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Properties

Label 20.4.14609322487...8784.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{20}\cdot 11^{18}\cdot 1583^{2}$
Root discriminant $36.16$
Ramified primes $2, 11, 1583$
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![-43, -126, 171, -1200, 727, -184, -694, 2412, 1489, 1166, 243, 946, 432, -48, 359, -80, 58, -2, 8, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 8*x^18 - 2*x^17 + 58*x^16 - 80*x^15 + 359*x^14 - 48*x^13 + 432*x^12 + 946*x^11 + 243*x^10 + 1166*x^9 + 1489*x^8 + 2412*x^7 - 694*x^6 - 184*x^5 + 727*x^4 - 1200*x^3 + 171*x^2 - 126*x - 43)
 
gp: K = bnfinit(x^20 - 2*x^19 + 8*x^18 - 2*x^17 + 58*x^16 - 80*x^15 + 359*x^14 - 48*x^13 + 432*x^12 + 946*x^11 + 243*x^10 + 1166*x^9 + 1489*x^8 + 2412*x^7 - 694*x^6 - 184*x^5 + 727*x^4 - 1200*x^3 + 171*x^2 - 126*x - 43, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 8 x^{18} - 2 x^{17} + 58 x^{16} - 80 x^{15} + 359 x^{14} - 48 x^{13} + 432 x^{12} + 946 x^{11} + 243 x^{10} + 1166 x^{9} + 1489 x^{8} + 2412 x^{7} - 694 x^{6} - 184 x^{5} + 727 x^{4} - 1200 x^{3} + 171 x^{2} - 126 x - 43 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(14609322487882432594514132598784=2^{20}\cdot 11^{18}\cdot 1583^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.16$
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:  $2, 11, 1583$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{43} a^{17} - \frac{11}{43} a^{16} + \frac{1}{43} a^{15} - \frac{5}{43} a^{14} - \frac{14}{43} a^{13} + \frac{18}{43} a^{12} - \frac{1}{43} a^{11} + \frac{17}{43} a^{10} + \frac{16}{43} a^{9} - \frac{12}{43} a^{8} + \frac{5}{43} a^{7} + \frac{1}{43} a^{6} - \frac{8}{43} a^{5} + \frac{18}{43} a^{4} - \frac{7}{43} a^{3} - \frac{16}{43} a^{2} - \frac{3}{43} a$, $\frac{1}{43} a^{18} + \frac{9}{43} a^{16} + \frac{6}{43} a^{15} + \frac{17}{43} a^{14} - \frac{7}{43} a^{13} - \frac{18}{43} a^{12} + \frac{6}{43} a^{11} - \frac{12}{43} a^{10} - \frac{8}{43} a^{9} + \frac{2}{43} a^{8} + \frac{13}{43} a^{7} + \frac{3}{43} a^{6} + \frac{16}{43} a^{5} + \frac{19}{43} a^{4} - \frac{7}{43} a^{3} - \frac{7}{43} a^{2} + \frac{10}{43} a$, $\frac{1}{5332332661108526204751739412908099} a^{19} + \frac{38973931482359835289221632485087}{5332332661108526204751739412908099} a^{18} - \frac{53259252396136976968318726863844}{5332332661108526204751739412908099} a^{17} + \frac{2410567701248705404036825250142288}{5332332661108526204751739412908099} a^{16} - \frac{2401560814180086316124175461528464}{5332332661108526204751739412908099} a^{15} - \frac{18330205306759057121605461030136}{231840550482979400206597365778613} a^{14} + \frac{2061241099884327667868802325174618}{5332332661108526204751739412908099} a^{13} - \frac{862577036885857284663539040056697}{5332332661108526204751739412908099} a^{12} - \frac{1943654560708147119292773092133838}{5332332661108526204751739412908099} a^{11} - \frac{853819680919358385030838941936318}{5332332661108526204751739412908099} a^{10} - \frac{1995925215855689302180656321065119}{5332332661108526204751739412908099} a^{9} + \frac{2368571628946009915877402808664756}{5332332661108526204751739412908099} a^{8} + \frac{1952127607027728280839327534470526}{5332332661108526204751739412908099} a^{7} - \frac{1325331594044273900354739892782222}{5332332661108526204751739412908099} a^{6} + \frac{1113932872330729472184209342399309}{5332332661108526204751739412908099} a^{5} - \frac{965336789997644313474118786900522}{5332332661108526204751739412908099} a^{4} + \frac{1339693673001625412394842115429678}{5332332661108526204751739412908099} a^{3} - \frac{14941536771957541487067364611271}{231840550482979400206597365778613} a^{2} - \frac{401698271087789633678158837745523}{5332332661108526204751739412908099} a + \frac{7430417449173910283420484241414}{124007736304849446622133474718793}$

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:  $11$
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:  \( 44707610.3959 \) (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
$2$2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
$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}$
1583Data not computed