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Properties

Label 20.0.36487627390...3376.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{10}\cdot 11^{17}\cdot 89^{3}$
Root discriminant $21.29$
Ramified primes $2, 11, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T846

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![89, 296, 183, -1207, 1713, -984, 1121, -962, 1234, -369, 241, -151, 142, 18, 7, 0, 5, -1, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^17 + 5*x^16 + 7*x^14 + 18*x^13 + 142*x^12 - 151*x^11 + 241*x^10 - 369*x^9 + 1234*x^8 - 962*x^7 + 1121*x^6 - 984*x^5 + 1713*x^4 - 1207*x^3 + 183*x^2 + 296*x + 89)
 
gp: K = bnfinit(x^20 - x^17 + 5*x^16 + 7*x^14 + 18*x^13 + 142*x^12 - 151*x^11 + 241*x^10 - 369*x^9 + 1234*x^8 - 962*x^7 + 1121*x^6 - 984*x^5 + 1713*x^4 - 1207*x^3 + 183*x^2 + 296*x + 89, 1)
 

Normalized defining polynomial

\( x^{20} - x^{17} + 5 x^{16} + 7 x^{14} + 18 x^{13} + 142 x^{12} - 151 x^{11} + 241 x^{10} - 369 x^{9} + 1234 x^{8} - 962 x^{7} + 1121 x^{6} - 984 x^{5} + 1713 x^{4} - 1207 x^{3} + 183 x^{2} + 296 x + 89 \)

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:  \(364876273903737477578853376=2^{10}\cdot 11^{17}\cdot 89^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.29$
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:  $2, 11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} + \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{13} - \frac{1}{4} a^{12} - \frac{1}{8} a^{10} + \frac{1}{8} a^{9} - \frac{1}{2} a^{8} + \frac{1}{8} a^{7} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} + \frac{1}{8} a^{2} - \frac{1}{8}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{14} - \frac{1}{4} a^{13} - \frac{1}{8} a^{11} + \frac{1}{8} a^{10} - \frac{1}{2} a^{9} + \frac{1}{8} a^{8} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3} - \frac{1}{8} a$, $\frac{1}{16} a^{18} - \frac{1}{16} a^{16} - \frac{1}{16} a^{15} - \frac{1}{8} a^{14} + \frac{1}{16} a^{13} + \frac{1}{16} a^{12} - \frac{7}{16} a^{11} + \frac{5}{16} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} + \frac{7}{16} a^{7} + \frac{3}{8} a^{6} - \frac{1}{16} a^{5} - \frac{5}{16} a^{4} + \frac{1}{16} a^{3} + \frac{3}{8} a^{2} - \frac{1}{2} a - \frac{7}{16}$, $\frac{1}{53514234511744517895766210064} a^{19} + \frac{106742955361510936917663575}{53514234511744517895766210064} a^{18} - \frac{1067313280096563570257401361}{53514234511744517895766210064} a^{17} + \frac{612730217220403795430446397}{26757117255872258947883105032} a^{16} + \frac{4753770077647948683120850231}{53514234511744517895766210064} a^{15} - \frac{97306342823644226859224345}{53514234511744517895766210064} a^{14} - \frac{2319707654127946180562644773}{26757117255872258947883105032} a^{13} - \frac{726134687282263429861810838}{3344639656984032368485388129} a^{12} - \frac{582187055368475939512197796}{3344639656984032368485388129} a^{11} - \frac{17349474315211459492943961627}{53514234511744517895766210064} a^{10} + \frac{8424126019556288253829841665}{26757117255872258947883105032} a^{9} - \frac{17120035962447813808425074437}{53514234511744517895766210064} a^{8} - \frac{19959990612946742362197630167}{53514234511744517895766210064} a^{7} - \frac{10368117030084245415319980191}{53514234511744517895766210064} a^{6} + \frac{1106865003909059346067172715}{6689279313968064736970776258} a^{5} + \frac{8242439601022786178281757665}{26757117255872258947883105032} a^{4} + \frac{19389379287808407600134045583}{53514234511744517895766210064} a^{3} - \frac{6020421553249039845126206719}{13378558627936129473941552516} a^{2} + \frac{22045540458693902583943705841}{53514234511744517895766210064} a + \frac{8333984835671611566158948753}{53514234511744517895766210064}$

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:  \( -1 \) (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:  \( 125700.219074 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T846:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 163840
The 649 conjugacy class representatives for t20n846 are not computed
Character table for t20n846 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.2.19077940409.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ $20$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.10.10$x^{10} - 11 x^{8} + 10 x^{6} - 62 x^{4} + 21 x^{2} - 55$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
89Data not computed