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Properties

Label 20.0.44462223590...7184.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 13^{4}\cdot 347^{4}$
Root discriminant $15.22$
Ramified primes $2, 13, 347$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T279

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 4, 30, 26, 8, -18, 49, 44, 26, -30, 28, 18, 14, -14, 8, 2, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 2*x^18 + 2*x^17 + 8*x^16 - 14*x^15 + 14*x^14 + 18*x^13 + 28*x^12 - 30*x^11 + 26*x^10 + 44*x^9 + 49*x^8 - 18*x^7 + 8*x^6 + 26*x^5 + 30*x^4 + 4*x^3 + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 + 2*x^18 + 2*x^17 + 8*x^16 - 14*x^15 + 14*x^14 + 18*x^13 + 28*x^12 - 30*x^11 + 26*x^10 + 44*x^9 + 49*x^8 - 18*x^7 + 8*x^6 + 26*x^5 + 30*x^4 + 4*x^3 + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 2 x^{18} + 2 x^{17} + 8 x^{16} - 14 x^{15} + 14 x^{14} + 18 x^{13} + 28 x^{12} - 30 x^{11} + 26 x^{10} + 44 x^{9} + 49 x^{8} - 18 x^{7} + 8 x^{6} + 26 x^{5} + 30 x^{4} + 4 x^{3} + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(444622235903180809437184=2^{30}\cdot 13^{4}\cdot 347^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.22$
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:  $2, 13, 347$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2467251559} a^{19} + \frac{51450321}{2467251559} a^{18} - \frac{769011446}{2467251559} a^{17} - \frac{107115158}{2467251559} a^{16} + \frac{199048510}{2467251559} a^{15} + \frac{417795013}{2467251559} a^{14} + \frac{949504495}{2467251559} a^{13} - \frac{684211120}{2467251559} a^{12} + \frac{285707572}{2467251559} a^{11} + \frac{240511030}{2467251559} a^{10} + \frac{748730843}{2467251559} a^{9} - \frac{840291962}{2467251559} a^{8} + \frac{924928827}{2467251559} a^{7} + \frac{207210636}{2467251559} a^{6} + \frac{950759020}{2467251559} a^{5} - \frac{735784536}{2467251559} a^{4} + \frac{1211961290}{2467251559} a^{3} - \frac{1175712100}{2467251559} a^{2} - \frac{1219772570}{2467251559} a - \frac{1096885826}{2467251559}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{185986777}{2467251559} a^{19} + \frac{356327866}{2467251559} a^{18} - \frac{679193826}{2467251559} a^{17} + \frac{181756459}{2467251559} a^{16} - \frac{1636212498}{2467251559} a^{15} + \frac{523812804}{2467251559} a^{14} - \frac{3622533843}{2467251559} a^{13} + \frac{284569919}{2467251559} a^{12} - \frac{6972696458}{2467251559} a^{11} - \frac{8397606150}{2467251559} a^{10} - \frac{4582115725}{2467251559} a^{9} - \frac{702400414}{2467251559} a^{8} - \frac{13026499996}{2467251559} a^{7} - \frac{23277714470}{2467251559} a^{6} - \frac{704015138}{2467251559} a^{5} + \frac{343340907}{2467251559} a^{4} - \frac{5587701400}{2467251559} a^{3} - \frac{13190487165}{2467251559} a^{2} - \frac{1225346246}{2467251559} a + \frac{121319396}{2467251559} \) (order $4$)
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:  \( 5076.15673611 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T279:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3840
The 36 conjugacy class representatives for t20n279
Character table for t20n279 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 5.3.4511.1, 10.0.20837499904.1, 10.0.20837499904.2, 10.6.20837499904.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 30 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
347Data not computed