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Properties

Label 20.0.11562819469...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 269^{6}$
Root discriminant $17.91$
Ramified primes $5, 269$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T369

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, -40, 130, -240, 415, -965, 2135, -3635, 4910, -5560, 5355, -4420, 3199, -2035, 1124, -547, 235, -86, 26, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 26*x^18 - 86*x^17 + 235*x^16 - 547*x^15 + 1124*x^14 - 2035*x^13 + 3199*x^12 - 4420*x^11 + 5355*x^10 - 5560*x^9 + 4910*x^8 - 3635*x^7 + 2135*x^6 - 965*x^5 + 415*x^4 - 240*x^3 + 130*x^2 - 40*x + 5)
 
gp: K = bnfinit(x^20 - 6*x^19 + 26*x^18 - 86*x^17 + 235*x^16 - 547*x^15 + 1124*x^14 - 2035*x^13 + 3199*x^12 - 4420*x^11 + 5355*x^10 - 5560*x^9 + 4910*x^8 - 3635*x^7 + 2135*x^6 - 965*x^5 + 415*x^4 - 240*x^3 + 130*x^2 - 40*x + 5, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 26 x^{18} - 86 x^{17} + 235 x^{16} - 547 x^{15} + 1124 x^{14} - 2035 x^{13} + 3199 x^{12} - 4420 x^{11} + 5355 x^{10} - 5560 x^{9} + 4910 x^{8} - 3635 x^{7} + 2135 x^{6} - 965 x^{5} + 415 x^{4} - 240 x^{3} + 130 x^{2} - 40 x + 5 \)

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:  \(11562819469661895751953125=5^{15}\cdot 269^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.91$
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:  $5, 269$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{86415126783302870299} a^{19} + \frac{32249457988649647062}{86415126783302870299} a^{18} + \frac{18188245685479440141}{86415126783302870299} a^{17} - \frac{5871728021083478379}{86415126783302870299} a^{16} + \frac{30466566336648192140}{86415126783302870299} a^{15} + \frac{26687902332612139786}{86415126783302870299} a^{14} + \frac{40956908099212517035}{86415126783302870299} a^{13} - \frac{5154605472731774088}{86415126783302870299} a^{12} + \frac{20804420331104091728}{86415126783302870299} a^{11} + \frac{8980691626008401090}{86415126783302870299} a^{10} - \frac{2073683977970974384}{86415126783302870299} a^{9} - \frac{12126667257771112420}{86415126783302870299} a^{8} - \frac{15452395355150431310}{86415126783302870299} a^{7} - \frac{22513252190656030671}{86415126783302870299} a^{6} + \frac{28230879774067906837}{86415126783302870299} a^{5} + \frac{34369309786855982584}{86415126783302870299} a^{4} + \frac{11351500427360974333}{86415126783302870299} a^{3} + \frac{30982126705646946168}{86415126783302870299} a^{2} + \frac{35352716172130565644}{86415126783302870299} a + \frac{2399901201192727122}{86415126783302870299}$

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:  \( -1 \) (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:  \( 15620.7540228 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T369:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7680
The 72 conjugacy class representatives for t20n369 are not computed
Character table for t20n369 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.1.33625.1, 10.2.5653203125.1

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

Sibling fields

Degree 20 sibling: 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 $20$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
5Data not computed
269Data not computed