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Properties

Label 20.20.1256906660...8944.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{30}\cdot 11^{18}\cdot 1451^{2}$
Root discriminant $50.70$
Ramified primes $2, 11, 1451$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T262

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1583, 3666, 37814, -128586, -59820, 552758, -292220, -612082, 438377, 338096, -249732, -109934, 74182, 21818, -12444, -2560, 1170, 160, -56, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 56*x^18 + 160*x^17 + 1170*x^16 - 2560*x^15 - 12444*x^14 + 21818*x^13 + 74182*x^12 - 109934*x^11 - 249732*x^10 + 338096*x^9 + 438377*x^8 - 612082*x^7 - 292220*x^6 + 552758*x^5 - 59820*x^4 - 128586*x^3 + 37814*x^2 + 3666*x - 1583)
 
gp: K = bnfinit(x^20 - 4*x^19 - 56*x^18 + 160*x^17 + 1170*x^16 - 2560*x^15 - 12444*x^14 + 21818*x^13 + 74182*x^12 - 109934*x^11 - 249732*x^10 + 338096*x^9 + 438377*x^8 - 612082*x^7 - 292220*x^6 + 552758*x^5 - 59820*x^4 - 128586*x^3 + 37814*x^2 + 3666*x - 1583, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 56 x^{18} + 160 x^{17} + 1170 x^{16} - 2560 x^{15} - 12444 x^{14} + 21818 x^{13} + 74182 x^{12} - 109934 x^{11} - 249732 x^{10} + 338096 x^{9} + 438377 x^{8} - 612082 x^{7} - 292220 x^{6} + 552758 x^{5} - 59820 x^{4} - 128586 x^{3} + 37814 x^{2} + 3666 x - 1583 \)

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:  \(12569066605710630176407970532818944=2^{30}\cdot 11^{18}\cdot 1451^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.70$
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:  $2, 11, 1451$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{43} a^{18} + \frac{5}{43} a^{17} + \frac{10}{43} a^{16} + \frac{11}{43} a^{15} + \frac{17}{43} a^{14} + \frac{17}{43} a^{13} + \frac{20}{43} a^{12} - \frac{5}{43} a^{11} - \frac{5}{43} a^{10} - \frac{4}{43} a^{9} - \frac{11}{43} a^{7} - \frac{21}{43} a^{6} - \frac{10}{43} a^{5} - \frac{7}{43} a^{4} + \frac{21}{43} a^{3} - \frac{8}{43} a^{2} + \frac{9}{43} a + \frac{16}{43}$, $\frac{1}{292957280242035146135032959375767} a^{19} - \frac{1563340395470941501756488639932}{292957280242035146135032959375767} a^{18} + \frac{131454731143478366415036189871048}{292957280242035146135032959375767} a^{17} + \frac{49957603088187878733478454212584}{292957280242035146135032959375767} a^{16} - \frac{105564730274892807077389522700468}{292957280242035146135032959375767} a^{15} + \frac{84850712277839130109834590054662}{292957280242035146135032959375767} a^{14} + \frac{102741306569814470685912993309386}{292957280242035146135032959375767} a^{13} + \frac{91784003245098212471391098263406}{292957280242035146135032959375767} a^{12} - \frac{136627602076444269571469074945510}{292957280242035146135032959375767} a^{11} + \frac{39301916015689565032244442649890}{292957280242035146135032959375767} a^{10} - \frac{111828494832791584989302070086423}{292957280242035146135032959375767} a^{9} - \frac{54458138490342423892487142743578}{292957280242035146135032959375767} a^{8} - \frac{45886612601747487481745325206447}{292957280242035146135032959375767} a^{7} - \frac{109790753470371641756876727950476}{292957280242035146135032959375767} a^{6} - \frac{142302410648017849223027220204237}{292957280242035146135032959375767} a^{5} - \frac{83245912914943759355941739214341}{292957280242035146135032959375767} a^{4} - \frac{83201645965470100579057474744783}{292957280242035146135032959375767} a^{3} - \frac{91709610097033862840814018831152}{292957280242035146135032959375767} a^{2} - \frac{74717043747090276633296740826598}{292957280242035146135032959375767} a - \frac{14821694040281756758458075189411}{292957280242035146135032959375767}$

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:  \( 46268761767.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T262:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2560
The 40 conjugacy class representatives for t20n262
Character table for t20n262 is not computed

Intermediate fields

\(\Q(\sqrt{11}) \), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{44})^+\)

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 32 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
1451Data not computed