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Properties

Label 20.10.8095471366...6875.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,5^{10}\cdot 211^{9}$
Root discriminant $24.85$
Ramified primes $5, 211$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T647

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 28, -559, 2000, -3530, 3055, 649, -5379, 7220, -4525, -282, 3407, -3456, 1735, -195, -400, 356, -165, 50, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 50*x^18 - 165*x^17 + 356*x^16 - 400*x^15 - 195*x^14 + 1735*x^13 - 3456*x^12 + 3407*x^11 - 282*x^10 - 4525*x^9 + 7220*x^8 - 5379*x^7 + 649*x^6 + 3055*x^5 - 3530*x^4 + 2000*x^3 - 559*x^2 + 28*x + 1)
 
gp: K = bnfinit(x^20 - 10*x^19 + 50*x^18 - 165*x^17 + 356*x^16 - 400*x^15 - 195*x^14 + 1735*x^13 - 3456*x^12 + 3407*x^11 - 282*x^10 - 4525*x^9 + 7220*x^8 - 5379*x^7 + 649*x^6 + 3055*x^5 - 3530*x^4 + 2000*x^3 - 559*x^2 + 28*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 50 x^{18} - 165 x^{17} + 356 x^{16} - 400 x^{15} - 195 x^{14} + 1735 x^{13} - 3456 x^{12} + 3407 x^{11} - 282 x^{10} - 4525 x^{9} + 7220 x^{8} - 5379 x^{7} + 649 x^{6} + 3055 x^{5} - 3530 x^{4} + 2000 x^{3} - 559 x^{2} + 28 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-8095471366604708725498046875=-\,5^{10}\cdot 211^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.85$
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:  $5, 211$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{1617529607} a^{18} - \frac{9}{1617529607} a^{17} + \frac{592986742}{1617529607} a^{16} + \frac{108695089}{1617529607} a^{15} - \frac{126593082}{1617529607} a^{14} - \frac{207248394}{1617529607} a^{13} - \frac{246578114}{1617529607} a^{12} + \frac{222914169}{1617529607} a^{11} - \frac{587047678}{1617529607} a^{10} - \frac{285927676}{1617529607} a^{9} + \frac{170552399}{1617529607} a^{8} - \frac{433089773}{1617529607} a^{7} - \frac{321670276}{1617529607} a^{6} + \frac{147409500}{1617529607} a^{5} - \frac{180115383}{1617529607} a^{4} + \frac{353186294}{1617529607} a^{3} + \frac{263878447}{1617529607} a^{2} + \frac{528647744}{1617529607} a - \frac{618706340}{1617529607}$, $\frac{1}{389824635287} a^{19} + \frac{111}{389824635287} a^{18} - \frac{36610195299}{389824635287} a^{17} + \frac{187729235833}{389824635287} a^{16} - \frac{79282370001}{389824635287} a^{15} + \frac{8864525871}{389824635287} a^{14} + \frac{94580805524}{389824635287} a^{13} - \frac{30983989118}{389824635287} a^{12} + \frac{102186544131}{389824635287} a^{11} - \frac{77201767464}{389824635287} a^{10} - \frac{26053120686}{389824635287} a^{9} - \frac{170835295519}{389824635287} a^{8} - \frac{154196808277}{389824635287} a^{7} + \frac{99036992975}{389824635287} a^{6} - \frac{34251922807}{389824635287} a^{5} + \frac{61233350291}{389824635287} a^{4} - \frac{77050957191}{389824635287} a^{3} - \frac{9861708398}{389824635287} a^{2} - \frac{34232753480}{389824635287} a - \frac{181001714862}{389824635287}$

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:  $14$
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:  \( 3143353.78813 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T647:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 51200
The 152 conjugacy class representatives for t20n647 are not computed
Character table for t20n647 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.6194123253125.1

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 $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
211Data not computed