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Properties

Label 20.12.3143662735...3241.1
Degree $20$
Signature $[12, 4]$
Discriminant $3^{4}\cdot 19^{2}\cdot 401^{10}$
Root discriminant $33.49$
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![-83, 451, 279, -3165, 2298, 5463, -8754, 2795, 2555, -4679, 5694, -4202, 1714, -294, -249, 254, -94, 20, 1, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + x^18 + 20*x^17 - 94*x^16 + 254*x^15 - 249*x^14 - 294*x^13 + 1714*x^12 - 4202*x^11 + 5694*x^10 - 4679*x^9 + 2555*x^8 + 2795*x^7 - 8754*x^6 + 5463*x^5 + 2298*x^4 - 3165*x^3 + 279*x^2 + 451*x - 83)
 
gp: K = bnfinit(x^20 - 4*x^19 + x^18 + 20*x^17 - 94*x^16 + 254*x^15 - 249*x^14 - 294*x^13 + 1714*x^12 - 4202*x^11 + 5694*x^10 - 4679*x^9 + 2555*x^8 + 2795*x^7 - 8754*x^6 + 5463*x^5 + 2298*x^4 - 3165*x^3 + 279*x^2 + 451*x - 83, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + x^{18} + 20 x^{17} - 94 x^{16} + 254 x^{15} - 249 x^{14} - 294 x^{13} + 1714 x^{12} - 4202 x^{11} + 5694 x^{10} - 4679 x^{9} + 2555 x^{8} + 2795 x^{7} - 8754 x^{6} + 5463 x^{5} + 2298 x^{4} - 3165 x^{3} + 279 x^{2} + 451 x - 83 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3143662735061279800077532193241=3^{4}\cdot 19^{2}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.49$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{15} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{15} + \frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{9} a^{18} + \frac{2}{9} a^{15} + \frac{4}{9} a^{14} - \frac{2}{9} a^{13} - \frac{4}{9} a^{11} - \frac{1}{3} a^{10} + \frac{2}{9} a^{9} - \frac{1}{9} a^{8} + \frac{4}{9} a^{7} - \frac{2}{9} a^{6} + \frac{2}{9} a^{5} + \frac{4}{9} a^{4} - \frac{4}{9} a^{3} + \frac{1}{9} a^{2} + \frac{2}{9} a - \frac{2}{9}$, $\frac{1}{1312284928033617169097882727} a^{19} - \frac{35261298683956572007123208}{1312284928033617169097882727} a^{18} + \frac{45529068819761571523025006}{437428309344539056365960909} a^{17} - \frac{155931644970846398219776042}{1312284928033617169097882727} a^{16} - \frac{108704122302098838552356215}{437428309344539056365960909} a^{15} + \frac{84152659442516204238161762}{1312284928033617169097882727} a^{14} + \frac{60339408256258824749623546}{1312284928033617169097882727} a^{13} + \frac{178904876313541785437685890}{1312284928033617169097882727} a^{12} + \frac{228475305972951318513114344}{1312284928033617169097882727} a^{11} + \frac{67205327883926565002027003}{1312284928033617169097882727} a^{10} + \frac{331944059633641557206636875}{1312284928033617169097882727} a^{9} - \frac{20480644346685721155851045}{48603145482726561818440101} a^{8} + \frac{212122712222723774136166028}{1312284928033617169097882727} a^{7} - \frac{189678366065686137087107960}{437428309344539056365960909} a^{6} + \frac{1784817998709590749200922}{16201048494242187272813367} a^{5} - \frac{53231103615357456394725059}{145809436448179685455320303} a^{4} + \frac{167076780112903946386557184}{437428309344539056365960909} a^{3} - \frac{41244629332356439247295593}{145809436448179685455320303} a^{2} - \frac{13760502082034342067901723}{48603145482726561818440101} a + \frac{504154555321220785880277049}{1312284928033617169097882727}$

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:  $15$
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:  \( 120469299.615 \) (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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$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}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{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}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
401Data not computed