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Properties

Label 20.8.34929585945...9249.1
Degree $20$
Signature $[8, 6]$
Discriminant $3^{2}\cdot 19^{2}\cdot 401^{10}$
Root discriminant $30.00$
Ramified primes $3, 19, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T347

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, 567, 414, -3195, -1557, 7116, -2192, -1844, -928, 1685, 583, -1114, 183, 117, 78, -82, -22, 37, -9, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 9*x^18 + 37*x^17 - 22*x^16 - 82*x^15 + 78*x^14 + 117*x^13 + 183*x^12 - 1114*x^11 + 583*x^10 + 1685*x^9 - 928*x^8 - 1844*x^7 - 2192*x^6 + 7116*x^5 - 1557*x^4 - 3195*x^3 + 414*x^2 + 567*x + 81)
 
gp: K = bnfinit(x^20 - 2*x^19 - 9*x^18 + 37*x^17 - 22*x^16 - 82*x^15 + 78*x^14 + 117*x^13 + 183*x^12 - 1114*x^11 + 583*x^10 + 1685*x^9 - 928*x^8 - 1844*x^7 - 2192*x^6 + 7116*x^5 - 1557*x^4 - 3195*x^3 + 414*x^2 + 567*x + 81, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 9 x^{18} + 37 x^{17} - 22 x^{16} - 82 x^{15} + 78 x^{14} + 117 x^{13} + 183 x^{12} - 1114 x^{11} + 583 x^{10} + 1685 x^{9} - 928 x^{8} - 1844 x^{7} - 2192 x^{6} + 7116 x^{5} - 1557 x^{4} - 3195 x^{3} + 414 x^{2} + 567 x + 81 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(349295859451253311119725799249=3^{2}\cdot 19^{2}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.00$
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:  $3, 19, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{16} + \frac{1}{6} a^{15} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{6} a^{5} + \frac{1}{3} a^{4} + \frac{1}{6} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{17} - \frac{1}{6} a^{15} - \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{6} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{6} a^{8} + \frac{1}{3} a^{7} + \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2} a$, $\frac{1}{18} a^{18} + \frac{1}{18} a^{17} - \frac{1}{9} a^{15} + \frac{4}{9} a^{14} + \frac{1}{9} a^{13} - \frac{1}{6} a^{12} - \frac{1}{3} a^{11} + \frac{1}{6} a^{10} + \frac{1}{9} a^{9} + \frac{2}{9} a^{8} - \frac{1}{18} a^{7} + \frac{1}{9} a^{6} + \frac{1}{18} a^{5} - \frac{4}{9} a^{4} - \frac{1}{6} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{1847646632142592759283323986} a^{19} + \frac{9896628692372487083222714}{923823316071296379641661993} a^{18} - \frac{321977284554763808689565}{7895925778387148543945829} a^{17} - \frac{51004471127853587605172459}{1847646632142592759283323986} a^{16} - \frac{263840625557933735853781069}{1847646632142592759283323986} a^{15} - \frac{277330275739436297122918117}{1847646632142592759283323986} a^{14} - \frac{144638622343111075148882833}{615882210714197586427774662} a^{13} + \frac{622155099378039802268391}{68431356746021954047530518} a^{12} + \frac{73662474183125823880561856}{307941105357098793213887331} a^{11} + \frac{113811021531027147250358165}{1847646632142592759283323986} a^{10} - \frac{286054637674820634464778208}{923823316071296379641661993} a^{9} + \frac{731544862461163520457310391}{1847646632142592759283323986} a^{8} + \frac{256714421575703327171510080}{923823316071296379641661993} a^{7} - \frac{6368170573866909798492691}{923823316071296379641661993} a^{6} + \frac{186140077458169723341142531}{1847646632142592759283323986} a^{5} + \frac{235386233307604873264948643}{615882210714197586427774662} a^{4} + \frac{39815535315165864732845531}{205294070238065862142591554} a^{3} + \frac{9829510309376595384269902}{34215678373010977023765259} a^{2} - \frac{13283513636643596683575040}{102647035119032931071295777} a - \frac{10047453995346760513910885}{68431356746021954047530518}$

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:  $13$
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:  \( 20024791.7315 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T347:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 104 conjugacy class representatives for t20n347 are not computed
Character table for t20n347 is not computed

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.10.10368641602001.1

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
401Data not computed