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Properties

Label 20.0.15583578925...2688.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{5}\cdot 11^{19}$
Root discriminant $25.68$
Ramified primes $2, 3, 11$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T427

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2047, -4050, 4997, -1104, -721, 1490, 1041, -162, 207, 408, -55, -296, -13, 80, -15, -6, 5, -4, 3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 3*x^18 - 4*x^17 + 5*x^16 - 6*x^15 - 15*x^14 + 80*x^13 - 13*x^12 - 296*x^11 - 55*x^10 + 408*x^9 + 207*x^8 - 162*x^7 + 1041*x^6 + 1490*x^5 - 721*x^4 - 1104*x^3 + 4997*x^2 - 4050*x + 2047)
 
gp: K = bnfinit(x^20 - 2*x^19 + 3*x^18 - 4*x^17 + 5*x^16 - 6*x^15 - 15*x^14 + 80*x^13 - 13*x^12 - 296*x^11 - 55*x^10 + 408*x^9 + 207*x^8 - 162*x^7 + 1041*x^6 + 1490*x^5 - 721*x^4 - 1104*x^3 + 4997*x^2 - 4050*x + 2047, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 3 x^{18} - 4 x^{17} + 5 x^{16} - 6 x^{15} - 15 x^{14} + 80 x^{13} - 13 x^{12} - 296 x^{11} - 55 x^{10} + 408 x^{9} + 207 x^{8} - 162 x^{7} + 1041 x^{6} + 1490 x^{5} - 721 x^{4} - 1104 x^{3} + 4997 x^{2} - 4050 x + 2047 \)

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:  \(15583578925526925703866482688=2^{20}\cdot 3^{5}\cdot 11^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.68$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{717248699691262922320917868866255806899} a^{19} + \frac{12316254596646792058262579237215289362}{717248699691262922320917868866255806899} a^{18} + \frac{177401447491415757603681585637270185779}{717248699691262922320917868866255806899} a^{17} + \frac{175978054765948023658338463269912938179}{717248699691262922320917868866255806899} a^{16} + \frac{330289677649217551185625063821651058026}{717248699691262922320917868866255806899} a^{15} - \frac{330683165787873406777152683236517838311}{717248699691262922320917868866255806899} a^{14} + \frac{215242205715468885670947945603970261757}{717248699691262922320917868866255806899} a^{13} + \frac{301449731057858226894920676111873328725}{717248699691262922320917868866255806899} a^{12} - \frac{132340048152177632922634264440302849666}{717248699691262922320917868866255806899} a^{11} + \frac{116790730491832183100402106067974675607}{717248699691262922320917868866255806899} a^{10} + \frac{19526564503867605770453696122939822847}{717248699691262922320917868866255806899} a^{9} + \frac{203474071596271960684859444436242953867}{717248699691262922320917868866255806899} a^{8} + \frac{319596082150848101529082301433808847062}{717248699691262922320917868866255806899} a^{7} + \frac{35821119060115634826954991912410747818}{717248699691262922320917868866255806899} a^{6} + \frac{335437126925568946337205491689799892056}{717248699691262922320917868866255806899} a^{5} - \frac{258964526065375832165943918604670819504}{717248699691262922320917868866255806899} a^{4} + \frac{226747763042692423650334677058733221076}{717248699691262922320917868866255806899} a^{3} - \frac{284864335706832024963953492416597255}{2336314982707696815377582634743504257} a^{2} - \frac{217390563421090592249511020612096005762}{717248699691262922320917868866255806899} a - \frac{349936426312268923318969447487427559727}{717248699691262922320917868866255806899}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $9$
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:  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:  \( 248797.644395 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T427:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 136 conjugacy class representatives for t20n427 are not computed
Character table for t20n427 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.4.2414538435584.1

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 R ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ $20$ R $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$3$3.10.0.1$x^{10} - x^{3} - x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11Data not computed