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Properties

Label 20.6.84961403747...0000.1
Degree $20$
Signature $[6, 7]$
Discriminant $-\,2^{34}\cdot 5^{12}\cdot 1193^{4}$
Root discriminant $35.19$
Ramified primes $2, 5, 1193$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T925

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17, 142, 13, 198, 894, 212, -1567, 1226, -2526, -2164, 3404, 1436, -783, -490, -216, 108, 75, -8, -2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 2*x^18 - 8*x^17 + 75*x^16 + 108*x^15 - 216*x^14 - 490*x^13 - 783*x^12 + 1436*x^11 + 3404*x^10 - 2164*x^9 - 2526*x^8 + 1226*x^7 - 1567*x^6 + 212*x^5 + 894*x^4 + 198*x^3 + 13*x^2 + 142*x + 17)
 
gp: K = bnfinit(x^20 - 4*x^19 - 2*x^18 - 8*x^17 + 75*x^16 + 108*x^15 - 216*x^14 - 490*x^13 - 783*x^12 + 1436*x^11 + 3404*x^10 - 2164*x^9 - 2526*x^8 + 1226*x^7 - 1567*x^6 + 212*x^5 + 894*x^4 + 198*x^3 + 13*x^2 + 142*x + 17, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 2 x^{18} - 8 x^{17} + 75 x^{16} + 108 x^{15} - 216 x^{14} - 490 x^{13} - 783 x^{12} + 1436 x^{11} + 3404 x^{10} - 2164 x^{9} - 2526 x^{8} + 1226 x^{7} - 1567 x^{6} + 212 x^{5} + 894 x^{4} + 198 x^{3} + 13 x^{2} + 142 x + 17 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-8496140374773858304000000000000=-\,2^{34}\cdot 5^{12}\cdot 1193^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.19$
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:  $2, 5, 1193$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{9309878116649938806103310021429125} a^{19} - \frac{3633955344872460586860915365638291}{9309878116649938806103310021429125} a^{18} + \frac{385519115968125862949591726533928}{1861975623329987761220662004285825} a^{17} - \frac{86033760630229439825135946726101}{716144470511533754315639232417625} a^{16} + \frac{1526329333180727772926385681762156}{9309878116649938806103310021429125} a^{15} + \frac{485557833981238173780664295599836}{9309878116649938806103310021429125} a^{14} - \frac{1998456344123864747889736857845023}{9309878116649938806103310021429125} a^{13} + \frac{1812609350106345046987811079040611}{9309878116649938806103310021429125} a^{12} + \frac{589422957741507110116446177145022}{1861975623329987761220662004285825} a^{11} + \frac{3743213221045196884468362123240991}{9309878116649938806103310021429125} a^{10} + \frac{525068054666944077950477022895612}{9309878116649938806103310021429125} a^{9} + \frac{1408996744638013519187314042271}{9065119879892832333109357372375} a^{8} - \frac{70235785254987572168871669419926}{1861975623329987761220662004285825} a^{7} + \frac{1148040184870141772728206394498161}{9309878116649938806103310021429125} a^{6} + \frac{1027076984520990851582401608452101}{9309878116649938806103310021429125} a^{5} - \frac{86161503922892794856845640408331}{372395124665997552244132400857165} a^{4} - \frac{2078767937577506551320059571515931}{9309878116649938806103310021429125} a^{3} - \frac{677709006507933112636962079595721}{1861975623329987761220662004285825} a^{2} + \frac{1568766254065842921945849963138398}{9309878116649938806103310021429125} a + \frac{3092889587667794829106239550708291}{9309878116649938806103310021429125}$

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:  $12$
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:  \( 128106669.391 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T925:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 409600
The 190 conjugacy class representatives for t20n925 are not computed
Character table for t20n925 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.728703488000000.1

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 R $20$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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$2.4.10.2$x^{4} + 2 x^{2} - 1$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
1193Data not computed