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Properties

Label 20.0.82962908926...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 5^{10}\cdot 11^{9}$
Root discriminant $44.25$
Ramified primes $2, 5, 11$
Class number Not computed
Class group Not computed
Galois group $C_5\times C_5:D_4$ (as 20T53)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3437500000, 0, -1375000000, 0, 206250000, 0, -27500000, 0, 5500000, 0, -500000, 0, 57500, 0, -5000, 0, 450, 0, -20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 20*x^18 + 450*x^16 - 5000*x^14 + 57500*x^12 - 500000*x^10 + 5500000*x^8 - 27500000*x^6 + 206250000*x^4 - 1375000000*x^2 + 3437500000)
 
gp: K = bnfinit(x^20 - 20*x^18 + 450*x^16 - 5000*x^14 + 57500*x^12 - 500000*x^10 + 5500000*x^8 - 27500000*x^6 + 206250000*x^4 - 1375000000*x^2 + 3437500000, 1)
 

Normalized defining polynomial

\( x^{20} - 20 x^{18} + 450 x^{16} - 5000 x^{14} + 57500 x^{12} - 500000 x^{10} + 5500000 x^{8} - 27500000 x^{6} + 206250000 x^{4} - 1375000000 x^{2} + 3437500000 \)

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:  \(829629089261462612869120000000000=2^{55}\cdot 5^{10}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.25$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $\frac{1}{5} a^{2}$, $\frac{1}{5} a^{3}$, $\frac{1}{50} a^{4}$, $\frac{1}{50} a^{5}$, $\frac{1}{250} a^{6}$, $\frac{1}{250} a^{7}$, $\frac{1}{2500} a^{8}$, $\frac{1}{2500} a^{9}$, $\frac{1}{12500} a^{10}$, $\frac{1}{12500} a^{11}$, $\frac{1}{125000} a^{12}$, $\frac{1}{125000} a^{13}$, $\frac{1}{625000} a^{14}$, $\frac{1}{625000} a^{15}$, $\frac{1}{6250000} a^{16}$, $\frac{1}{6250000} a^{17}$, $\frac{1}{628885656250000} a^{18} + \frac{837739}{31444282812500} a^{16} - \frac{250309}{314442828125} a^{14} + \frac{854403}{1257771312500} a^{12} + \frac{203587}{125777131250} a^{10} - \frac{65328}{503108525} a^{8} - \frac{7047079}{5031085250} a^{6} + \frac{1097881}{201243410} a^{4} + \frac{194177}{20124341} a^{2} - \frac{4788209}{20124341}$, $\frac{1}{628885656250000} a^{19} + \frac{837739}{31444282812500} a^{17} - \frac{250309}{314442828125} a^{15} + \frac{854403}{1257771312500} a^{13} + \frac{203587}{125777131250} a^{11} - \frac{65328}{503108525} a^{9} - \frac{7047079}{5031085250} a^{7} + \frac{1097881}{201243410} a^{5} + \frac{194177}{20124341} a^{3} - \frac{4788209}{20124341} a$

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

Class group and class number

Not computed

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:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times C_5:D_4$ (as 20T53):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed
Character table for $C_5\times C_5:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-2}) \), 4.0.563200.2, 10.0.479756288.1

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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
2Data not computed
5Data not computed
$11$11.10.9.4$x^{10} - 99$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.0.1$x^{10} + x^{2} - x + 6$$1$$10$$0$$C_{10}$$[\ ]^{10}$