2$ (as 20T50)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -20, 50, 230, 425, 254, 30, 100, 790, 150, 2, -20, 275, 10, 0, -2, 30, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 30*x^16 - 2*x^15 + 10*x^13 + 275*x^12 - 20*x^11 + 2*x^10 + 150*x^9 + 790*x^8 + 100*x^7 + 30*x^6 + 254*x^5 + 425*x^4 + 230*x^3 + 50*x^2 - 20*x + 4)
 
gp: K = bnfinit(x^20 + 30*x^16 - 2*x^15 + 10*x^13 + 275*x^12 - 20*x^11 + 2*x^10 + 150*x^9 + 790*x^8 + 100*x^7 + 30*x^6 + 254*x^5 + 425*x^4 + 230*x^3 + 50*x^2 - 20*x + 4, 1)
 

Normalized defining polynomial

\( x^{20} + 30 x^{16} - 2 x^{15} + 10 x^{13} + 275 x^{12} - 20 x^{11} + 2 x^{10} + 150 x^{9} + 790 x^{8} + 100 x^{7} + 30 x^{6} + 254 x^{5} + 425 x^{4} + 230 x^{3} + 50 x^{2} - 20 x + 4 \)

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:  \(250000000000000000000000000000=2^{28}\cdot 5^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.51$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{6}$, $\frac{1}{2746435794793670492986} a^{19} + \frac{194966415927848752216}{1373217897396835246493} a^{18} + \frac{88742343382685123042}{1373217897396835246493} a^{17} + \frac{310215568696987247049}{1373217897396835246493} a^{16} + \frac{118300641652718023549}{1373217897396835246493} a^{15} + \frac{253753558137742935559}{2746435794793670492986} a^{14} + \frac{297276560772433635891}{2746435794793670492986} a^{13} - \frac{142390233200446524069}{1373217897396835246493} a^{12} + \frac{183738398502804114498}{1373217897396835246493} a^{11} + \frac{73030021558381526449}{2746435794793670492986} a^{10} + \frac{380736233260280249719}{2746435794793670492986} a^{9} - \frac{506334282344487611714}{1373217897396835246493} a^{8} - \frac{36844003841638819129}{249675981344879135726} a^{7} + \frac{663536098676934523107}{2746435794793670492986} a^{6} - \frac{85409255224855313705}{249675981344879135726} a^{5} - \frac{13001422336738101620}{72274626178780802447} a^{4} + \frac{562956277558979174612}{1373217897396835246493} a^{3} - \frac{485982813680113065637}{1373217897396835246493} a^{2} + \frac{519946350887456108976}{1373217897396835246493} a - \frac{131403378414044731563}{1373217897396835246493}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{93125681396339483125}{2746435794793670492986} a^{19} - \frac{38561665454154242805}{2746435794793670492986} a^{18} + \frac{4215412563479519625}{1373217897396835246493} a^{17} - \frac{1868782540638878825}{2746435794793670492986} a^{16} - \frac{1396713268349253508397}{1373217897396835246493} a^{15} - \frac{486748261465781947650}{1373217897396835246493} a^{14} + \frac{165459964725667412560}{1373217897396835246493} a^{13} - \frac{1004389177108070849625}{2746435794793670492986} a^{12} - \frac{25980705087253077942765}{2746435794793670492986} a^{11} - \frac{4373869345394606644412}{1373217897396835246493} a^{10} + \frac{1458516323508777421025}{1373217897396835246493} a^{9} - \frac{14733125454205472032655}{2746435794793670492986} a^{8} - \frac{3600908737504563308900}{124837990672439567863} a^{7} - \frac{39396828645473475395585}{2746435794793670492986} a^{6} + \frac{1851864750943251117}{124837990672439567863} a^{5} - \frac{670319392847835212575}{72274626178780802447} a^{4} - \frac{49348399492192410148305}{2746435794793670492986} a^{3} - \frac{19600058213840978743050}{1373217897396835246493} a^{2} - \frac{4942890374903875839260}{1373217897396835246493} a + \frac{515184946854607886331}{1373217897396835246493} \) (order $4$)
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:  \( 12937342.471 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5\wr C_2$ (as 20T50):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 14 conjugacy class representatives for $D_5\wr C_2$
Character table for $D_5\wr C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-5}) \), \(\Q(i, \sqrt{5})\), 10.6.7812500000000.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 25 sibling: 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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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
$2$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.8.12.14$x^{8} + 12 x^{4} + 144$$4$$2$$12$$D_4$$[2, 2]^{2}$
2.8.12.14$x^{8} + 12 x^{4} + 144$$4$$2$$12$$D_4$$[2, 2]^{2}$
$5$5.10.15.7$x^{10} - 10 x^{6} - 20 x^{5} + 5$$10$$1$$15$$C_5^2 : C_4$$[5/4, 7/4]_{4}$
5.10.15.7$x^{10} - 10 x^{6} - 20 x^{5} + 5$$10$$1$$15$$C_5^2 : C_4$$[5/4, 7/4]_{4}$