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Properties

Label 20.0.60020399738...3961.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{14}\cdot 17^{6}\cdot 151^{4}$
Root discriminant $13.77$
Ramified primes $3, 17, 151$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T368

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 25, -80, 207, -437, 759, -1095, 1314, -1308, 1071, -696, 339, -114, 39, -58, 84, -70, 34, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^19 + 34*x^18 - 70*x^17 + 84*x^16 - 58*x^15 + 39*x^14 - 114*x^13 + 339*x^12 - 696*x^11 + 1071*x^10 - 1308*x^9 + 1314*x^8 - 1095*x^7 + 759*x^6 - 437*x^5 + 207*x^4 - 80*x^3 + 25*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^20 - 9*x^19 + 34*x^18 - 70*x^17 + 84*x^16 - 58*x^15 + 39*x^14 - 114*x^13 + 339*x^12 - 696*x^11 + 1071*x^10 - 1308*x^9 + 1314*x^8 - 1095*x^7 + 759*x^6 - 437*x^5 + 207*x^4 - 80*x^3 + 25*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 9 x^{19} + 34 x^{18} - 70 x^{17} + 84 x^{16} - 58 x^{15} + 39 x^{14} - 114 x^{13} + 339 x^{12} - 696 x^{11} + 1071 x^{10} - 1308 x^{9} + 1314 x^{8} - 1095 x^{7} + 759 x^{6} - 437 x^{5} + 207 x^{4} - 80 x^{3} + 25 x^{2} - 6 x + 1 \)

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:  \(60020399738333350163961=3^{14}\cdot 17^{6}\cdot 151^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.77$
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:  $3, 17, 151$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{15} - \frac{1}{3} a^{13} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{9} a^{18} - \frac{1}{9} a^{17} + \frac{1}{9} a^{15} + \frac{2}{9} a^{14} + \frac{4}{9} a^{13} + \frac{1}{9} a^{12} - \frac{1}{3} a^{11} + \frac{1}{9} a^{10} + \frac{2}{9} a^{9} - \frac{1}{9} a^{8} - \frac{1}{9} a^{6} + \frac{4}{9} a^{5} - \frac{2}{9} a^{4} + \frac{1}{9} a^{3} - \frac{1}{3} a^{2} - \frac{4}{9} a - \frac{1}{9}$, $\frac{1}{2563857} a^{19} + \frac{61048}{2563857} a^{18} - \frac{155435}{2563857} a^{17} - \frac{135743}{2563857} a^{16} + \frac{34795}{2563857} a^{15} - \frac{674323}{2563857} a^{14} - \frac{99767}{854619} a^{13} - \frac{672940}{2563857} a^{12} + \frac{959914}{2563857} a^{11} + \frac{552001}{2563857} a^{10} + \frac{67166}{854619} a^{9} - \frac{517919}{2563857} a^{8} - \frac{1106323}{2563857} a^{7} + \frac{321254}{2563857} a^{6} + \frac{338438}{854619} a^{5} - \frac{298769}{854619} a^{4} + \frac{850865}{2563857} a^{3} + \frac{114707}{2563857} a^{2} - \frac{88000}{284873} a + \frac{332617}{2563857}$

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:  \( -\frac{4175659}{2563857} a^{19} + \frac{36303559}{2563857} a^{18} - \frac{43767647}{854619} a^{17} + \frac{255359777}{2563857} a^{16} - \frac{281727350}{2563857} a^{15} + \frac{166993343}{2563857} a^{14} - \frac{113052934}{2563857} a^{13} + \frac{47929394}{284873} a^{12} - \frac{1282960594}{2563857} a^{11} + \frac{2549790640}{2563857} a^{10} - \frac{3775140155}{2563857} a^{9} + \frac{1474270033}{854619} a^{8} - \frac{4242358217}{2563857} a^{7} + \frac{3336576953}{2563857} a^{6} - \frac{2151642448}{2563857} a^{5} + \frac{1122491759}{2563857} a^{4} - \frac{154590352}{854619} a^{3} + \frac{146878438}{2563857} a^{2} - \frac{36813647}{2563857} a + \frac{823888}{284873} \) (order $6$)
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:  \( 3551.87507638 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T368:

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 t20n368 are not computed
Character table for t20n368 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.3.23103.1, 10.2.81663537177.1, 10.4.244990611531.1, 10.0.1601245827.1

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$17$17.6.0.1$x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
17.6.0.1$x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$151$151.3.0.1$x^{3} - x + 5$$1$$3$$0$$C_3$$[\ ]^{3}$
151.3.0.1$x^{3} - x + 5$$1$$3$$0$$C_3$$[\ ]^{3}$
151.3.0.1$x^{3} - x + 5$$1$$3$$0$$C_3$$[\ ]^{3}$
151.3.0.1$x^{3} - x + 5$$1$$3$$0$$C_3$$[\ ]^{3}$
151.4.2.1$x^{4} + 3473 x^{2} + 3283344$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
151.4.2.1$x^{4} + 3473 x^{2} + 3283344$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$