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Properties

Label 20.2.18079489015...9712.2
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{10}\cdot 11^{16}\cdot 727^{3}$
Root discriminant $25.87$
Ramified primes $2, 11, 727$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T747

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![131, -514, 1481, -3053, 4901, -7868, 12870, -19510, 24238, -24620, 20510, -14742, 9144, -5063, 2406, -1024, 364, -115, 30, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 30*x^18 - 115*x^17 + 364*x^16 - 1024*x^15 + 2406*x^14 - 5063*x^13 + 9144*x^12 - 14742*x^11 + 20510*x^10 - 24620*x^9 + 24238*x^8 - 19510*x^7 + 12870*x^6 - 7868*x^5 + 4901*x^4 - 3053*x^3 + 1481*x^2 - 514*x + 131)
 
gp: K = bnfinit(x^20 - 6*x^19 + 30*x^18 - 115*x^17 + 364*x^16 - 1024*x^15 + 2406*x^14 - 5063*x^13 + 9144*x^12 - 14742*x^11 + 20510*x^10 - 24620*x^9 + 24238*x^8 - 19510*x^7 + 12870*x^6 - 7868*x^5 + 4901*x^4 - 3053*x^3 + 1481*x^2 - 514*x + 131, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 30 x^{18} - 115 x^{17} + 364 x^{16} - 1024 x^{15} + 2406 x^{14} - 5063 x^{13} + 9144 x^{12} - 14742 x^{11} + 20510 x^{10} - 24620 x^{9} + 24238 x^{8} - 19510 x^{7} + 12870 x^{6} - 7868 x^{5} + 4901 x^{4} - 3053 x^{3} + 1481 x^{2} - 514 x + 131 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-18079489015266793067734899712=-\,2^{10}\cdot 11^{16}\cdot 727^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.87$
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:  $2, 11, 727$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{481157131551923022042043409949043} a^{19} + \frac{224715468035793912760841776431485}{481157131551923022042043409949043} a^{18} - \frac{192573024785580001809177609915560}{481157131551923022042043409949043} a^{17} - \frac{237156106891935438998217547471775}{481157131551923022042043409949043} a^{16} - \frac{92729727491714667516478416653227}{481157131551923022042043409949043} a^{15} + \frac{204241444142167101366351691257500}{481157131551923022042043409949043} a^{14} - \frac{68512265494447552910707650854493}{481157131551923022042043409949043} a^{13} - \frac{85190761631129786079548035084939}{481157131551923022042043409949043} a^{12} - \frac{88860299648340353875673781724286}{481157131551923022042043409949043} a^{11} + \frac{9444871351454293298355812130529}{481157131551923022042043409949043} a^{10} + \frac{200277802819348478936744212906915}{481157131551923022042043409949043} a^{9} + \frac{79516392377559706298171341190349}{481157131551923022042043409949043} a^{8} - \frac{88451871982859088941750663413492}{481157131551923022042043409949043} a^{7} - \frac{29919180497389399660152648294972}{481157131551923022042043409949043} a^{6} + \frac{145334534208885577903234075559364}{481157131551923022042043409949043} a^{5} + \frac{75307066863433058475937367570449}{481157131551923022042043409949043} a^{4} + \frac{202024004846561714428503990600380}{481157131551923022042043409949043} a^{3} - \frac{85837901963042056933452795968353}{481157131551923022042043409949043} a^{2} + \frac{205609426218253834804525925008127}{481157131551923022042043409949043} a - \frac{205173780188024937610752370441969}{481157131551923022042043409949043}$

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:  $10$
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:  \( 720668.779677 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T747:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n747 are not computed
Character table for t20n747 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.8.155838906487.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $20$ $20$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ $20$

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$2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
2.10.10.6$x^{10} - 5 x^{8} - 18 x^{6} - 46 x^{4} + 49 x^{2} - 13$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
11Data not computed
727Data not computed