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Properties

Label 20.20.1834106146...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{30}\cdot 5^{10}\cdot 211^{10}$
Root discriminant $91.87$
Ramified primes $2, 5, 211$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T538

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![45589504, 0, -179873280, 0, 274923776, 0, -210697472, 0, 87358912, 0, -19830688, 0, 2453616, 0, -169432, 0, 6492, 0, -128, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 128*x^18 + 6492*x^16 - 169432*x^14 + 2453616*x^12 - 19830688*x^10 + 87358912*x^8 - 210697472*x^6 + 274923776*x^4 - 179873280*x^2 + 45589504)
 
gp: K = bnfinit(x^20 - 128*x^18 + 6492*x^16 - 169432*x^14 + 2453616*x^12 - 19830688*x^10 + 87358912*x^8 - 210697472*x^6 + 274923776*x^4 - 179873280*x^2 + 45589504, 1)
 

Normalized defining polynomial

\( x^{20} - 128 x^{18} + 6492 x^{16} - 169432 x^{14} + 2453616 x^{12} - 19830688 x^{10} + 87358912 x^{8} - 210697472 x^{6} + 274923776 x^{4} - 179873280 x^{2} + 45589504 \)

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:  \(1834106146368079565753952501760000000000=2^{30}\cdot 5^{10}\cdot 211^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.87$
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:  $2, 5, 211$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{4471535432883181552921088} a^{18} + \frac{1326902892125725070083}{1117883858220795388230272} a^{16} + \frac{15151412522131936775}{558941929110397694115136} a^{14} - \frac{3812339087793086433819}{558941929110397694115136} a^{12} + \frac{433848657127123395955}{279470964555198847057568} a^{10} - \frac{751330483523143024595}{139735482277599423528784} a^{8} + \frac{2697151535596013577929}{69867741138799711764392} a^{6} + \frac{725271096739775349509}{34933870569399855882196} a^{4} - \frac{1034810741748644511752}{8733467642349963970549} a^{2} + \frac{322393546322147017}{41390841906871867159}$, $\frac{1}{4471535432883181552921088} a^{19} + \frac{1326902892125725070083}{1117883858220795388230272} a^{17} + \frac{15151412522131936775}{558941929110397694115136} a^{15} - \frac{3812339087793086433819}{558941929110397694115136} a^{13} + \frac{433848657127123395955}{279470964555198847057568} a^{11} - \frac{751330483523143024595}{139735482277599423528784} a^{9} + \frac{2697151535596013577929}{69867741138799711764392} a^{7} + \frac{725271096739775349509}{34933870569399855882196} a^{5} - \frac{1034810741748644511752}{8733467642349963970549} a^{3} + \frac{322393546322147017}{41390841906871867159} a$

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:  \( 14784482481600 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T538:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 25600
The 88 conjugacy class representatives for t20n538 are not computed
Character table for t20n538 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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
211Data not computed